Optimal. Leaf size=149 \[ \frac{(A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt [3]{b}}-\frac{(A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt [3]{b}}+\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} \sqrt [3]{b}}-\frac{A}{2 a x^2} \]
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Rubi [A] time = 0.237478, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{(A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt [3]{b}}-\frac{(A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt [3]{b}}+\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} \sqrt [3]{b}}-\frac{A}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^3)/(x^3*(a + b*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 33.2997, size = 138, normalized size = 0.93 \[ - \frac{A}{2 a x^{2}} - \frac{\left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} \sqrt [3]{b}} + \frac{\left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} \sqrt [3]{b}} + \frac{\sqrt{3} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} \sqrt [3]{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)/x**3/(b*x**3+a),x)
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Mathematica [A] time = 0.222422, size = 135, normalized size = 0.91 \[ \frac{\frac{(A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-\frac{3 a^{2/3} A}{x^2}+\frac{2 (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{2 \sqrt{3} (A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{6 a^{5/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^3)/(x^3*(a + b*x^3)),x]
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Maple [A] time = 0.005, size = 195, normalized size = 1.3 \[ -{\frac{A}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{A}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}A}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}B}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A}{2\,a{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)/x^3/(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^3),x, algorithm="maxima")
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Fricas [A] time = 0.252096, size = 198, normalized size = 1.33 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left (B a - A b\right )} x^{2} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left (B a - A b\right )} x^{2} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 6 \,{\left (B a - A b\right )} x^{2} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} A\right )}}{18 \, \left (-a^{2} b\right )^{\frac{1}{3}} a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)*x^3),x, algorithm="fricas")
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Sympy [A] time = 2.32981, size = 73, normalized size = 0.49 \[ - \frac{A}{2 a x^{2}} + \operatorname{RootSum}{\left (27 t^{3} a^{5} b + A^{3} b^{3} - 3 A^{2} B a b^{2} + 3 A B^{2} a^{2} b - B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{3 t a^{2}}{- A b + B a} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)/x**3/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.219927, size = 217, normalized size = 1.46 \[ -\frac{{\left (B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b} - \frac{A}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)*x^3),x, algorithm="giac")
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