3.5.44 \(\int \frac {(c+d x^2)^3}{\sqrt {x} (a+b x^2)} \, dx\) [444]

Optimal. Leaf size=304 \[ \frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}} \]

[Out]

2/5*d^2*(-a*d+3*b*c)*x^(5/2)/b^2+2/9*d^3*x^(9/2)/b-1/2*(-a*d+b*c)^3*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/
a^(3/4)/b^(13/4)*2^(1/2)+1/2*(-a*d+b*c)^3*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/b^(13/4)*2^(1/2)-1
/4*(-a*d+b*c)^3*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/b^(13/4)*2^(1/2)+1/4*(-a*d+b*c)^
3*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/b^(13/4)*2^(1/2)+2*d*(a^2*d^2-3*a*b*c*d+3*b^2*
c^2)*x^(1/2)/b^3

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Rubi [A]
time = 0.17, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {472, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-a d)^3}{\sqrt {2} a^{3/4} b^{13/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-a d)^3}{\sqrt {2} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {2 d \sqrt {x} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac {2 d^2 x^{5/2} (3 b c-a d)}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Sqrt[x])/b^3 + (2*d^2*(3*b*c - a*d)*x^(5/2))/(5*b^2) + (2*d^3*x^(9/2))/
(9*b) - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*b^(13/4)) + ((b*c - a*d
)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*b^(13/4)) - ((b*c - a*d)^3*Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*Log[Sqrt[a] + Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*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 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

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 )} \, dx &=\int \left (\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )}{b^3 \sqrt {x}}+\frac {d^2 (3 b c-a d) x^{3/2}}{b^2}+\frac {d^3 x^{7/2}}{b}+\frac {b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3}{b^3 \sqrt {x} \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}+\frac {(b c-a d)^3 \int \frac {1}{\sqrt {x} \left (a+b x^2\right )} \, dx}{b^3}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}+\frac {\left (2 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a} b^3}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a} b^3}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}+\frac {(b c-a d)^3 \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 )}{2 \sqrt {a} b^{7/2}}+\frac {(b c-a d)^3 \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 )}{2 \sqrt {a} b^{7/2}}-\frac {(b c-a d)^3 \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 )}{2 \sqrt {2} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \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 )}{2 \sqrt {2} a^{3/4} b^{13/4}}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}-\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{13/4}}\\ &=\frac {2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {x}}{b^3}+\frac {2 d^2 (3 b c-a d) x^{5/2}}{5 b^2}+\frac {2 d^3 x^{9/2}}{9 b}-\frac {(b c-a d)^3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} b^{13/4}}-\frac {(b c-a d)^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}+\frac {(b c-a d)^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} b^{13/4}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 188, normalized size = 0.62 \begin {gather*} \frac {4 \sqrt [4]{b} d \sqrt {x} \left (45 a^2 d^2-9 a b d \left (15 c+d x^2\right )+b^2 \left (135 c^2+27 c d x^2+5 d^2 x^4\right )\right )+\frac {45 \sqrt {2} (-b c+a d)^3 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {45 \sqrt {2} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}}{90 b^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b^(1/4)*d*Sqrt[x]*(45*a^2*d^2 - 9*a*b*d*(15*c + d*x^2) + b^2*(135*c^2 + 27*c*d*x^2 + 5*d^2*x^4)) + (45*Sqrt
[2]*(-(b*c) + a*d)^3*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(3/4) + (45*Sqrt[2]*(b
*c - a*d)^3*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(3/4))/(90*b^(13/4))

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Maple [A]
time = 0.10, size = 214, normalized size = 0.70

