3.5.57 \(\int \frac {(c+d x^2)^3}{x^{3/2} (a+b x^2)^2} \, dx\) [457]

Optimal. Leaf size=368 \[ -\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (5 b c+7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}} \]

[Out]

-1/6*d^2*(-7*a*d+3*b*c)*x^(3/2)/a/b^2+1/8*(-a*d+b*c)^2*(7*a*d+5*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))
/a^(9/4)/b^(11/4)*2^(1/2)-1/8*(-a*d+b*c)^2*(7*a*d+5*b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/b^(
11/4)*2^(1/2)-1/16*(-a*d+b*c)^2*(7*a*d+5*b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/b^
(11/4)*2^(1/2)+1/16*(-a*d+b*c)^2*(7*a*d+5*b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/b
^(11/4)*2^(1/2)-1/2*c^2*(-a*d+5*b*c)/a^2/b/x^(1/2)+1/2*(-a*d+b*c)*(d*x^2+c)^2/a/b/(b*x^2+a)/x^(1/2)

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Rubi [A]
time = 0.29, antiderivative size = 368, 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, 479, 584, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^2 (7 a d+5 b c)}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (7 a d+5 b c)}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (7 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (7 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 x^{3/2} (3 b c-7 a d)}{6 a b^2}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{2 a b \sqrt {x} \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(c^2*(5*b*c - a*d))/(a^2*b*Sqrt[x]) - (d^2*(3*b*c - 7*a*d)*x^(3/2))/(6*a*b^2) + ((b*c - a*d)*(c + d*x^2)^
2)/(2*a*b*Sqrt[x]*(a + b*x^2)) + ((b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])
/(4*Sqrt[2]*a^(9/4)*b^(11/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/
(4*Sqrt[2]*a^(9/4)*b^(11/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(11/4)) + ((b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(11/4))

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 303

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

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 479

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-(c*b -
a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)),
Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(
p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 584

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

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}{x^{3/2} \left (a+b x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (-c (5 b c-a d)+d (3 b c-7 a d) x^4\right )}{x^2 \left (a+b x^4\right )} \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {c^2 (-5 b c+a d)}{a x^2}+\frac {d^2 (3 b c-7 a d) x^2}{b}+\frac {(-b c+a d)^2 (5 b c+7 a d) x^2}{a b \left (a+b x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 a b}\\ &=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (5 b c+7 a d)\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2 b^2}\\ &=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 (5 b c+7 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^2 b^{5/2}}-\frac {\left ((b c-a d)^2 (5 b c+7 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^2 b^{5/2}}\\ &=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {\left ((b c-a d)^2 (5 b c+7 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^2 b^3}-\frac {\left ((b c-a d)^2 (5 b c+7 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^2 b^3}-\frac {\left ((b c-a d)^2 (5 b c+7 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^{9/4} b^{11/4}}-\frac {\left ((b c-a d)^2 (5 b c+7 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^{9/4} b^{11/4}}\\ &=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}-\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}-\frac {\left ((b c-a d)^2 (5 b c+7 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^{9/4} b^{11/4}}+\frac {\left ((b c-a d)^2 (5 b c+7 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^{9/4} b^{11/4}}\\ &=-\frac {c^2 (5 b c-a d)}{2 a^2 b \sqrt {x}}-\frac {d^2 (3 b c-7 a d) x^{3/2}}{6 a b^2}+\frac {(b c-a d) \left (c+d x^2\right )^2}{2 a b \sqrt {x} \left (a+b x^2\right )}+\frac {(b c-a d)^2 (5 b c+7 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}\\ \end {align*}

