3.5.56 \(\int \frac {(c+d x^2)^3}{\sqrt {x} (a+b x^2)^2} \, dx\) [456]

Optimal. Leaf size=340 \[ \frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{13/4}}-\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}} \]

[Out]

2/5*d^3*x^(5/2)/b^2-3/8*(-a*d+b*c)^2*(3*a*d+b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(13/4)*2^
(1/2)+3/8*(-a*d+b*c)^2*(3*a*d+b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/b^(13/4)*2^(1/2)-3/16*(-a
*d+b*c)^2*(3*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/b^(13/4)*2^(1/2)+3/16*(-a*
d+b*c)^2*(3*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(7/4)/b^(13/4)*2^(1/2)+2*d^2*(-2*
a*d+3*b*c)*x^(1/2)/b^3+1/2*(-a*d+b*c)^3*x^(1/2)/a/b^3/(b*x^2+a)

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Rubi [A]
time = 0.26, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 398, 393, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (3 a d+b c)}{4 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (3 a d+b c)}{4 \sqrt {2} a^{7/4} b^{13/4}}-\frac {3 (b c-a d)^2 (3 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (3 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {2 d^2 \sqrt {x} (3 b c-2 a d)}{b^3}+\frac {\sqrt {x} (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac {2 d^3 x^{5/2}}{5 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(Sqrt[x]*(a + b*x^2)^2),x]

[Out]

(2*d^2*(3*b*c - 2*a*d)*Sqrt[x])/b^3 + (2*d^3*x^(5/2))/(5*b^2) + ((b*c - a*d)^3*Sqrt[x])/(2*a*b^3*(a + b*x^2))
- (3*(b*c - a*d)^2*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*b^(13/4)) +
(3*(b*c - a*d)^2*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*b^(13/4)) -
(3*(b*c - a*d)^2*(b*c + 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*
b^(13/4)) + (3*(b*c - a*d)^2*(b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt
[2]*a^(7/4)*b^(13/4))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^3}{\sqrt {x} \left (a+b x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {d^2 (3 b c-2 a d)}{b^3}+\frac {d^3 x^4}{b^2}+\frac {(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^4}{b^3 \left (a+b x^4\right )^2}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {2 \text {Subst}\left (\int \frac {(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^4}{\left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a b^3}\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{3/2} b^3}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{3/2} b^3}\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^{3/2} b^{7/2}}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{8 a^{3/2} b^{7/2}}-\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}-\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{13/4}}-\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{13/4}}\\ &=\frac {2 d^2 (3 b c-2 a d) \sqrt {x}}{b^3}+\frac {2 d^3 x^{5/2}}{5 b^2}+\frac {(b c-a d)^3 \sqrt {x}}{2 a b^3 \left (a+b x^2\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} b^{13/4}}-\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}\\ \end {align*}

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Mathematica [A]
time = 0.47, size = 227, normalized size = 0.67 \begin {gather*} \frac {\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (5 b^3 c^3-45 a^3 d^3+3 a^2 b d^2 \left (25 c-12 d x^2\right )+a b^2 d \left (-15 c^2+60 c d x^2+4 d^2 x^4\right )\right )}{a+b x^2}-15 \sqrt {2} (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+15 \sqrt {2} (b c-a d)^2 (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{40 a^{7/4} b^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^3/(Sqrt[x]*(a + b*x^2)^2),x]

[Out]

((4*a^(3/4)*b^(1/4)*Sqrt[x]*(5*b^3*c^3 - 45*a^3*d^3 + 3*a^2*b*d^2*(25*c - 12*d*x^2) + a*b^2*d*(-15*c^2 + 60*c*
d*x^2 + 4*d^2*x^4)))/(a + b*x^2) - 15*Sqrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2
]*a^(1/4)*b^(1/4)*Sqrt[x])] + 15*Sqrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])
/(Sqrt[a] + Sqrt[b]*x)])/(40*a^(7/4)*b^(13/4))