method	result	size
derivativedivides	\(\frac {2 d \left (\frac {b^{2} x^{\frac {9}{2}} d^{2}}{9}-\frac {a b \,d^{2} x^{\frac {5}{2}}}{5}+\frac {3 b^{2} c d \,x^{\frac {5}{2}}}{5}+a^{2} d^{2} \sqrt {x}-3 a b c d \sqrt {x}+3 b^{2} c^{2} \sqrt {x}\right )}{b^{3}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{3} a}\)	\(214\)
default	\(\frac {2 d \left (\frac {b^{2} x^{\frac {9}{2}} d^{2}}{9}-\frac {a b \,d^{2} x^{\frac {5}{2}}}{5}+\frac {3 b^{2} c d \,x^{\frac {5}{2}}}{5}+a^{2} d^{2} \sqrt {x}-3 a b c d \sqrt {x}+3 b^{2} c^{2} \sqrt {x}\right )}{b^{3}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{3} a}\)	\(214\)
risch	\(\frac {2 \left (5 b^{2} d^{2} x^{4}-9 a b \,d^{2} x^{2}+27 b^{2} c d \,x^{2}+45 a^{2} d^{2}-135 a b c d +135 b^{2} c^{2}\right ) d \sqrt {x}}{45 b^{3}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) d^{3}}{2 b^{3}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right ) c \,d^{2}}{2 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 ) c^{2} d}{2 b}+\frac {\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}}{2 a}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) d^{3}}{2 b^{3}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right ) c \,d^{2}}{2 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 ) c^{2} d}{2 b}+\frac {\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}}{2 a}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} \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}}{4 b^{3}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} a \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 \,d^{2}}{4 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 ) c^{2} d}{4 b}+\frac {\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}}{4 a}\)	\(639\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d/b^3*(1/9*b^2*x^(9/2)*d^2-1/5*a*b*d^2*x^(5/2)+3/5*b^2*c*d*x^(5/2)+a^2*d^2*x^(1/2)-3*a*b*c*d*x^(1/2)+3*b^2*c
^2*x^(1/2))+1/4*(-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+b^3*c^3)/b^3*(a/b)^(1/4)/a*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 = 0.51, size = 390, normalized size = 1.28 \begin {gather*} \frac {2 \, {\left (5 \, b^{2} d^{3} x^{\frac {9}{2}} + 9 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{\frac {5}{2}} + 45 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {x}\right )}}{45 \, b^{3}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 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} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 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} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 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} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 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}}}}{4 \, 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)/x^(1/2),x, algorithm="maxima")

[Out]

2/45*(5*b^2*d^3*x^(9/2) + 9*(3*b^2*c*d^2 - a*b*d^3)*x^(5/2) + 45*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*sqrt(x)
)/b^3 + 1/4*(2*sqrt(2)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - 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))) + 2*sqrt(2)*(b^3*c^3 - 3*
a*b^2*c^2*d + 3*a^2*b*c*d^2 - 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 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - 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 - 3*a*b^2*c^
2*d + 3*a^2*b*c*d^2 - 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)))/
b^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1862 vs. \(2 (227) = 454\).
time = 0.58, size = 1862, normalized size = 6.12 \begin {gather*} -\frac {180 \, b^{3} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{3} b^{13}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{2} b^{6} \sqrt {-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{3} b^{13}}} + {\left (b^{6} c^{6} - 6 \, a b^{5} c^{5} d + 15 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} + 15 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} + a^{6} d^{6}\right )} x} a^{2} b^{10} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{3} b^{13}}\right )^{\frac {3}{4}} + {\left (a^{2} b^{13} c^{3} - 3 \, a^{3} b^{12} c^{2} d + 3 \, a^{4} b^{11} c d^{2} - a^{5} b^{10} d^{3}\right )} \sqrt {x} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{3} b^{13}}\right )^{\frac {3}{4}}}{b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}\right ) + 45 \, b^{3} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{3} b^{13}}\right )^{\frac {1}{4}} \log \left (a b^{3} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{3} b^{13}}\right )^{\frac {1}{4}} - {\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}\right ) - 45 \, b^{3} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{3} b^{13}}\right )^{\frac {1}{4}} \log \left (-a b^{3} \left (-\frac {b^{12} c^{12} - 12 \, a b^{11} c^{11} d + 66 \, a^{2} b^{10} c^{10} d^{2} - 220 \, a^{3} b^{9} c^{9} d^{3} + 495 \, a^{4} b^{8} c^{8} d^{4} - 792 \, a^{5} b^{7} c^{7} d^{5} + 924 \, a^{6} b^{6} c^{6} d^{6} - 792 \, a^{7} b^{5} c^{5} d^{7} + 495 \, a^{8} b^{4} c^{4} d^{8} - 220 \, a^{9} b^{3} c^{3} d^{9} + 66 \, a^{10} b^{2} c^{2} d^{10} - 12 \, a^{11} b c d^{11} + a^{12} d^{12}}{a^{3} b^{13}}\right )^{\frac {1}{4}} - {\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}\right ) - 4 \, {\left (5 \, b^{2} d^{3} x^{4} + 135 \, b^{2} c^{2} d - 135 \, a b c d^{2} + 45 \, a^{2} d^{3} + 9 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{2}\right )} \sqrt {x}}{90 \, 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)/x^(1/2),x, algorithm="fricas")