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Mathematica [A]
time = 0.48, size = 229, normalized size = 0.62 \begin {gather*} \frac {\frac {4 \sqrt [4]{a} b^{3/4} \left (-15 b^3 c^3 x^2+7 a^3 d^3 x^2+3 a b^2 c^2 \left (-4 c+3 d x^2\right )+a^2 b d^2 x^2 \left (-9 c+4 d x^2\right )\right )}{\sqrt {x} \left (a+b x^2\right )}+3 \sqrt {2} (b c-a d)^2 (5 b c+7 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} (b c-a d)^2 (5 b c+7 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{24 a^{9/4} b^{11/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a^(1/4)*b^(3/4)*(-15*b^3*c^3*x^2 + 7*a^3*d^3*x^2 + 3*a*b^2*c^2*(-4*c + 3*d*x^2) + a^2*b*d^2*x^2*(-9*c + 4*
d*x^2)))/(Sqrt[x]*(a + b*x^2)) + 3*Sqrt[2]*(b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]
*a^(1/4)*b^(1/4)*Sqrt[x])] + 3*Sqrt[2]*(b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])
/(Sqrt[a] + Sqrt[b]*x)])/(24*a^(9/4)*b^(11/4))

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Maple [A]
time = 0.15, size = 225, normalized size = 0.61

method	result	size
derivativedivides	\(\frac {2 d^{3} x^{\frac {3}{2}}}{3 b^{2}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {7}{4} a^{3} d^{3}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {5}{4} b^{3} c^{3}-\frac {9}{4} a^{2} b c \,d^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{2}}-\frac {2 c^{3}}{a^{2} \sqrt {x}}\)	\(225\)
default	\(\frac {2 d^{3} x^{\frac {3}{2}}}{3 b^{2}}-\frac {2 \left (\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {7}{4} a^{3} d^{3}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {5}{4} b^{3} c^{3}-\frac {9}{4} a^{2} b c \,d^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{2}}-\frac {2 c^{3}}{a^{2} \sqrt {x}}\)	\(225\)
risch	\(\frac {-2 b^{2} c^{3}+\frac {2 a^{2} d^{3} x^{2}}{3}}{a^{2} \sqrt {x}\, b^{2}}+\frac {a \,x^{\frac {3}{2}} d^{3}}{2 b^{2} \left (b \,x^{2}+a \right )}-\frac {3 x^{\frac {3}{2}} d^{2} c}{2 b \left (b \,x^{2}+a \right )}+\frac {3 x^{\frac {3}{2}} d \,c^{2}}{2 a \left (b \,x^{2}+a \right )}-\frac {b \,x^{\frac {3}{2}} c^{3}}{2 a^{2} \left (b \,x^{2}+a \right )}-\frac {7 a \sqrt {2}\, d^{3} \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 )}{16 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {9 \sqrt {2}\, d^{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}{16 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, d \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^{2}}{16 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {7 a \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {9 \sqrt {2}\, d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) c}{8 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) c^{2}}{8 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {7 a \sqrt {2}\, d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {9 \sqrt {2}\, d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) c}{8 b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {3 \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) c^{2}}{8 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, c^{3} \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 )}{16 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {5 \sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(690\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*d^3*x^(3/2)/b^2-2/a^2/b^2*((-1/4*a^3*d^3+3/4*a^2*b*c*d^2-3/4*a*b^2*c^2*d+1/4*b^3*c^3)*x^(3/2)/(b*x^2+a)+1/
8*(7/4*a^3*d^3-3/4*a*b^2*c^2*d+5/4*b^3*c^3-9/4*a^2*b*c*d^2)/b/(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*arc
tan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))-2*c^3/a^2/x^(1/2)