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Maple [A]
time = 0.11, size = 231, normalized size = 0.68

method	result	size
derivativedivides	\(-\frac {2 d^{2} \left (-\frac {b d \,x^{\frac {5}{2}}}{5}+2 a d \sqrt {x}-3 b c \sqrt {x}\right )}{b^{3}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {x}}{2 a \left (b \,x^{2}+a \right )}+\frac {3 \left (3 a^{3} d^{3}-5 a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a^{2}}}{b^{3}}\)	\(231\)
default	\(-\frac {2 d^{2} \left (-\frac {b d \,x^{\frac {5}{2}}}{5}+2 a d \sqrt {x}-3 b c \sqrt {x}\right )}{b^{3}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {x}}{2 a \left (b \,x^{2}+a \right )}+\frac {3 \left (3 a^{3} d^{3}-5 a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a^{2}}}{b^{3}}\)	\(231\)
risch	\(-\frac {2 d^{2} \left (-b d \,x^{2}+10 a d -15 b c \right ) \sqrt {x}}{5 b^{3}}-\frac {\sqrt {x}\, d^{3} a^{2}}{2 b^{3} \left (b \,x^{2}+a \right )}+\frac {3 \sqrt {x}\, d^{2} a c}{2 b^{2} \left (b \,x^{2}+a \right )}-\frac {3 \sqrt {x}\, d \,c^{2}}{2 b \left (b \,x^{2}+a \right )}+\frac {\sqrt {x}\, c^{3}}{2 a \left (b \,x^{2}+a \right )}+\frac {9 a \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{3}}{8 b^{3}}-\frac {15 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{2} c}{8 b^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d \,c^{2}}{8 b a}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) c^{3}}{8 a^{2}}+\frac {9 a \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) d^{3}}{16 b^{3}}-\frac {15 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) d^{2} c}{16 b^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) d \,c^{2}}{16 b a}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right ) c^{3}}{16 a^{2}}+\frac {9 a \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{3}}{8 b^{3}}-\frac {15 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{2} c}{8 b^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d \,c^{2}}{8 b a}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) c^{3}}{8 a^{2}}\)	\(689\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^3/(b*x^2+a)^2/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*d^2/b^3*(-1/5*b*d*x^(5/2)+2*a*d*x^(1/2)-3*b*c*x^(1/2))+2/b^3*(-1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3
*c^3)/a*x^(1/2)/(b*x^2+a)+3/32*(3*a^3*d^3-5*a^2*b*c*d^2+a*b^2*c^2*d+b^3*c^3)/a^2*(a/b)^(1/4)*2^(1/2)*(ln((x+(a
/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/
4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))

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Maxima [A]
time = 0.50, size = 412, normalized size = 1.21 \begin {gather*} \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {x}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {2 \, {\left (b d^{3} x^{\frac {5}{2}} + 5 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} \sqrt {x}\right )}}{5 \, b^{3}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{16 \, a b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/(b*x^2+a)^2/x^(1/2),x, algorithm="maxima")

[Out]

1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)/(a*b^4*x^2 + a^2*b^3) + 2/5*(b*d^3*x^(5/2) + 5
*(3*b*c*d^2 - 2*a*d^3)*sqrt(x))/b^3 + 3/16*(2*sqrt(2)*(b^3*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*arct
an(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt
(b))) + 2*sqrt(2)*(b^3*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(
1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(b^3*c^3 + a*b^2*c^
2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4))
- sqrt(2)*(b^3*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x
+ sqrt(a))/(a^(3/4)*b^(1/4)))/(a*b^3)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1944 vs. \(2 (258) = 516\).
time = 0.55, size = 1944, normalized size = 5.72 \begin {gather*} \frac {60 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{4} b^{6} \sqrt {-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}} + {\left (b^{6} c^{6} + 2 \, a b^{5} c^{5} d - 9 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{3} b^{3} c^{3} d^{3} + 31 \, a^{4} b^{2} c^{2} d^{4} - 30 \, a^{5} b c d^{5} + 9 \, a^{6} d^{6}\right )} x} a^{5} b^{10} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {3}{4}} - {\left (a^{5} b^{13} c^{3} + a^{6} b^{12} c^{2} d - 5 \, a^{7} b^{11} c d^{2} + 3 \, a^{8} b^{10} d^{3}\right )} \sqrt {x} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {3}{4}}}{b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}\right ) + 15 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \log \left (3 \, a^{2} b^{3} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} + 3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \sqrt {x}\right ) - 15 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \log \left (-3 \, a^{2} b^{3} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} + 3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \sqrt {x}\right ) + 4 \, {\left (4 \, a b^{2} d^{3} x^{4} + 5 \, b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 75 \, a^{2} b c d^{2} - 45 \, a^{3} d^{3} + 12 \, {\left (5 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {x}}{40 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/(b*x^2+a)^2/x^(1/2),x, algorithm="fricas")

