Optimal. Leaf size=220 \[ -\frac{(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}}+\frac{2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac{2 (A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{2/3} b^{10/3}}+\frac{x^4 (A b-7 a B)}{18 a b^2 \left (a+b x^3\right )}-\frac{2 x (A b-7 a B)}{9 a b^3}+\frac{x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.125802, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {457, 288, 321, 200, 31, 634, 617, 204, 628} \[ -\frac{(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}}+\frac{2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac{2 (A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{2/3} b^{10/3}}+\frac{x^4 (A b-7 a B)}{18 a b^2 \left (a+b x^3\right )}-\frac{2 x (A b-7 a B)}{9 a b^3}+\frac{x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 457
Rule 288
Rule 321
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac{(-A b+7 a B) \int \frac{x^6}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}-\frac{(2 (A b-7 a B)) \int \frac{x^3}{a+b x^3} \, dx}{9 a b^2}\\ &=-\frac{2 (A b-7 a B) x}{9 a b^3}+\frac{(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}+\frac{(2 (A b-7 a B)) \int \frac{1}{a+b x^3} \, dx}{9 b^3}\\ &=-\frac{2 (A b-7 a B) x}{9 a b^3}+\frac{(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}+\frac{(2 (A b-7 a B)) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{2/3} b^3}+\frac{(2 (A b-7 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}{27 a^{2/3} b^3}\\ &=-\frac{2 (A b-7 a B) x}{9 a b^3}+\frac{(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}+\frac{2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac{(A b-7 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}{27 a^{2/3} b^{10/3}}+\frac{(A b-7 a B) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 \sqrt [3]{a} b^3}\\ &=-\frac{2 (A b-7 a B) x}{9 a b^3}+\frac{(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}+\frac{2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac{(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}}+\frac{(2 (A b-7 a B)) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{2/3} b^{10/3}}\\ &=-\frac{2 (A b-7 a B) x}{9 a b^3}+\frac{(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}-\frac{2 (A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{2/3} b^{10/3}}+\frac{2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac{(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}}\\ \end{align*}
Mathematica [A] time = 0.155371, size = 188, normalized size = 0.85 \[ \frac{\frac{2 (7 a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{4 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{4 \sqrt{3} (7 a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{3 \sqrt [3]{b} x (7 A b-13 a B)}{a+b x^3}+\frac{9 a \sqrt [3]{b} x (A b-a B)}{\left (a+b x^3\right )^2}+54 \sqrt [3]{b} B x}{54 b^{10/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 268, normalized size = 1.2 \begin{align*}{\frac{Bx}{{b}^{3}}}-{\frac{7\,A{x}^{4}}{18\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{13\,B{x}^{4}a}{18\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{2\,aAx}{9\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,{a}^{2}Bx}{9\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{2\,A}{27\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A}{27\,{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{2\,A\sqrt{3}}{27\,{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{14\,Ba}{27\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{7\,Ba}{27\,{b}^{4}}\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{14\,Ba\sqrt{3}}{27\,{b}^{4}}\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 [B] time = 1.60266, size = 1712, normalized size = 7.78 \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 = 2.22335, size = 141, normalized size = 0.64 \begin{align*} \frac{B x}{b^{3}} + \frac{x^{4} \left (- 7 A b^{2} + 13 B a b\right ) + x \left (- 4 A a b + 10 B a^{2}\right )}{18 a^{2} b^{3} + 36 a b^{4} x^{3} + 18 b^{5} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{2} b^{10} - 8 A^{3} b^{3} + 168 A^{2} B a b^{2} - 1176 A B^{2} a^{2} b + 2744 B^{3} a^{3}, \left ( t \mapsto t \log{\left (- \frac{27 t a b^{3}}{- 2 A b + 14 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 = 282, normalized size = 1.28 \begin{align*} \frac{B x}{b^{3}} + \frac{2 \,{\left (7 \, 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 )}{27 \, a b^{3}} - \frac{2 \, \sqrt{3}{\left (7 \, \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 )}{27 \, a b^{4}} - \frac{{\left (7 \, \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 )}{27 \, a b^{4}} + \frac{13 \, B a b x^{4} - 7 \, A b^{2} x^{4} + 10 \, B a^{2} x - 4 \, A a b x}{18 \,{\left (b x^{3} + a\right )}^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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