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Maxima [A]
time = 0.51, size = 304, normalized size = 0.83 \begin {gather*} \frac {2 \, d^{3} x^{\frac {3}{2}}}{3 \, b^{2}} - \frac {4 \, a b^{2} c^{3} + {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{2 \, {\left (a^{2} b^{3} x^{\frac {5}{2}} + a^{3} b^{2} \sqrt {x}\right )}} - \frac {{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a^{2} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*d^3*x^(3/2)/b^2 - 1/2*(4*a*b^2*c^3 + (5*b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^2)/(a^2*b^3*x
^(5/2) + a^3*b^2*sqrt(x)) - 1/16*(5*b^3*c^3 - 3*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 7*a^3*d^3)*(2*sqrt(2)*arctan(1/2
*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b))
+ 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqr
t(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4))
+ sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/(a^2*b^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2547 vs. \(2 (284) = 568\).
time = 0.56, size = 2547, normalized size = 6.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(12*(a^2*b^3*x^3 + a^3*b^2*x)*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*
b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 3
7665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/
(a^9*b^11))^(1/4)*arctan((sqrt((15625*b^18*c^18 - 56250*a*b^17*c^17*d - 84375*a^2*b^16*c^16*d^2 + 570000*a^3*b
^15*c^15*d^3 - 211500*a^4*b^14*c^14*d^4 - 2174040*a^5*b^13*c^13*d^5 + 2720004*a^6*b^12*c^12*d^6 + 3321072*a^7*
b^11*c^11*d^7 - 8368866*a^8*b^10*c^10*d^8 + 640420*a^9*b^9*c^9*d^9 + 11255310*a^10*b^8*c^8*d^10 - 8509968*a^11
*b^7*c^7*d^11 - 4831644*a^12*b^6*c^6*d^12 + 9537192*a^13*b^5*c^5*d^13 - 3095820*a^14*b^4*c^4*d^14 - 2551920*a^
15*b^3*c^3*d^15 + 2614689*a^16*b^2*c^2*d^16 - 907578*a^17*b*c*d^17 + 117649*a^18*d^18)*x - (625*a^5*b^17*c^12
- 1500*a^6*b^16*c^11*d - 3150*a^7*b^15*c^10*d^2 + 11060*a^8*b^14*c^9*d^3 + 1071*a^9*b^13*c^8*d^4 - 28728*a^10*
b^12*c^7*d^5 + 19068*a^11*b^11*c^6*d^6 + 27144*a^12*b^10*c^5*d^7 - 37665*a^13*b^9*c^4*d^8 + 2324*a^14*b^8*c^3*
d^9 + 19698*a^15*b^7*c^2*d^10 - 12348*a^16*b^6*c*d^11 + 2401*a^17*b^5*d^12)*sqrt(-(625*b^12*c^12 - 1500*a*b^11
*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 1906
8*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*
d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11)))*a^2*b^3*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150
*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*
d^6 + 27144*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a
^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11))^(1/4) - (125*a^2*b^12*c^9 - 225*a^3*b^11*c^8*d - 540*a^4*b^10*c^7*d
^2 + 1308*a^5*b^9*c^6*d^3 + 342*a^6*b^8*c^5*d^4 - 2430*a^7*b^7*c^4*d^5 + 1140*a^8*b^6*c^3*d^6 + 1260*a^9*b^5*c
^2*d^7 - 1323*a^10*b^4*c*d^8 + 343*a^11*b^3*d^9)*sqrt(x)*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10
*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 271
44*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d
^11 + 2401*a^12*d^12)/(a^9*b^11))^(1/4))/(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*
a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7
- 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^
12)) - 3*(a^2*b^3*x^3 + a^3*b^2*x)*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*
b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 3
7665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/
(a^9*b^11))^(1/4)*log(a^7*b^8*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c
^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 37665*
a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*
b^11))^(3/4) + (125*b^9*c^9 - 225*a*b^8*c^8*d - 540*a^2*b^7*c^7*d^2 + 1308*a^3*b^6*c^6*d^3 + 342*a^4*b^5*c^5*d
^4 - 2430*a^5*b^4*c^4*d^5 + 1140*a^6*b^3*c^3*d^6 + 1260*a^7*b^2*c^2*d^7 - 1323*a^8*b*c*d^8 + 343*a^9*d^9)*sqrt
(x)) + 3*(a^2*b^3*x^3 + a^3*b^2*x)*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*
b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 3
7665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/
(a^9*b^11))^(1/4)*log(-a^7*b^8*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*
c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 37665
*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9
*b^11))^(3/4) + (125*b^9*c^9 - 225*a*b^8*c^8*d - 540*a^2*b^7*c^7*d^2 + 1308*a^3*b^6*c^6*d^3 + 342*a^4*b^5*c^5*
d^4 - 2430*a^5*b^4*c^4*d^5 + 1140*a^6*b^3*c^3*d^6 + 1260*a^7*b^2*c^2*d^7 - 1323*a^8*b*c*d^8 + 343*a^9*d^9)*sqr
t(x)) + 4*(4*a^2*b*d^3*x^4 - 12*a*b^2*c^3 - (15*b^3*c^3 - 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 7*a^3*d^3)*x^2)*sqrt
(x))/(a^2*b^3*x^3 + a^3*b^2*x)

