Optimal. Leaf size=123 \[ -\frac {5 a^2 \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{8/3}}-\frac {5 a^2 \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b} \]
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Rubi [A] time = 0.09, antiderivative size = 176, normalized size of antiderivative = 1.43, number of steps used = 9, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {321, 331, 292, 31, 634, 617, 204, 628} \[ -\frac {5 a^2 \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac {5 a^2 \log \left (\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{54 b^{8/3}}-\frac {5 a^2 \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 321
Rule 331
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac {(5 a) \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx}{6 b}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}+\frac {\left (5 a^2\right ) \int \frac {x}{\left (a+b x^3\right )^{2/3}} \, dx}{9 b^2}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {x}{1-b x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 b^2}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{b} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{7/3}}-\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{7/3}}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac {5 a^2 \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{54 b^{8/3}}-\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{18 b^{7/3}}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac {5 a^2 \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac {5 a^2 \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{54 b^{8/3}}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{8/3}}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac {5 a^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {5 a^2 \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac {5 a^2 \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{54 b^{8/3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 69, normalized size = 0.56 \[ \frac {x^2 \left (5 a^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {b x^3}{b x^3+a}\right )-5 a^2-2 a b x^3+3 b^2 x^6\right )}{18 b^2 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 212, normalized size = 1.72 \[ \frac {10 \, \sqrt {3} a^{2} b \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} b x - 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}}}{3 \, b^{2} x}\right ) - 10 \, \left (-b^{2}\right )^{\frac {2}{3}} a^{2} \log \left (-\frac {\left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + 5 \, \left (-b^{2}\right )^{\frac {2}{3}} a^{2} \log \left (-\frac {\left (-b^{2}\right )^{\frac {1}{3}} b x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right ) + 3 \, {\left (3 \, b^{3} x^{5} - 5 \, a b^{2} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{54 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{7}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {x^{7}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 184, normalized size = 1.50 \[ \frac {5 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{27 \, b^{\frac {8}{3}}} + \frac {5 \, a^{2} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{54 \, b^{\frac {8}{3}}} - \frac {5 \, a^{2} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{27 \, b^{\frac {8}{3}}} + \frac {\frac {8 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2} b}{x} - \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a^{2}}{x^{4}}}{18 \, {\left (b^{4} - \frac {2 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^7}{{\left (b\,x^3+a\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.84, size = 37, normalized size = 0.30 \[ \frac {x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {11}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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