Optimal. Leaf size=215 \[ \frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}-\frac{7 A b-4 a B}{12 a^2 b x^4}+\frac{7 A b-4 a B}{3 a^3 x}-\frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}-\frac{\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{10/3}}+\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.126611, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {457, 325, 292, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}-\frac{7 A b-4 a B}{12 a^2 b x^4}+\frac{7 A b-4 a B}{3 a^3 x}-\frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}-\frac{\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{10/3}}+\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 457
Rule 325
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^5 \left (a+b x^3\right )^2} \, dx &=\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )}+\frac{(7 A b-4 a B) \int \frac{1}{x^5 \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac{7 A b-4 a B}{12 a^2 b x^4}+\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac{(7 A b-4 a B) \int \frac{1}{x^2 \left (a+b x^3\right )} \, dx}{3 a^2}\\ &=-\frac{7 A b-4 a B}{12 a^2 b x^4}+\frac{7 A b-4 a B}{3 a^3 x}+\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )}+\frac{(b (7 A b-4 a B)) \int \frac{x}{a+b x^3} \, dx}{3 a^3}\\ &=-\frac{7 A b-4 a B}{12 a^2 b x^4}+\frac{7 A b-4 a B}{3 a^3 x}+\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac{\left (b^{2/3} (7 A b-4 a B)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{10/3}}+\frac{\left (b^{2/3} (7 A b-4 a B)\right ) \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{10/3}}\\ &=-\frac{7 A b-4 a B}{12 a^2 b x^4}+\frac{7 A b-4 a B}{3 a^3 x}+\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}+\frac{\left (\sqrt [3]{b} (7 A b-4 a B)\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{10/3}}+\frac{\left (b^{2/3} (7 A b-4 a B)\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^3}\\ &=-\frac{7 A b-4 a B}{12 a^2 b x^4}+\frac{7 A b-4 a B}{3 a^3 x}+\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}+\frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}+\frac{\left (\sqrt [3]{b} (7 A b-4 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{10/3}}\\ &=-\frac{7 A b-4 a B}{12 a^2 b x^4}+\frac{7 A b-4 a B}{3 a^3 x}+\frac{A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac{\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{10/3}}-\frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}+\frac{\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}\\ \end{align*}
Mathematica [A] time = 0.136187, size = 185, normalized size = 0.86 \[ \frac{2 \sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{9 a^{4/3} A}{x^4}-\frac{12 \sqrt [3]{a} b x^2 (a B-A b)}{a+b x^3}-\frac{36 \sqrt [3]{a} (a B-2 A b)}{x}+4 \sqrt [3]{b} (4 a B-7 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt{3} \sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{36 a^{10/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 257, normalized size = 1.2 \begin{align*}{\frac{A{x}^{2}{b}^{2}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{bB{x}^{2}}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{7\,Ab}{9\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,Ab}{18\,{a}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,Ab\sqrt{3}}{9\,{a}^{3}}\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{4\,B}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,B}{9\,{a}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{4\,B\sqrt{3}}{9\,{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{A}{4\,{a}^{2}{x}^{4}}}+2\,{\frac{Ab}{{a}^{3}x}}-{\frac{B}{{a}^{2}x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81541, size = 585, normalized size = 2.72 \begin{align*} -\frac{12 \,{\left (4 \, B a b - 7 \, A b^{2}\right )} x^{6} + 9 \,{\left (4 \, B a^{2} - 7 \, A a b\right )} x^{3} + 9 \, A a^{2} + 4 \, \sqrt{3}{\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} +{\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 2 \,{\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} +{\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \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 \,{\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} +{\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{36 \,{\left (a^{3} b x^{7} + a^{4} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.24492, size = 153, normalized size = 0.71 \begin{align*} \operatorname{RootSum}{\left (729 t^{3} a^{10} + 343 A^{3} b^{4} - 588 A^{2} B a b^{3} + 336 A B^{2} a^{2} b^{2} - 64 B^{3} a^{3} b, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{7}}{49 A^{2} b^{3} - 56 A B a b^{2} + 16 B^{2} a^{2} b} + x \right )} \right )\right )} - \frac{3 A a^{2} + x^{6} \left (- 28 A b^{2} + 16 B a b\right ) + x^{3} \left (- 21 A a b + 12 B a^{2}\right )}{12 a^{4} x^{4} + 12 a^{3} b x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.65287, size = 312, normalized size = 1.45 \begin{align*} \frac{{\left (4 \, B a b \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 7 \, A b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{4}} + \frac{\sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 7 \, \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 )}{9 \, a^{4} b} - \frac{B a b x^{2} - A b^{2} x^{2}}{3 \,{\left (b x^{3} + a\right )} a^{3}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{4} b} - \frac{4 \, B a x^{3} - 8 \, A b x^{3} + A a}{4 \, a^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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