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Sympy [A]
time = 202.38, size = 1443, normalized size = 3.92 \begin {gather*} c^{3} \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{9 b^{2} x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {5 a \sqrt {x} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} + \frac {5 a \sqrt {x} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {10 a \sqrt {x} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {16 a \sqrt [4]{- \frac {a}{b}}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {5 b x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} + \frac {5 b x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {10 b x^{\frac {5}{2}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} - \frac {20 b x^{2} \sqrt [4]{- \frac {a}{b}}}{8 a^{3} \sqrt {x} \sqrt [4]{- \frac {a}{b}} + 8 a^{2} b x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases}\right ) + \frac {6 c^{2} d x^{\frac {3}{2}}}{4 a^{2} + 4 a b x^{2}} + 6 c^{2} d \operatorname {RootSum} {\left (65536 t^{4} a^{5} b^{3} + 1, \left ( t \mapsto t \log {\left (4096 t^{3} a^{4} b^{2} + \sqrt {x} \right )} \right )\right )} + 3 c d^{2} \left (\begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a^{2}} & \text {for}\: b = 0 \\- \frac {2}{b^{2} \sqrt {x}} & \text {for}\: a = 0 \\\frac {3 a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {3 a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {6 a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {4 b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {3 b x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {3 b x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {6 b x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a b^{2} \sqrt [4]{- \frac {a}{b}} + 8 b^{3} x^{2} \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases}\right ) + d^{3} \left (\begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {11}{2}}}{11 a^{2}} & \text {for}\: b = 0 \\\frac {2 x^{\frac {3}{2}}}{3 b^{2}} & \text {for}\: a = 0 \\- \frac {21 a^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{24 a b^{3} \sqrt [4]{- \frac {a}{b}} + 24 b^{4} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {21 a^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{24 a b^{3} \sqrt [4]{- \frac {a}{b}} + 24 b^{4} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {42 a^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{24 a b^{3} \sqrt [4]{- \frac {a}{b}} + 24 b^{4} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {28 a b x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{b}}}{24 a b^{3} \sqrt [4]{- \frac {a}{b}} + 24 b^{4} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {21 a b x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{24 a b^{3} \sqrt [4]{- \frac {a}{b}} + 24 b^{4} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {21 a b x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{24 a b^{3} \sqrt [4]{- \frac {a}{b}} + 24 b^{4} x^{2} \sqrt [4]{- \frac {a}{b}}} - \frac {42 a b x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{24 a b^{3} \sqrt [4]{- \frac {a}{b}} + 24 b^{4} x^{2} \sqrt [4]{- \frac {a}{b}}} + \frac {16 b^{2} x^{\frac {7}{2}} \sqrt [4]{- \frac {a}{b}}}{24 a b^{3} \sqrt [4]{- \frac {a}{b}} + 24 b^{4} x^{2} \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*Piecewise((zoo/x**(9/2), Eq(a, 0) & Eq(b, 0)), (-2/(9*b**2*x**(9/2)), Eq(a, 0)), (-2/(a**2*sqrt(x)), Eq(b