[Out]

1/40*(60*(a*b^4*x^2 + a^2*b^3)*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 12
7*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1
932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(1/4)*arctan((sqr
t(a^4*b^6*sqrt(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4
+ 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d
^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13)) + (b^6*c^6 + 2*a*b^5*c^5*d - 9*a^2
*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 - 30*a^5*b*c*d^5 + 9*a^6*d^6)*x)*a^5*b^10*(-(b^12*c^12 +
4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644
*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10
- 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(3/4) - (a^5*b^13*c^3 + a^6*b^12*c^2*d - 5*a^7*b^11*c*d^2 + 3*
a^8*b^10*d^3)*sqrt(x)*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8
*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b
^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(3/4))/(b^12*c^12 + 4*a*b^
11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^
6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a
^11*b*c*d^11 + 81*a^12*d^12)) + 15*(a*b^4*x^2 + a^2*b^3)*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2
- 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7
+ 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^
7*b^13))^(1/4)*log(3*a^2*b^3*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*
a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 193
2*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(1/4) + 3*(b^3*c^3
+ a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*sqrt(x)) - 15*(a*b^4*x^2 + a^2*b^3)*(-(b^12*c^12 + 4*a*b^11*c^11*d
- 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6
+ 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^
11 + 81*a^12*d^12)/(a^7*b^13))^(1/4)*log(-3*a^2*b^3*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44
*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 103
9*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^1
3))^(1/4) + 3*(b^3*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*sqrt(x)) + 4*(4*a*b^2*d^3*x^4 + 5*b^3*c^3 -
15*a*b^2*c^2*d + 75*a^2*b*c*d^2 - 45*a^3*d^3 + 12*(5*a*b^2*c*d^2 - 3*a^2*b*d^3)*x^2)*sqrt(x))/(a*b^4*x^2 + a^2
*b^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1775 vs. \(2 (323) = 646\).
time = 56.92, size = 1775, normalized size = 5.22 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 c^{3}}{7 x^{\frac {7}{2}}} - \frac {2 c^{2} d}{x^{\frac {3}{2}}} + 6 c d^{2} \sqrt {x} + \frac {2 d^{3} x^{\frac {5}{2}}}{5}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 c^{3}}{7 x^{\frac {7}{2}}} - \frac {2 c^{2} d}{x^{\frac {3}{2}}} + 6 c d^{2} \sqrt {x} + \frac {2 d^{3} x^{\frac {5}{2}}}{5}}{b^{2}} & \text {for}\: a = 0 \\\frac {2 c^{3} \sqrt {x} + \frac {6 c^{2} d x^{\frac {5}{2}}}{5} + \frac {2 c d^{2} x^{\frac {9}{2}}}{3} + \frac {2 d^{3} x^{\frac {13}{2}}}{13}}{a^{2}} & \text {for}\: b = 0 \\- \frac {180 a^{4} d^{3} \sqrt {x}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {45 a^{4} d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {45 a^{4} d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {90 a^{4} d^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {300 a^{3} b c d^{2} \sqrt {x}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {75 a^{3} b c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {75 a^{3} b c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {150 a^{3} b c d^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {144 a^{3} b d^{3} x^{\frac {5}{2}}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {45 a^{3} b d^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {45 a^{3} b d^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {90 a^{3} b d^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {60 a^{2} b^{2} c^{2} d \sqrt {x}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {15 a^{2} b^{2} c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {15 a^{2} b^{2} c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {30 a^{2} b^{2} c^{2} d \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {240 a^{2} b^{2} c d^{2} x^{\frac {5}{2}}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {75 a^{2} b^{2} c d^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {75 a^{2} b^{2} c d^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {150 a^{2} b^{2} c d^{2} x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {16 a^{2} b^{2} d^{3} x^{\frac {9}{2}}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {20 a b^{3} c^{3} \sqrt {x}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {15 a b^{3} c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {15 a b^{3} c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {30 a b^{3} c^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {15 a b^{3} c^{2} d x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {15 a b^{3} c^{2} d x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {30 a b^{3} c^{2} d x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} - \frac {15 b^{4} c^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {15 b^{4} c^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} + \frac {30 b^{4} c^{3} x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{40 a^{3} b^{3} + 40 a^{2} b^{4} x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**3/(b*x**2+a)**2/x**(1/2),x)

