Optimal. Leaf size=196 \[ \frac{(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{5 A b-2 a B}{6 a^2 b x^2}-\frac{(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{(5 A b-2 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{A b-a B}{3 a b x^2 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.102756, antiderivative size = 196, 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, Rules used = {457, 325, 200, 31, 634, 617, 204, 628} \[ \frac{(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{5 A b-2 a B}{6 a^2 b x^2}-\frac{(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{(5 A b-2 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3} \sqrt [3]{b}}+\frac{A b-a B}{3 a b x^2 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 457
Rule 325
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^3 \left (a+b x^3\right )^2} \, dx &=\frac{A b-a B}{3 a b x^2 \left (a+b x^3\right )}+\frac{(5 A b-2 a B) \int \frac{1}{x^3 \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac{5 A b-2 a B}{6 a^2 b x^2}+\frac{A b-a B}{3 a b x^2 \left (a+b x^3\right )}-\frac{(5 A b-2 a B) \int \frac{1}{a+b x^3} \, dx}{3 a^2}\\ &=-\frac{5 A b-2 a B}{6 a^2 b x^2}+\frac{A b-a B}{3 a b x^2 \left (a+b x^3\right )}-\frac{(5 A b-2 a B) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{8/3}}-\frac{(5 A b-2 a B) \int \frac{2 \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^{8/3}}\\ &=-\frac{5 A b-2 a B}{6 a^2 b x^2}+\frac{A b-a B}{3 a b x^2 \left (a+b x^3\right )}-\frac{(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}-\frac{(5 A b-2 a B) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{7/3}}+\frac{(5 A b-2 a B) \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^{8/3} \sqrt [3]{b}}\\ &=-\frac{5 A b-2 a B}{6 a^2 b x^2}+\frac{A b-a B}{3 a b x^2 \left (a+b x^3\right )}-\frac{(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{(5 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{8/3} \sqrt [3]{b}}\\ &=-\frac{5 A b-2 a B}{6 a^2 b x^2}+\frac{A b-a B}{3 a b x^2 \left (a+b x^3\right )}+\frac{(5 A b-2 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3} \sqrt [3]{b}}-\frac{(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.12531, size = 163, normalized size = 0.83 \[ \frac{\frac{(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac{6 a^{2/3} x (a B-A b)}{a+b x^3}-\frac{9 a^{2/3} A}{x^2}+\frac{2 (2 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{2 \sqrt{3} (5 A b-2 a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{18 a^{8/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 237, normalized size = 1.2 \begin{align*} -{\frac{Abx}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{xB}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{5\,A}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,A}{18\,{a}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,A\sqrt{3}}{9\,{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{2\,B}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{9\,ab}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,B\sqrt{3}}{9\,ab}\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}^{2}{x}^{2}}} \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 = 1.77428, size = 1396, normalized size = 7.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.944601, size = 109, normalized size = 0.56 \begin{align*} \frac{- 3 A a + x^{3} \left (- 5 A b + 2 B a\right )}{6 a^{3} x^{2} + 6 a^{2} b x^{5}} + \operatorname{RootSum}{\left (729 t^{3} a^{8} b + 125 A^{3} b^{3} - 150 A^{2} B a b^{2} + 60 A B^{2} a^{2} b - 8 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{9 t a^{3}}{- 5 A b + 2 B a} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14669, size = 254, normalized size = 1.3 \begin{align*} -\frac{{\left (2 \, B a - 5 \, A b\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^{3}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 5 \, \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 )}{9 \, a^{3} b} + \frac{B a x - A b x}{3 \,{\left (b x^{3} + a\right )} a^{2}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{1}{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^{3} b} - \frac{A}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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