Optimal. Leaf size=264 \[ -\frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} \left (b c^3-a d^3\right )}-\frac{\sqrt [3]{a} d \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{2/3} \left (a^{2/3} d^2+\sqrt [3]{a} \sqrt [3]{b} c d+b^{2/3} c^2\right )}+\frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}+\frac{c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )}-\frac{c^2 \log (c+d x)}{b c^3-a d^3} \]
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Rubi [A] time = 0.960649, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} \left (b c^3-a d^3\right )}-\frac{\sqrt [3]{a} d \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{2/3} \left (a^{2/3} d^2+\sqrt [3]{a} \sqrt [3]{b} c d+b^{2/3} c^2\right )}+\frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}+\frac{c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )}-\frac{c^2 \log (c+d x)}{b c^3-a d^3} \]
Antiderivative was successfully verified.
[In] Int[x^2/((c + d*x)*(a + b*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 113.591, size = 231, normalized size = 0.88 \[ - \frac{\sqrt{3} \sqrt [3]{a} d \left (\sqrt [3]{a} d - \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{2}{3}} \left (a d^{3} - b c^{3}\right )} - \frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d + \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{2}{3}} \left (a d^{3} - b c^{3}\right )} + \frac{\sqrt [3]{a} d \left (\sqrt [3]{a} d + \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{2}{3}} \left (a d^{3} - b c^{3}\right )} - \frac{c^{2} \log{\left (a + b x^{3} \right )}}{3 \left (a d^{3} - b c^{3}\right )} + \frac{c^{2} \log{\left (c + d x \right )}}{a d^{3} - b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(d*x+c)/(b*x**3+a),x)
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Mathematica [A] time = 0.169607, size = 228, normalized size = 0.86 \[ \frac{-\sqrt [3]{a} \sqrt [3]{b} c d \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-a^{2/3} d^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 b^{2/3} c^2 \log \left (a+b x^3\right )+2 \sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} d \left (\sqrt [3]{a} d-\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-6 b^{2/3} c^2 \log (c+d x)}{6 b^{2/3} \left (b c^3-a d^3\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((c + d*x)*(a + b*x^3)),x]
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Maple [A] time = 0.008, size = 336, normalized size = 1.3 \[ -{\frac{acd}{ \left ( 3\,a{d}^{3}-3\,b{c}^{3} \right ) b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{acd}{ \left ( 6\,a{d}^{3}-6\,b{c}^{3} \right ) b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{acd\sqrt{3}}{ \left ( 3\,a{d}^{3}-3\,b{c}^{3} \right ) b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a{d}^{2}}{ \left ( 3\,a{d}^{3}-3\,b{c}^{3} \right ) b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{a{d}^{2}}{ \left ( 6\,a{d}^{3}-6\,b{c}^{3} \right ) b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{a{d}^{2}\sqrt{3}}{ \left ( 3\,a{d}^{3}-3\,b{c}^{3} \right ) b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{c}^{2}\ln \left ( b{x}^{3}+a \right ) }{3\,a{d}^{3}-3\,b{c}^{3}}}+{\frac{{c}^{2}\ln \left ( dx+c \right ) }{a{d}^{3}-b{c}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(d*x+c)/(b*x^3+a),x)
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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(x^2/((b*x^3 + a)*(d*x + c)),x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^3 + a)*(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(d*x+c)/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.270792, size = 432, normalized size = 1.64 \[ -\frac{c^{2} d{\rm ln}\left ({\left | d x + c \right |}\right )}{b c^{3} d - a d^{4}} + \frac{c^{2}{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \,{\left (b c^{3} - a d^{3}\right )}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}} d \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{2} c^{2} - \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} b c d + \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} d^{2}} + \frac{{\left (a b^{2} c^{3} d^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{2} b d^{5} \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{2} c^{4} d + a^{2} b c d^{4}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b^{3} c^{6} - 2 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b d^{6}\right )}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b c d - \left (-a b^{2}\right )^{\frac{2}{3}} d^{2}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b^{3} c^{3} - a b^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x^3 + a)*(d*x + c)),x, algorithm="giac")
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