Optimal. Leaf size=165 \[ \frac{\sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3}}-\frac{\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}-\frac{\sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}+\frac{A b-a B}{a^2 x}-\frac{A}{4 a x^4} \]
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Rubi [A] time = 0.286374, antiderivative size = 165, 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 \[ \frac{\sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3}}-\frac{\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}-\frac{\sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}+\frac{A b-a B}{a^2 x}-\frac{A}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^3)/(x^5*(a + b*x^3)),x]
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Rubi in Sympy [A] time = 37.821, size = 150, normalized size = 0.91 \[ - \frac{A}{4 a x^{4}} + \frac{A b - B a}{a^{2} x} - \frac{\sqrt [3]{b} \left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{7}{3}}} + \frac{\sqrt [3]{b} \left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{7}{3}}} - \frac{\sqrt{3} \sqrt [3]{b} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{7}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)/x**5/(b*x**3+a),x)
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Mathematica [A] time = 0.233049, size = 154, normalized size = 0.93 \[ \frac{2 \sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{3 a^{4/3} A}{x^4}+\frac{12 \sqrt [3]{a} (A b-a B)}{x}+4 \sqrt [3]{b} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt{3} \sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{12 a^{7/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^3)/(x^5*(a + b*x^3)),x]
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Maple [A] time = 0.009, size = 216, normalized size = 1.3 \[ -{\frac{A}{4\,a{x}^{4}}}+{\frac{Ab}{x{a}^{2}}}-{\frac{B}{ax}}-{\frac{Ab}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{Ab}{6\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{3}A}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}B}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)/x^5/(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)/((b*x^3 + a)*x^5),x, algorithm="maxima")
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Fricas [A] time = 0.230054, size = 248, normalized size = 1.5 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (B a - A b\right )} x^{4} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 4 \, \sqrt{3}{\left (B a - A b\right )} x^{4} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 12 \,{\left (B a - A b\right )} x^{4} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b}{a}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (4 \,{\left (B a - A b\right )} x^{3} + A a\right )}\right )}}{36 \, a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)*x^5),x, algorithm="fricas")
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Sympy [A] time = 2.64998, size = 112, normalized size = 0.68 \[ \operatorname{RootSum}{\left (27 t^{3} a^{7} + A^{3} b^{4} - 3 A^{2} B a b^{3} + 3 A B^{2} a^{2} b^{2} - B^{3} a^{3} b, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{5}}{A^{2} b^{3} - 2 A B a b^{2} + B^{2} a^{2} b} + x \right )} \right )\right )} - \frac{A a + x^{3} \left (- 4 A b + 4 B a\right )}{4 a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)/x**5/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.221523, size = 266, normalized size = 1.61 \[ \frac{{\left (B a b \left (-\frac{a}{b}\right )^{\frac{1}{3}} - A b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3} b} - \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{3} b} - \frac{4 \, B a x^{3} - 4 \, A b x^{3} + A a}{4 \, a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)*x^5),x, algorithm="giac")
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