, 0)), (-5*a*sqrt(x)*log(sqrt(x) - (-a/b)**(1/4))/(8*a**3*sqrt(x)*(-a/b)**(1/4) + 8*a**2*b*x**(5/2)*(-a/b)**(1
/4)) + 5*a*sqrt(x)*log(sqrt(x) + (-a/b)**(1/4))/(8*a**3*sqrt(x)*(-a/b)**(1/4) + 8*a**2*b*x**(5/2)*(-a/b)**(1/4
)) - 10*a*sqrt(x)*atan(sqrt(x)/(-a/b)**(1/4))/(8*a**3*sqrt(x)*(-a/b)**(1/4) + 8*a**2*b*x**(5/2)*(-a/b)**(1/4))
- 16*a*(-a/b)**(1/4)/(8*a**3*sqrt(x)*(-a/b)**(1/4) + 8*a**2*b*x**(5/2)*(-a/b)**(1/4)) - 5*b*x**(5/2)*log(sqrt
(x) - (-a/b)**(1/4))/(8*a**3*sqrt(x)*(-a/b)**(1/4) + 8*a**2*b*x**(5/2)*(-a/b)**(1/4)) + 5*b*x**(5/2)*log(sqrt(
x) + (-a/b)**(1/4))/(8*a**3*sqrt(x)*(-a/b)**(1/4) + 8*a**2*b*x**(5/2)*(-a/b)**(1/4)) - 10*b*x**(5/2)*atan(sqrt
(x)/(-a/b)**(1/4))/(8*a**3*sqrt(x)*(-a/b)**(1/4) + 8*a**2*b*x**(5/2)*(-a/b)**(1/4)) - 20*b*x**2*(-a/b)**(1/4)/
(8*a**3*sqrt(x)*(-a/b)**(1/4) + 8*a**2*b*x**(5/2)*(-a/b)**(1/4)), True)) + 6*c**2*d*x**(3/2)/(4*a**2 + 4*a*b*x
**2) + 6*c**2*d*RootSum(65536*_t**4*a**5*b**3 + 1, Lambda(_t, _t*log(4096*_t**3*a**4*b**2 + sqrt(x)))) + 3*c*d
**2*Piecewise((zoo/sqrt(x), Eq(a, 0) & Eq(b, 0)), (2*x**(7/2)/(7*a**2), Eq(b, 0)), (-2/(b**2*sqrt(x)), Eq(a, 0
)), (3*a*log(sqrt(x) - (-a/b)**(1/4))/(8*a*b**2*(-a/b)**(1/4) + 8*b**3*x**2*(-a/b)**(1/4)) - 3*a*log(sqrt(x) +
(-a/b)**(1/4))/(8*a*b**2*(-a/b)**(1/4) + 8*b**3*x**2*(-a/b)**(1/4)) + 6*a*atan(sqrt(x)/(-a/b)**(1/4))/(8*a*b*
*2*(-a/b)**(1/4) + 8*b**3*x**2*(-a/b)**(1/4)) - 4*b*x**(3/2)*(-a/b)**(1/4)/(8*a*b**2*(-a/b)**(1/4) + 8*b**3*x*
*2*(-a/b)**(1/4)) + 3*b*x**2*log(sqrt(x) - (-a/b)**(1/4))/(8*a*b**2*(-a/b)**(1/4) + 8*b**3*x**2*(-a/b)**(1/4))
- 3*b*x**2*log(sqrt(x) + (-a/b)**(1/4))/(8*a*b**2*(-a/b)**(1/4) + 8*b**3*x**2*(-a/b)**(1/4)) + 6*b*x**2*atan(
sqrt(x)/(-a/b)**(1/4))/(8*a*b**2*(-a/b)**(1/4) + 8*b**3*x**2*(-a/b)**(1/4)), True)) + d**3*Piecewise((zoo*x**(
3/2), Eq(a, 0) & Eq(b, 0)), (2*x**(11/2)/(11*a**2), Eq(b, 0)), (2*x**(3/2)/(3*b**2), Eq(a, 0)), (-21*a**2*log(
sqrt(x) - (-a/b)**(1/4))/(24*a*b**3*(-a/b)**(1/4) + 24*b**4*x**2*(-a/b)**(1/4)) + 21*a**2*log(sqrt(x) + (-a/b)
**(1/4))/(24*a*b**3*(-a/b)**(1/4) + 24*b**4*x**2*(-a/b)**(1/4)) - 42*a**2*atan(sqrt(x)/(-a/b)**(1/4))/(24*a*b*
*3*(-a/b)**(1/4) + 24*b**4*x**2*(-a/b)**(1/4)) + 28*a*b*x**(3/2)*(-a/b)**(1/4)/(24*a*b**3*(-a/b)**(1/4) + 24*b
**4*x**2*(-a/b)**(1/4)) - 21*a*b*x**2*log(sqrt(x) - (-a/b)**(1/4))/(24*a*b**3*(-a/b)**(1/4) + 24*b**4*x**2*(-a
/b)**(1/4)) + 21*a*b*x**2*log(sqrt(x) + (-a/b)**(1/4))/(24*a*b**3*(-a/b)**(1/4) + 24*b**4*x**2*(-a/b)**(1/4))
- 42*a*b*x**2*atan(sqrt(x)/(-a/b)**(1/4))/(24*a*b**3*(-a/b)**(1/4) + 24*b**4*x**2*(-a/b)**(1/4)) + 16*b**2*x**
(7/2)*(-a/b)**(1/4)/(24*a*b**3*(-a/b)**(1/4) + 24*b**4*x**2*(-a/b)**(1/4)), True))

