Optimal. Leaf size=162 \[ \frac{\sqrt [3]{a} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{7/3}}-\frac{\sqrt [3]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac{\sqrt [3]{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{7/3}}+\frac{x (A b-a B)}{b^2}+\frac{B x^4}{4 b} \]
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Rubi [A] time = 0.289126, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{\sqrt [3]{a} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{7/3}}-\frac{\sqrt [3]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac{\sqrt [3]{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{7/3}}+\frac{x (A b-a B)}{b^2}+\frac{B x^4}{4 b} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(A + B*x^3))/(a + b*x^3),x]
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Rubi in Sympy [A] time = 40.1321, size = 150, normalized size = 0.93 \[ \frac{B x^{4}}{4 b} - \frac{\sqrt [3]{a} \left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{7}{3}}} + \frac{\sqrt [3]{a} \left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{7}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{7}{3}}} + \frac{x \left (A b - B a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x**3+A)/(b*x**3+a),x)
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Mathematica [A] time = 0.150298, size = 152, normalized size = 0.94 \[ \frac{-2 \sqrt [3]{a} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+12 \sqrt [3]{b} x (A b-a B)+4 \sqrt [3]{a} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt{3} \sqrt [3]{a} (a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+3 b^{4/3} B x^4}{12 b^{7/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(A + B*x^3))/(a + b*x^3),x]
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Maple [A] time = 0.003, size = 221, normalized size = 1.4 \[{\frac{B{x}^{4}}{4\,b}}+{\frac{Ax}{b}}-{\frac{Bxa}{{b}^{2}}}-{\frac{aA}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}B}{3\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{aA}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}B}{6\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a\sqrt{3}A}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{a}^{2}\sqrt{3}B}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x^3+A)/(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((B*x^3 + A)*x^3/(b*x^3 + a),x, algorithm="maxima")
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Fricas [A] time = 0.236106, size = 213, normalized size = 1.31 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 4 \, \sqrt{3}{\left (B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 12 \,{\left (B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (B b x^{4} - 4 \,{\left (B a - A b\right )} x\right )}\right )}}{36 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^3/(b*x^3 + a),x, algorithm="fricas")
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Sympy [A] time = 2.26818, size = 87, normalized size = 0.54 \[ \frac{B x^{4}}{4 b} + \operatorname{RootSum}{\left (27 t^{3} b^{7} + A^{3} a b^{3} - 3 A^{2} B a^{2} b^{2} + 3 A B^{2} a^{3} b - B^{3} a^{4}, \left ( t \mapsto t \log{\left (\frac{3 t b^{2}}{- A b + B a} + x \right )} \right )\right )} - \frac{x \left (- A b + B a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x**3+A)/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.216273, size = 251, normalized size = 1.55 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{3}} - \frac{{\left (B a^{2} b^{2} - A a b^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{4}} + \frac{B b^{3} x^{4} - 4 \, B a b^{2} x + 4 \, A b^{3} x}{4 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^3/(b*x^3 + a),x, algorithm="giac")
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