[Out]

-1/90*(180*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*
d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*
d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/4)*arctan((sqrt(a^2*b^6*sqrt(-(b^12*
c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d
^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d
^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13)) + (b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c
^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*x)*a^2*b^10*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^
10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*
b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)
/(a^3*b^13))^(3/4) + (a^2*b^13*c^3 - 3*a^3*b^12*c^2*d + 3*a^4*b^11*c*d^2 - a^5*b^10*d^3)*sqrt(x)*(-(b^12*c^12
- 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 +
924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 -
12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(3/4))/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a
^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a
^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)) + 45*b^3*(-(b^12*
c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d
^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d
^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/4)*log(a*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*
c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5
*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a
^3*b^13))^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 45*b^3*(-(b^12*c^12 - 12*a*b^
11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b
^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*
b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/4)*log(-a*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 22
0*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 49
5*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/
4) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) - 4*(5*b^2*d^3*x^4 + 135*b^2*c^2*d - 135*a*b
*c*d^2 + 45*a^2*d^3 + 9*(3*b^2*c*d^2 - a*b*d^3)*x^2)*sqrt(x))/b^3

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (289) = 578\).
time = 13.44, size = 649, normalized size = 2.13 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 c^{3}}{3 x^{\frac {3}{2}}} + 6 c^{2} d \sqrt {x} + \frac {6 c d^{2} x^{\frac {5}{2}}}{5} + \frac {2 d^{3} x^{\frac {9}{2}}}{9}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 c^{3}}{3 x^{\frac {3}{2}}} + 6 c^{2} d \sqrt {x} + \frac {6 c d^{2} x^{\frac {5}{2}}}{5} + \frac {2 d^{3} x^{\frac {9}{2}}}{9}}{b} & \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} & \text {for}\: b = 0 \\\frac {2 a^{2} d^{3} \sqrt {x}}{b^{3}} + \frac {a^{2} d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {a^{2} d^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {a^{2} d^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{3}} - \frac {6 a c d^{2} \sqrt {x}}{b^{2}} - \frac {3 a c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {3 a c d^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {3 a c d^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b^{2}} - \frac {2 a d^{3} x^{\frac {5}{2}}}{5 b^{2}} + \frac {6 c^{2} d \sqrt {x}}{b} + \frac {3 c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {3 c^{2} d \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 b} - \frac {3 c^{2} d \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{b} + \frac {6 c d^{2} x^{\frac {5}{2}}}{5 b} + \frac {2 d^{3} x^{\frac {9}{2}}}{9 b} - \frac {c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {c^{3} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{2 a} + \frac {c^{3} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{a} & \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)/x**(1/2),x)

[Out]

Piecewise((zoo*(-2*c**3/(3*x**(3/2)) + 6*c**2*d*sqrt(x) + 6*c*d**2*x**(5/2)/5 + 2*d**3*x**(9/2)/9), Eq(a, 0) &
Eq(b, 0)), ((-2*c**3/(3*x**(3/2)) + 6*c**2*d*sqrt(x) + 6*c*d**2*x**(5/2)/5 + 2*d**3*x**(9/2)/9)/b, 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, Eq(b, 0)), (2*a**2*d**
3*sqrt(x)/b**3 + a**2*d**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b**3) - a**2*d**3*(-a/b)**(1/4)*log(s
qrt(x) + (-a/b)**(1/4))/(2*b**3) - a**2*d**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**3 - 6*a*c*d**2*sqrt(
x)/b**2 - 3*a*c*d**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b**2) + 3*a*c*d**2*(-a/b)**(1/4)*log(sqrt(x
) + (-a/b)**(1/4))/(2*b**2) + 3*a*c*d**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b**2 - 2*a*d**3*x**(5/2)/(5
*b**2) + 6*c**2*d*sqrt(x)/b + 3*c**2*d*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*b) - 3*c**2*d*(-a/b)**(1/
4)*log(sqrt(x) + (-a/b)**(1/4))/(2*b) - 3*c**2*d*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/b + 6*c*d**2*x**(5/
2)/(5*b) + 2*d**3*x**(9/2)/(9*b) - c**3*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(2*a) + c**3*(-a/b)**(1/4)*
log(sqrt(x) + (-a/b)**(1/4))/(2*a) + c**3*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/a, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (227) = 454\).
time = 1.30, size = 490, normalized size = 1.61 \begin {gather*} \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \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 )}{2 \, a b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \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 )}{2 \, a b^{4}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \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 )}{4 \, a b^{4}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - \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 )}{4 \, a b^{4}} + \frac {2 \, {\left (5 \, b^{8} d^{3} x^{\frac {9}{2}} + 27 \, b^{8} c d^{2} x^{\frac {5}{2}} - 9 \, a b^{7} d^{3} x^{\frac {5}{2}} + 135 \, b^{8} c^{2} d \sqrt {x} - 135 \, a b^{7} c d^{2} \sqrt {x} + 45 \, a^{2} b^{6} d^{3} \sqrt {x}\right )}}{45 \, b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (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*b^4) + 1/2*sqrt(2)*((a*b^3)^(1/
4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*arctan(-1/2*sq
rt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^4) + 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^
(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x +
sqrt(a/b))/(a*b^4) - 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b
*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^4) + 2/45*(5*b^8*d^3*x^
(9/2) + 27*b^8*c*d^2*x^(5/2) - 9*a*b^7*d^3*x^(5/2) + 135*b^8*c^2*d*sqrt(x) - 135*a*b^7*c*d^2*sqrt(x) + 45*a^2*
b^6*d^3*sqrt(x))/b^9

