Optimal. Leaf size=222 \[ \frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}+\frac{x^5 (A b-4 a B)}{9 a b^2 \left (a+b x^3\right )}-\frac{5 x^2 (A b-4 a B)}{18 a b^3}-\frac{5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} \sqrt [3]{a} b^{11/3}}+\frac{x^8 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.139961, antiderivative size = 222, 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, 292, 31, 634, 617, 204, 628} \[ \frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}+\frac{x^5 (A b-4 a B)}{9 a b^2 \left (a+b x^3\right )}-\frac{5 x^2 (A b-4 a B)}{18 a b^3}-\frac{5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} \sqrt [3]{a} b^{11/3}}+\frac{x^8 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 457
Rule 288
Rule 321
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac{(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac{(-2 A b+8 a B) \int \frac{x^7}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac{(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}-\frac{(5 (A b-4 a B)) \int \frac{x^4}{a+b x^3} \, dx}{9 a b^2}\\ &=-\frac{5 (A b-4 a B) x^2}{18 a b^3}+\frac{(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}+\frac{(5 (A b-4 a B)) \int \frac{x}{a+b x^3} \, dx}{9 b^3}\\ &=-\frac{5 (A b-4 a B) x^2}{18 a b^3}+\frac{(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}-\frac{(5 (A b-4 a B)) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 \sqrt [3]{a} b^{10/3}}+\frac{(5 (A b-4 a B)) \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}{27 \sqrt [3]{a} b^{10/3}}\\ &=-\frac{5 (A b-4 a B) x^2}{18 a b^3}+\frac{(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}-\frac{5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}+\frac{(5 (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}{54 \sqrt [3]{a} b^{11/3}}+\frac{(5 (A b-4 a B)) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^{10/3}}\\ &=-\frac{5 (A b-4 a B) x^2}{18 a b^3}+\frac{(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}-\frac{5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}+\frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}+\frac{(5 (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 )}{9 \sqrt [3]{a} b^{11/3}}\\ &=-\frac{5 (A b-4 a B) x^2}{18 a b^3}+\frac{(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac{(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}-\frac{5 (A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} \sqrt [3]{a} b^{11/3}}-\frac{5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}+\frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}\\ \end{align*}
Mathematica [A] time = 0.154885, size = 194, normalized size = 0.87 \[ \frac{\frac{5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}-\frac{6 b^{2/3} x^2 (4 A b-7 a B)}{a+b x^3}+\frac{9 a b^{2/3} x^2 (A b-a B)}{\left (a+b x^3\right )^2}+\frac{10 (4 a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac{10 \sqrt{3} (4 a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}+27 b^{2/3} B x^2}{54 b^{11/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 275, normalized size = 1.2 \begin{align*}{\frac{B{x}^{2}}{2\,{b}^{3}}}-{\frac{4\,A{x}^{5}}{9\,b \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{7\,B{x}^{5}a}{9\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,aA{x}^{2}}{18\,{b}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{11\,B{x}^{2}{a}^{2}}{18\,{b}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,A}{27\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{5\,A}{54\,{b}^{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{5\,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 ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{20\,Ba}{27\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{10\,Ba}{27\,{b}^{4}}\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{20\,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 ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \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 = 1.57092, size = 1705, normalized size = 7.68 \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 = 3.23634, size = 162, normalized size = 0.73 \begin{align*} \frac{B x^{2}}{2 b^{3}} + \frac{x^{5} \left (- 8 A b^{2} + 14 B a b\right ) + x^{2} \left (- 5 A a b + 11 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 b^{11} + 125 A^{3} b^{3} - 1500 A^{2} B a b^{2} + 6000 A B^{2} a^{2} b - 8000 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a b^{7}}{25 A^{2} b^{2} - 200 A B a b + 400 B^{2} a^{2}} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14368, size = 313, normalized size = 1.41 \begin{align*} \frac{B x^{2}}{2 \, b^{3}} + \frac{5 \,{\left (4 \, B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} - A b \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 )}{27 \, a b^{3}} + \frac{5 \, \sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - \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 )}{27 \, a b^{5}} + \frac{14 \, B a b x^{5} - 8 \, A b^{2} x^{5} + 11 \, B a^{2} x^{2} - 5 \, A a b x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} b^{3}} - \frac{5 \,{\left (4 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a - \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 )}{54 \, a b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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