[Out]

Piecewise((zoo*(-2*c**3/(7*x**(7/2)) - 2*c**2*d/x**(3/2) + 6*c*d**2*sqrt(x) + 2*d**3*x**(5/2)/5), Eq(a, 0) & E
q(b, 0)), ((-2*c**3/(7*x**(7/2)) - 2*c**2*d/x**(3/2) + 6*c*d**2*sqrt(x) + 2*d**3*x**(5/2)/5)/b**2, Eq(a, 0)),
((2*c**3*sqrt(x) + 6*c**2*d*x**(5/2)/5 + 2*c*d**2*x**(9/2)/3 + 2*d**3*x**(13/2)/13)/a**2, Eq(b, 0)), (-180*a**
4*d**3*sqrt(x)/(40*a**3*b**3 + 40*a**2*b**4*x**2) - 45*a**4*d**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(4
0*a**3*b**3 + 40*a**2*b**4*x**2) + 45*a**4*d**3*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(40*a**3*b**3 + 40*
a**2*b**4*x**2) + 90*a**4*d**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) +
300*a**3*b*c*d**2*sqrt(x)/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 75*a**3*b*c*d**2*(-a/b)**(1/4)*log(sqrt(x) - (-
a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) - 75*a**3*b*c*d**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/
(40*a**3*b**3 + 40*a**2*b**4*x**2) - 150*a**3*b*c*d**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(40*a**3*b**3
+ 40*a**2*b**4*x**2) - 144*a**3*b*d**3*x**(5/2)/(40*a**3*b**3 + 40*a**2*b**4*x**2) - 45*a**3*b*d**3*x**2*(-a/
b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 45*a**3*b*d**3*x**2*(-a/b)**(1/4)*
log(sqrt(x) + (-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 90*a**3*b*d**3*x**2*(-a/b)**(1/4)*atan(sqrt(
x)/(-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) - 60*a**2*b**2*c**2*d*sqrt(x)/(40*a**3*b**3 + 40*a**2*b**
4*x**2) - 15*a**2*b**2*c**2*d*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) +
15*a**2*b**2*c**2*d*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 30*a**2*b*
*2*c**2*d*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 240*a**2*b**2*c*d**2*
x**(5/2)/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 75*a**2*b**2*c*d**2*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/
4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) - 75*a**2*b**2*c*d**2*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(
40*a**3*b**3 + 40*a**2*b**4*x**2) - 150*a**2*b**2*c*d**2*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(40*a*
*3*b**3 + 40*a**2*b**4*x**2) + 16*a**2*b**2*d**3*x**(9/2)/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 20*a*b**3*c**3*
sqrt(x)/(40*a**3*b**3 + 40*a**2*b**4*x**2) - 15*a*b**3*c**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(40*a**
3*b**3 + 40*a**2*b**4*x**2) + 15*a*b**3*c**3*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(40*a**3*b**3 + 40*a**
2*b**4*x**2) + 30*a*b**3*c**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) - 1
5*a*b**3*c**2*d*x**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 15*a*b**3
*c**2*d*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 30*a*b**3*c**2*d*
x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) - 15*b**4*c**3*x**2*(-a/b)**
(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 15*b**4*c**3*x**2*(-a/b)**(1/4)*log(sq
rt(x) + (-a/b)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2) + 30*b**4*c**3*x**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b
)**(1/4))/(40*a**3*b**3 + 40*a**2*b**4*x**2), True))