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Mupad [B]
time = 0.21, size = 1460, normalized size = 4.80 \begin {gather*} \sqrt {x}\,\left (\frac {6\,c^2\,d}{b}+\frac {a\,\left (\frac {2\,a\,d^3}{b^2}-\frac {6\,c\,d^2}{b}\right )}{b}\right )-x^{5/2}\,\left (\frac {2\,a\,d^3}{5\,b^2}-\frac {6\,c\,d^2}{5\,b}\right )+\frac {2\,d^3\,x^{9/2}}{9\,b}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {8\,\sqrt {x}\,\left (a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{b^3}-\frac {{\left (a\,d-b\,c\right )}^3\,\left (16\,a^4\,d^3-48\,a^3\,b\,c\,d^2+48\,a^2\,b^2\,c^2\,d-16\,a\,b^3\,c^3\right )}{2\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{3/4}\,b^{13/4}}+\frac {\left (\frac {8\,\sqrt {x}\,\left (a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{b^3}+\frac {{\left (a\,d-b\,c\right )}^3\,\left (16\,a^4\,d^3-48\,a^3\,b\,c\,d^2+48\,a^2\,b^2\,c^2\,d-16\,a\,b^3\,c^3\right )}{2\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{3/4}\,b^{13/4}}}{\frac {\left (\frac {8\,\sqrt {x}\,\left (a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{b^3}-\frac {{\left (a\,d-b\,c\right )}^3\,\left (16\,a^4\,d^3-48\,a^3\,b\,c\,d^2+48\,a^2\,b^2\,c^2\,d-16\,a\,b^3\,c^3\right )}{2\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{3/4}\,b^{13/4}}-\frac {\left (\frac {8\,\sqrt {x}\,\left (a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{b^3}+\frac {{\left (a\,d-b\,c\right )}^3\,\left (16\,a^4\,d^3-48\,a^3\,b\,c\,d^2+48\,a^2\,b^2\,c^2\,d-16\,a\,b^3\,c^3\right )}{2\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{3/4}\,b^{13/4}}}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{3/4}\,b^{13/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {8\,\sqrt {x}\,\left (a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{b^3}-\frac {{\left (a\,d-b\,c\right )}^3\,\left (16\,a^4\,d^3-48\,a^3\,b\,c\,d^2+48\,a^2\,b^2\,c^2\,d-16\,a\,b^3\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{3/4}\,b^{13/4}}+\frac {\left (\frac {8\,\sqrt {x}\,\left (a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{b^3}+\frac {{\left (a\,d-b\,c\right )}^3\,\left (16\,a^4\,d^3-48\,a^3\,b\,c\,d^2+48\,a^2\,b^2\,c^2\,d-16\,a\,b^3\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{3/4}\,b^{13/4}}}{\frac {\left (\frac {8\,\sqrt {x}\,\left (a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{b^3}-\frac {{\left (a\,d-b\,c\right )}^3\,\left (16\,a^4\,d^3-48\,a^3\,b\,c\,d^2+48\,a^2\,b^2\,c^2\,d-16\,a\,b^3\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{3/4}\,b^{13/4}}-\frac {\left (\frac {8\,\sqrt {x}\,\left (a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{b^3}+\frac {{\left (a\,d-b\,c\right )}^3\,\left (16\,a^4\,d^3-48\,a^3\,b\,c\,d^2+48\,a^2\,b^2\,c^2\,d-16\,a\,b^3\,c^3\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{3/4}\,b^{13/4}}\right )\,{\left (a\,d-b\,c\right )}^3\,1{}\mathrm {i}}{{\left (-a\right )}^{3/4}\,b^{13/4}}}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (-a\right )}^{3/4}\,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)),x)

