Optimal. Leaf size=288 \[ -\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} (b c-a d)}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} (b c-a d)}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} (b c-a d)}+\frac{d^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)}-\frac{d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)}+\frac{d^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.348858, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} (b c-a d)}+\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} (b c-a d)}-\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} (b c-a d)}+\frac{d^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} (b c-a d)}-\frac{d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} (b c-a d)}+\frac{d^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^3)*(c + d*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 66.5821, size = 260, normalized size = 0.9 \[ \frac{d^{\frac{2}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{2}{3}} \left (a d - b c\right )} - \frac{d^{\frac{2}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{2}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} d^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{2}{3}} \left (a d - b c\right )} - \frac{b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} \left (a d - b c\right )} + \frac{b^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**3+a)/(d*x**3+c),x)
[Out]
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Mathematica [A] time = 0.222932, size = 224, normalized size = 0.78 \[ \frac{\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}-\frac{2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{d^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{2/3}}+\frac{2 d^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{2/3}}-\frac{2 \sqrt{3} d^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{2/3}}}{6 a d-6 b c} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^3)*(c + d*x^3)),x]
[Out]
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Maple [A] time = 0.002, size = 222, normalized size = 0.8 \[ -{\frac{1}{3\,ad-3\,bc}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{6\,ad-6\,bc}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}}{3\,ad-3\,bc}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{3\,ad-3\,bc}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,ad-6\,bc}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,ad-3\,bc}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^3+a)/(d*x^3+c),x)
[Out]
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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(1/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.302653, size = 377, normalized size = 1.31 \[ \frac{\sqrt{3}{\left (\sqrt{3} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) + \sqrt{3} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} + c^{2} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d x + c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}\right ) + 6 \, \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 6 \, \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} d x - \sqrt{3} c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}{3 \, c \left (\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}\right )\right )}}{18 \,{\left (b c - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 105.04, size = 447, normalized size = 1.55 \[ \operatorname{RootSum}{\left (t^{3} \left (27 a^{5} d^{3} - 81 a^{4} b c d^{2} + 81 a^{3} b^{2} c^{2} d - 27 a^{2} b^{3} c^{3}\right ) + b^{2}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{4} a^{7} c^{2} d^{5} - 243 t^{4} a^{6} b c^{3} d^{4} + 162 t^{4} a^{5} b^{2} c^{4} d^{3} + 162 t^{4} a^{4} b^{3} c^{5} d^{2} - 243 t^{4} a^{3} b^{4} c^{6} d + 81 t^{4} a^{2} b^{5} c^{7} - 3 t a^{4} d^{4} + 3 t a^{3} b c d^{3} + 3 t a b^{3} c^{3} d - 3 t b^{4} c^{4}}{a^{2} b d^{3} + b^{3} c^{2} d} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} c^{2} d^{3} - 81 a^{2} b c^{3} d^{2} + 81 a b^{2} c^{4} d - 27 b^{3} c^{5}\right ) - d^{2}, \left ( t \mapsto t \log{\left (x + \frac{81 t^{4} a^{7} c^{2} d^{5} - 243 t^{4} a^{6} b c^{3} d^{4} + 162 t^{4} a^{5} b^{2} c^{4} d^{3} + 162 t^{4} a^{4} b^{3} c^{5} d^{2} - 243 t^{4} a^{3} b^{4} c^{6} d + 81 t^{4} a^{2} b^{5} c^{7} - 3 t a^{4} d^{4} + 3 t a^{3} b c d^{3} + 3 t a b^{3} c^{3} d - 3 t b^{4} c^{4}}{a^{2} b d^{3} + b^{3} c^{2} d} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**3+a)/(d*x**3+c),x)
[Out]
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GIAC/XCAS [A] time = 0.226518, size = 375, normalized size = 1.3 \[ -\frac{b \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b c - a^{2} d\right )}} + \frac{d \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} - a c d\right )}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a b c - \sqrt{3} a^{2} d} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c^{2} - \sqrt{3} a c d} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a b c - a^{2} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{2} - a c d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^3 + a)*(d*x^3 + c)),x, algorithm="giac")
[Out]