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Giac [A]
time = 1.70, size = 511, normalized size = 1.50 \begin {gather*} \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{2} b^{4}} + \frac {b^{3} c^{3} \sqrt {x} - 3 \, a b^{2} c^{2} d \sqrt {x} + 3 \, a^{2} b c d^{2} \sqrt {x} - a^{3} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} a b^{3}} + \frac {2 \, {\left (b^{8} d^{3} x^{\frac {5}{2}} + 15 \, b^{8} c d^{2} \sqrt {x} - 10 \, a b^{7} d^{3} \sqrt {x}\right )}}{5 \, b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/(b*x^2+a)^2/x^(1/2),x, algorithm="giac")

[Out]

3/8*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 + (a*b^3)^(1/4)*a*b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 + 3*(a*b^3)^(1/4)
*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^4) + 3/8*sqrt(2)*((a*b^3)^(
1/4)*b^3*c^3 + (a*b^3)^(1/4)*a*b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 + 3*(a*b^3)^(1/4)*a^3*d^3)*arctan(-1/2*
sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^4) + 3/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 + (a*b^
3)^(1/4)*a*b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 + 3*(a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4)
+ x + sqrt(a/b))/(a^2*b^4) - 3/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 + (a*b^3)^(1/4)*a*b^2*c^2*d - 5*(a*b^3)^(1/4)
*a^2*b*c*d^2 + 3*(a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^4) + 1/2*(b^3
*c^3*sqrt(x) - 3*a*b^2*c^2*d*sqrt(x) + 3*a^2*b*c*d^2*sqrt(x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*a*b^3) + 2/5*(b^8
*d^3*x^(5/2) + 15*b^8*c*d^2*sqrt(x) - 10*a*b^7*d^3*sqrt(x))/b^10

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Mupad [B]
time = 0.13, size = 1636, normalized size = 4.81 \begin {gather*} \frac {2\,d^3\,x^{5/2}}{5\,b^2}-\sqrt {x}\,\left (\frac {4\,a\,d^3}{b^3}-\frac {6\,c\,d^2}{b^2}\right )-\frac {\sqrt {x}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,a\,\left (b^4\,x^2+a\,b^3\right )}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {9\,\sqrt {x}\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{a^2\,b^3}-\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,3{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}+\frac {\left (\frac {9\,\sqrt {x}\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{a^2\,b^3}+\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,3{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}}{\frac {3\,\left (\frac {9\,\sqrt {x}\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{a^2\,b^3}-\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}-\frac {3\,\left (\frac {9\,\sqrt {x}\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{a^2\,b^3}+\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,3{}\mathrm {i}}{4\,{\left (-a\right )}^{7/4}\,b^{13/4}}+\frac {3\,\mathrm {atan}\left (\frac {\frac {3\,\left (\frac {9\,\sqrt {x}\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{a^2\,b^3}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )\,3{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}+\frac {3\,\left (\frac {9\,\sqrt {x}\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{a^2\,b^3}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )\,3{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}}{\frac {\left (\frac {9\,\sqrt {x}\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{a^2\,b^3}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )\,3{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,3{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}-\frac {\left (\frac {9\,\sqrt {x}\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{a^2\,b^3}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (72\,a^3\,d^3-120\,a^2\,b\,c\,d^2+24\,a\,b^2\,c^2\,d+24\,b^3\,c^3\right )\,3{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,3{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )}{4\,{\left (-a\right )}^{7/4}\,b^{13/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^3/(x^(1/2)*(a + b*x^2)^2),x)

[Out]