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Giac [A]
time = 1.82, size = 504, normalized size = 1.37 \begin {gather*} \frac {2 \, d^{3} x^{\frac {3}{2}}}{3 \, b^{2}} - \frac {5 \, b^{3} c^{3} x^{2} - 3 \, a b^{2} c^{2} d x^{2} + 3 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 4 \, a b^{2} c^{3}}{2 \, {\left (b x^{\frac {5}{2}} + a \sqrt {x}\right )} a^{2} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{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^{3} b^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{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^{3} b^{5}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{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^{3} b^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{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^{3} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*d^3*x^(3/2)/b^2 - 1/2*(5*b^3*c^3*x^2 - 3*a*b^2*c^2*d*x^2 + 3*a^2*b*c*d^2*x^2 - a^3*d^3*x^2 + 4*a*b^2*c^3)/
((b*x^(5/2) + a*sqrt(x))*a^2*b^2) - 1/8*sqrt(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*
b^3)^(3/4)*a^2*b*c*d^2 + 7*(a*b^3)^(3/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(
1/4))/(a^3*b^5) - 1/8*sqrt(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2*b*c
*d^2 + 7*(a*b^3)^(3/4)*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^5) +
1/16*sqrt(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2*b*c*d^2 + 7*(a*b^3)
^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^5) - 1/16*sqrt(2)*(5*(a*b^3)^(3/4)*b^3
*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2*b*c*d^2 + 7*(a*b^3)^(3/4)*a^3*d^3)*log(-sqrt(2)*sqrt(
x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^5)

