Optimal. Leaf size=190 \[ -\frac{(A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{7/3}}+\frac{(A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{(A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{x (A b-4 a B)}{3 a b^2}+\frac{x^4 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.106447, antiderivative size = 190, 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, 321, 200, 31, 634, 617, 204, 628} \[ -\frac{(A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{7/3}}+\frac{(A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{(A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{x (A b-4 a B)}{3 a b^2}+\frac{x^4 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 457
Rule 321
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac{(A b-a B) x^4}{3 a b \left (a+b x^3\right )}+\frac{(-A b+4 a B) \int \frac{x^3}{a+b x^3} \, dx}{3 a b}\\ &=-\frac{(A b-4 a B) x}{3 a b^2}+\frac{(A b-a B) x^4}{3 a b \left (a+b x^3\right )}+\frac{(A b-4 a B) \int \frac{1}{a+b x^3} \, dx}{3 b^2}\\ &=-\frac{(A b-4 a B) x}{3 a b^2}+\frac{(A b-a B) x^4}{3 a b \left (a+b x^3\right )}+\frac{(A b-4 a B) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{2/3} b^2}+\frac{(A b-4 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^{2/3} b^2}\\ &=-\frac{(A b-4 a B) x}{3 a b^2}+\frac{(A b-a B) x^4}{3 a b \left (a+b x^3\right )}+\frac{(A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{(A b-4 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^{2/3} b^{7/3}}+\frac{(A b-4 a B) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 \sqrt [3]{a} b^2}\\ &=-\frac{(A b-4 a B) x}{3 a b^2}+\frac{(A b-a B) x^4}{3 a b \left (a+b x^3\right )}+\frac{(A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{(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^{2/3} b^{7/3}}+\frac{(A b-4 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^{2/3} b^{7/3}}\\ &=-\frac{(A b-4 a B) x}{3 a b^2}+\frac{(A b-a B) x^4}{3 a b \left (a+b x^3\right )}-\frac{(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^{2/3} b^{7/3}}+\frac{(A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{(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^{2/3} b^{7/3}}\\ \end{align*}
Mathematica [A] time = 0.121352, size = 160, normalized size = 0.84 \[ \frac{\frac{(4 a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{2 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{2 \sqrt{3} (4 a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{6 \sqrt [3]{b} x (A b-a B)}{a+b x^3}+18 \sqrt [3]{b} B x}{18 b^{7/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 228, normalized size = 1.2 \begin{align*}{\frac{Bx}{{b}^{2}}}-{\frac{xA}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{Bax}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{4\,Ba}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,Ba}{9\,{b}^{3}}\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{4\,Ba\sqrt{3}}{9\,{b}^{3}}\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}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A}{18\,{b}^{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{A\sqrt{3}}{9\,{b}^{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}}}} \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 = 1.83695, size = 1289, normalized size = 6.78 \begin{align*} \left [\frac{18 \, B a^{2} b^{2} x^{4} - 3 \, \sqrt{\frac{1}{3}}{\left (4 \, B a^{3} b - A a^{2} b^{2} +{\left (4 \, B a^{2} b^{2} - A a b^{3}\right )} x^{3}\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac{1}{3}} a x - a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac{2}{3}} x - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{b x^{3} + a}\right ) +{\left ({\left (4 \, B a b - A b^{2}\right )} x^{3} + 4 \, B a^{2} - A a b\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac{2}{3}} x + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) - 2 \,{\left ({\left (4 \, B a b - A b^{2}\right )} x^{3} + 4 \, B a^{2} - A a b\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right ) + 6 \,{\left (4 \, B a^{3} b - A a^{2} b^{2}\right )} x}{18 \,{\left (a^{2} b^{4} x^{3} + a^{3} b^{3}\right )}}, \frac{18 \, B a^{2} b^{2} x^{4} - 6 \, \sqrt{\frac{1}{3}}{\left (4 \, B a^{3} b - A a^{2} b^{2} +{\left (4 \, B a^{2} b^{2} - A a b^{3}\right )} x^{3}\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a^{2} b\right )^{\frac{2}{3}} x - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) +{\left ({\left (4 \, B a b - A b^{2}\right )} x^{3} + 4 \, B a^{2} - A a b\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac{2}{3}} x + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) - 2 \,{\left ({\left (4 \, B a b - A b^{2}\right )} x^{3} + 4 \, B a^{2} - A a b\right )} \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right ) + 6 \,{\left (4 \, B a^{3} b - A a^{2} b^{2}\right )} x}{18 \,{\left (a^{2} b^{4} x^{3} + a^{3} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.9163, size = 102, normalized size = 0.54 \begin{align*} \frac{B x}{b^{2}} + \frac{x \left (- A b + B a\right )}{3 a b^{2} + 3 b^{3} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{7} - A^{3} b^{3} + 12 A^{2} B a b^{2} - 48 A B^{2} a^{2} b + 64 B^{3} a^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t a b^{2}}{- A b + 4 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 = 1.13746, size = 254, normalized size = 1.34 \begin{align*} \frac{B x}{b^{2}} + \frac{{\left (4 \, B a - 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 b^{2}} - \frac{\sqrt{3}{\left (4 \, \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 )}{9 \, a b^{3}} + \frac{B a x - A b x}{3 \,{\left (b x^{3} + a\right )} b^{2}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - \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 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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