Optimal. Leaf size=168 \[ -\frac{b^{2/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3}}+\frac{b^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3}}-\frac{b^{2/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3}}+\frac{A b-a B}{2 a^2 x^2}-\frac{A}{5 a x^5} \]
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Rubi [A] time = 0.28673, antiderivative size = 168, 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 \[ -\frac{b^{2/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3}}+\frac{b^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3}}-\frac{b^{2/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3}}+\frac{A b-a B}{2 a^2 x^2}-\frac{A}{5 a x^5} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^3)/(x^6*(a + b*x^3)),x]
[Out]
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Rubi in Sympy [A] time = 38.7188, size = 153, normalized size = 0.91 \[ - \frac{A}{5 a x^{5}} + \frac{A b - B a}{2 a^{2} x^{2}} + \frac{b^{\frac{2}{3}} \left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{8}{3}}} - \frac{b^{\frac{2}{3}} \left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{8}{3}}} - \frac{\sqrt{3} b^{\frac{2}{3}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{8}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)/x**6/(b*x**3+a),x)
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Mathematica [A] time = 0.2404, size = 154, normalized size = 0.92 \[ \frac{5 b^{2/3} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{15 a^{2/3} (A b-a B)}{x^2}-\frac{6 a^{5/3} A}{x^5}+10 b^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt{3} b^{2/3} (A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{30 a^{8/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^3)/(x^6*(a + b*x^3)),x]
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Maple [A] time = 0.011, size = 217, normalized size = 1.3 \[ -{\frac{A}{5\,a{x}^{5}}}+{\frac{Ab}{2\,{a}^{2}{x}^{2}}}-{\frac{B}{2\,a{x}^{2}}}+{\frac{Ab}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{Ab}{6\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b\sqrt{3}A}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}B}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)/x^6/(b*x^3+a),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((B*x^3 + A)/((b*x^3 + a)*x^6),x, algorithm="maxima")
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Fricas [A] time = 0.236858, size = 266, normalized size = 1.58 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (B a - A b\right )} x^{5} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 10 \, \sqrt{3}{\left (B a - A b\right )} x^{5} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 30 \,{\left (B a - A b\right )} x^{5} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (5 \,{\left (B a - A b\right )} x^{3} + 2 \, A a\right )}\right )}}{90 \, a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)*x^6),x, algorithm="fricas")
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Sympy [A] time = 3.18139, size = 99, normalized size = 0.59 \[ \operatorname{RootSum}{\left (27 t^{3} a^{8} - A^{3} b^{5} + 3 A^{2} B a b^{4} - 3 A B^{2} a^{2} b^{3} + B^{3} a^{3} b^{2}, \left ( t \mapsto t \log{\left (- \frac{3 t a^{3}}{- A b^{2} + B a b} + x \right )} \right )\right )} - \frac{2 A a + x^{3} \left (- 5 A b + 5 B a\right )}{10 a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)/x**6/(b*x**3+a),x)
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GIAC/XCAS [A] time = 0.216897, size = 238, normalized size = 1.42 \[ -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3}} + \frac{{\left (B a b - A b^{2}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{3}} - \frac{5 \, B a x^{3} - 5 \, A b x^{3} + 2 \, A a}{10 \, a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)*x^6),x, algorithm="giac")
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