Optimal. Leaf size=288 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{b} (b c-a d)}-\frac{\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{d} (b c-a d)}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{b} (b c-a d)}+\frac{\sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{d} (b c-a d)}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{b} (b c-a d)}-\frac{\sqrt [3]{c} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} \sqrt [3]{d} (b c-a d)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.382143, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{b} (b c-a d)}-\frac{\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{d} (b c-a d)}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{b} (b c-a d)}+\frac{\sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{d} (b c-a d)}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{b} (b c-a d)}-\frac{\sqrt [3]{c} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} \sqrt [3]{d} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x^3)*(c + d*x^3)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 70.1417, size = 260, normalized size = 0.9 \[ \frac{\sqrt [3]{a} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 \sqrt [3]{b} \left (a d - b c\right )} - \frac{\sqrt [3]{a} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{b} \left (a d - b c\right )} - \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{b} \left (a d - b c\right )} - \frac{\sqrt [3]{c} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 \sqrt [3]{d} \left (a d - b c\right )} + \frac{\sqrt [3]{c} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{d} \left (a d - b c\right )} + \frac{\sqrt{3} \sqrt [3]{c} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 \sqrt [3]{d} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**3+a)/(d*x**3+c),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.186623, size = 224, normalized size = 0.78 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}-\frac{\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt [3]{d}}-\frac{2 \sqrt{3} \sqrt [3]{c} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{\sqrt [3]{d}}}{6 b c-6 a d} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((a + b*x^3)*(c + d*x^3)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 246, normalized size = 0.9 \[{\frac{a}{ \left ( 3\,ad-3\,bc \right ) b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a}{ \left ( 6\,ad-6\,bc \right ) b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ) b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{ \left ( 3\,ad-3\,bc \right ) d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{c}{ \left ( 6\,ad-6\,bc \right ) d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ) d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x^3+a)/(d*x^3+c),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.24978, size = 284, normalized size = 0.99 \[ \frac{\sqrt{3}{\left (\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) + \sqrt{3} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right ) + 6 \, \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 6 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{c}{d}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )\right )}}{18 \,{\left (b c - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 38.6851, size = 342, normalized size = 1.19 \[ \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} d^{4} - 81 a^{2} b c d^{3} + 81 a b^{2} c^{2} d^{2} - 27 b^{3} c^{3} d\right ) + c, \left ( t \mapsto t \log{\left (x + \frac{162 t^{4} a^{4} b d^{5} - 648 t^{4} a^{3} b^{2} c d^{4} + 972 t^{4} a^{2} b^{3} c^{2} d^{3} - 648 t^{4} a b^{4} c^{3} d^{2} + 162 t^{4} b^{5} c^{4} d - 3 t a^{2} d^{2} + 6 t a b c d - 3 t b^{2} c^{2}}{a d + b c} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} b d^{3} - 81 a^{2} b^{2} c d^{2} + 81 a b^{3} c^{2} d - 27 b^{4} c^{3}\right ) - a, \left ( t \mapsto t \log{\left (x + \frac{162 t^{4} a^{4} b d^{5} - 648 t^{4} a^{3} b^{2} c d^{4} + 972 t^{4} a^{2} b^{3} c^{2} d^{3} - 648 t^{4} a b^{4} c^{3} d^{2} + 162 t^{4} b^{5} c^{4} d - 3 t a^{2} d^{2} + 6 t a b c d - 3 t b^{2} c^{2}}{a d + b c} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x**3+a)/(d*x**3+c),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.227858, size = 375, normalized size = 1.3 \[ \frac{a \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b c - a^{2} d\right )}} - \frac{c \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} - a c d\right )}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{2} c - \sqrt{3} a b d} + \frac{\left (-c d^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c d - \sqrt{3} a d^{2}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b^{2} c - a b d\right )}} + \frac{\left (-c d^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c d - a d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="giac")
[Out]