(2*d^3*x^(5/2))/(5*b^2) - x^(1/2)*((4*a*d^3)/b^3 - (6*c*d^2)/b^2) - (x^(1/2)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*
d - 3*a^2*b*c*d^2))/(2*a*(a*b^3 + b^4*x^2)) + (atan(((((9*x^(1/2)*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4
*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(a^2*b^3) - (3*(a*d - b*c)^2*(3*a*d +
b*c)*(72*a^3*d^3 + 24*b^3*c^3 + 24*a*b^2*c^2*d - 120*a^2*b*c*d^2))/(8*(-a)^(7/4)*b^(13/4)))*(a*d - b*c)^2*(3*
a*d + b*c)*3i)/(8*(-a)^(7/4)*b^(13/4)) + (((9*x^(1/2)*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3
*d^3 + 31*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(a^2*b^3) + (3*(a*d - b*c)^2*(3*a*d + b*c)*(72*a^
3*d^3 + 24*b^3*c^3 + 24*a*b^2*c^2*d - 120*a^2*b*c*d^2))/(8*(-a)^(7/4)*b^(13/4)))*(a*d - b*c)^2*(3*a*d + b*c)*3
i)/(8*(-a)^(7/4)*b^(13/4)))/((3*((9*x^(1/2)*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*
a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(a^2*b^3) - (3*(a*d - b*c)^2*(3*a*d + b*c)*(72*a^3*d^3 + 24
*b^3*c^3 + 24*a*b^2*c^2*d - 120*a^2*b*c*d^2))/(8*(-a)^(7/4)*b^(13/4)))*(a*d - b*c)^2*(3*a*d + b*c))/(8*(-a)^(7
/4)*b^(13/4)) - (3*((9*x^(1/2)*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d
^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(a^2*b^3) + (3*(a*d - b*c)^2*(3*a*d + b*c)*(72*a^3*d^3 + 24*b^3*c^3 + 24
*a*b^2*c^2*d - 120*a^2*b*c*d^2))/(8*(-a)^(7/4)*b^(13/4)))*(a*d - b*c)^2*(3*a*d + b*c))/(8*(-a)^(7/4)*b^(13/4))
))*(a*d - b*c)^2*(3*a*d + b*c)*3i)/(4*(-a)^(7/4)*b^(13/4)) + (3*atan(((3*((9*x^(1/2)*(9*a^6*d^6 + b^6*c^6 - 9*
a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(a^2*b^3) - ((a*d
- b*c)^2*(3*a*d + b*c)*(72*a^3*d^3 + 24*b^3*c^3 + 24*a*b^2*c^2*d - 120*a^2*b*c*d^2)*3i)/(8*(-a)^(7/4)*b^(13/4)
))*(a*d - b*c)^2*(3*a*d + b*c))/(8*(-a)^(7/4)*b^(13/4)) + (3*((9*x^(1/2)*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*
d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(a^2*b^3) + ((a*d - b*c)^2*(3*
a*d + b*c)*(72*a^3*d^3 + 24*b^3*c^3 + 24*a*b^2*c^2*d - 120*a^2*b*c*d^2)*3i)/(8*(-a)^(7/4)*b^(13/4)))*(a*d - b*
c)^2*(3*a*d + b*c))/(8*(-a)^(7/4)*b^(13/4)))/((((9*x^(1/2)*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^
3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(a^2*b^3) - ((a*d - b*c)^2*(3*a*d + b*c)*(72
*a^3*d^3 + 24*b^3*c^3 + 24*a*b^2*c^2*d - 120*a^2*b*c*d^2)*3i)/(8*(-a)^(7/4)*b^(13/4)))*(a*d - b*c)^2*(3*a*d +
b*c)*3i)/(8*(-a)^(7/4)*b^(13/4)) - (((9*x^(1/2)*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 +
31*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(a^2*b^3) + ((a*d - b*c)^2*(3*a*d + b*c)*(72*a^3*d^3 +
24*b^3*c^3 + 24*a*b^2*c^2*d - 120*a^2*b*c*d^2)*3i)/(8*(-a)^(7/4)*b^(13/4)))*(a*d - b*c)^2*(3*a*d + b*c)*3i)/(8
*(-a)^(7/4)*b^(13/4))))*(a*d - b*c)^2*(3*a*d + b*c))/(4*(-a)^(7/4)*b^(13/4))

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