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Mupad [B]
time = 0.22, size = 657, normalized size = 1.79 \begin {gather*} \frac {\frac {x^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{2\,a^2}-\frac {2\,b^2\,c^3}{a}}{b^3\,x^{5/2}+a\,b^2\,\sqrt {x}}+\frac {2\,d^3\,x^{3/2}}{3\,b^2}-\frac {\mathrm {atan}\left (\frac {\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,\left (1568\,a^{13}\,b^8\,d^6-4032\,a^{12}\,b^9\,c\,d^5+1248\,a^{11}\,b^{10}\,c^2\,d^4+3968\,a^{10}\,b^{11}\,c^3\,d^3-2592\,a^9\,b^{12}\,c^4\,d^2-960\,a^8\,b^{13}\,c^5\,d+800\,a^7\,b^{14}\,c^6\right )}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}\,\left (2744\,a^{14}\,b^5\,d^9-10584\,a^{13}\,b^6\,c\,d^8+10080\,a^{12}\,b^7\,c^2\,d^7+9120\,a^{11}\,b^8\,c^3\,d^6-19440\,a^{10}\,b^9\,c^4\,d^5+2736\,a^9\,b^{10}\,c^5\,d^4+10464\,a^8\,b^{11}\,c^6\,d^3-4320\,a^7\,b^{12}\,c^7\,d^2-1800\,a^6\,b^{13}\,c^8\,d+1000\,a^5\,b^{14}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}}-\frac {\mathrm {atan}\left (\frac {\sqrt {x}\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,\left (1568\,a^{13}\,b^8\,d^6-4032\,a^{12}\,b^9\,c\,d^5+1248\,a^{11}\,b^{10}\,c^2\,d^4+3968\,a^{10}\,b^{11}\,c^3\,d^3-2592\,a^9\,b^{12}\,c^4\,d^2-960\,a^8\,b^{13}\,c^5\,d+800\,a^7\,b^{14}\,c^6\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}\,\left (2744\,a^{14}\,b^5\,d^9-10584\,a^{13}\,b^6\,c\,d^8+10080\,a^{12}\,b^7\,c^2\,d^7+9120\,a^{11}\,b^8\,c^3\,d^6-19440\,a^{10}\,b^9\,c^4\,d^5+2736\,a^9\,b^{10}\,c^5\,d^4+10464\,a^8\,b^{11}\,c^6\,d^3-4320\,a^7\,b^{12}\,c^7\,d^2-1800\,a^6\,b^{13}\,c^8\,d+1000\,a^5\,b^{14}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{9/4}\,b^{11/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^2*(a^3*d^3 - 5*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(2*a^2) - (2*b^2*c^3)/a)/(b^3*x^(5/2) + a*b^2*x^(
1/2)) + (2*d^3*x^(3/2))/(3*b^2) - (atan((x^(1/2)*(a*d - b*c)^2*(7*a*d + 5*b*c)*(800*a^7*b^14*c^6 + 1568*a^13*b
^8*d^6 - 960*a^8*b^13*c^5*d - 4032*a^12*b^9*c*d^5 - 2592*a^9*b^12*c^4*d^2 + 3968*a^10*b^11*c^3*d^3 + 1248*a^11
*b^10*c^2*d^4))/(4*(-a)^(9/4)*b^(11/4)*(1000*a^5*b^14*c^9 + 2744*a^14*b^5*d^9 - 1800*a^6*b^13*c^8*d - 10584*a^
13*b^6*c*d^8 - 4320*a^7*b^12*c^7*d^2 + 10464*a^8*b^11*c^6*d^3 + 2736*a^9*b^10*c^5*d^4 - 19440*a^10*b^9*c^4*d^5
+ 9120*a^11*b^8*c^3*d^6 + 10080*a^12*b^7*c^2*d^7)))*(a*d - b*c)^2*(7*a*d + 5*b*c))/(4*(-a)^(9/4)*b^(11/4)) -
(atan((x^(1/2)*(a*d - b*c)^2*(7*a*d + 5*b*c)*(800*a^7*b^14*c^6 + 1568*a^13*b^8*d^6 - 960*a^8*b^13*c^5*d - 4032
*a^12*b^9*c*d^5 - 2592*a^9*b^12*c^4*d^2 + 3968*a^10*b^11*c^3*d^3 + 1248*a^11*b^10*c^2*d^4)*1i)/(4*(-a)^(9/4)*b
^(11/4)*(1000*a^5*b^14*c^9 + 2744*a^14*b^5*d^9 - 1800*a^6*b^13*c^8*d - 10584*a^13*b^6*c*d^8 - 4320*a^7*b^12*c^
7*d^2 + 10464*a^8*b^11*c^6*d^3 + 2736*a^9*b^10*c^5*d^4 - 19440*a^10*b^9*c^4*d^5 + 9120*a^11*b^8*c^3*d^6 + 1008
0*a^12*b^7*c^2*d^7)))*(a*d - b*c)^2*(7*a*d + 5*b*c)*1i)/(4*(-a)^(9/4)*b^(11/4))

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