[Out]

x^(1/2)*((6*c^2*d)/b + (a*((2*a*d^3)/b^2 - (6*c*d^2)/b))/b) - x^(5/2)*((2*a*d^3)/(5*b^2) - (6*c*d^2)/(5*b)) +
(2*d^3*x^(9/2))/(9*b) - (atan(((((8*x^(1/2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*
a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))/b^3 - ((a*d - b*c)^3*(16*a^4*d^3 - 16*a*b^3*c^3 + 48*a^2*b^2
*c^2*d - 48*a^3*b*c*d^2))/(2*(-a)^(3/4)*b^(13/4)))*(a*d - b*c)^3*1i)/((-a)^(3/4)*b^(13/4)) + (((8*x^(1/2)*(a^6
*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)
)/b^3 + ((a*d - b*c)^3*(16*a^4*d^3 - 16*a*b^3*c^3 + 48*a^2*b^2*c^2*d - 48*a^3*b*c*d^2))/(2*(-a)^(3/4)*b^(13/4)
))*(a*d - b*c)^3*1i)/((-a)^(3/4)*b^(13/4)))/((((8*x^(1/2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3
*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))/b^3 - ((a*d - b*c)^3*(16*a^4*d^3 - 16*a*b^3*c^
3 + 48*a^2*b^2*c^2*d - 48*a^3*b*c*d^2))/(2*(-a)^(3/4)*b^(13/4)))*(a*d - b*c)^3)/((-a)^(3/4)*b^(13/4)) - (((8*x
^(1/2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a
^5*b*c*d^5))/b^3 + ((a*d - b*c)^3*(16*a^4*d^3 - 16*a*b^3*c^3 + 48*a^2*b^2*c^2*d - 48*a^3*b*c*d^2))/(2*(-a)^(3/
4)*b^(13/4)))*(a*d - b*c)^3)/((-a)^(3/4)*b^(13/4))))*(a*d - b*c)^3*1i)/((-a)^(3/4)*b^(13/4)) - (atan(((((8*x^(
1/2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5
*b*c*d^5))/b^3 - ((a*d - b*c)^3*(16*a^4*d^3 - 16*a*b^3*c^3 + 48*a^2*b^2*c^2*d - 48*a^3*b*c*d^2)*1i)/(2*(-a)^(3
/4)*b^(13/4)))*(a*d - b*c)^3)/((-a)^(3/4)*b^(13/4)) + (((8*x^(1/2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 2
0*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))/b^3 + ((a*d - b*c)^3*(16*a^4*d^3 - 16
*a*b^3*c^3 + 48*a^2*b^2*c^2*d - 48*a^3*b*c*d^2)*1i)/(2*(-a)^(3/4)*b^(13/4)))*(a*d - b*c)^3)/((-a)^(3/4)*b^(13/
4)))/((((8*x^(1/2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5
*c^5*d - 6*a^5*b*c*d^5))/b^3 - ((a*d - b*c)^3*(16*a^4*d^3 - 16*a*b^3*c^3 + 48*a^2*b^2*c^2*d - 48*a^3*b*c*d^2)*
1i)/(2*(-a)^(3/4)*b^(13/4)))*(a*d - b*c)^3*1i)/((-a)^(3/4)*b^(13/4)) - (((8*x^(1/2)*(a^6*d^6 + b^6*c^6 + 15*a^
2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5))/b^3 + ((a*d - b*c)^3
*(16*a^4*d^3 - 16*a*b^3*c^3 + 48*a^2*b^2*c^2*d - 48*a^3*b*c*d^2)*1i)/(2*(-a)^(3/4)*b^(13/4)))*(a*d - b*c)^3*1i
)/((-a)^(3/4)*b^(13/4))))*(a*d - b*c)^3)/((-a)^(3/4)*b^(13/4))

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