Optimal. Leaf size=336 \[ -\frac {\left (-a^2 d^2-3 a b c d+9 b^2 c^2\right ) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{5/3} d^3}-\frac {\left (-a^2 d^2-3 a b c d+9 b^2 c^2\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} b^{5/3} d^3}-\frac {c^{5/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^3}+\frac {c^{5/3} \sqrt [3]{b c-a d} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d^3}+\frac {c^{5/3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^3}-\frac {x^2 \sqrt [3]{a+b x^3} (6 b c-a d)}{18 b d^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 d} \]
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Rubi [C] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 0.19, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \begin {gather*} \frac {x^8 \sqrt [3]{a+b x^3} F_1\left (\frac {8}{3};-\frac {1}{3},1;\frac {11}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{8 c \sqrt [3]{\frac {b x^3}{a}+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {x^7 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \frac {x^7 \sqrt [3]{1+\frac {b x^3}{a}}}{c+d x^3} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {x^8 \sqrt [3]{a+b x^3} F_1\left (\frac {8}{3};-\frac {1}{3},1;\frac {11}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{8 c \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}
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Mathematica [C] time = 0.37, size = 226, normalized size = 0.67 \begin {gather*} \frac {5 c x^2 \left (a \left (\frac {b x^3}{a}+1\right )^{2/3} (6 b c-a d) \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {(a d-b c) x^3}{a \left (d x^3+c\right )}\right )+\left (a+b x^3\right ) \left (\frac {d x^3}{c}+1\right )^{2/3} \left (a d-6 b c+3 b d x^3\right )\right )-2 x^5 \left (\frac {b x^3}{a}+1\right )^{2/3} \left (\frac {d x^3}{c}+1\right )^{2/3} \left (a^2 d^2+3 a b c d-9 b^2 c^2\right ) F_1\left (\frac {5}{3};\frac {2}{3},1;\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{90 b c d^2 \left (a+b x^3\right )^{2/3} \left (\frac {d x^3}{c}+1\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [C] time = 5.50, size = 613, normalized size = 1.82 \begin {gather*} \frac {\left (a^2 d^2+3 a b c d-9 b^2 c^2\right ) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{27 b^{5/3} d^3}-\frac {\left (-a^2 d^2-3 a b c d+9 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{2 \sqrt [3]{a+b x^3}+\sqrt [3]{b} x}\right )}{9 \sqrt {3} b^{5/3} d^3}+\frac {\left (-a^2 d^2-3 a b c d+9 b^2 c^2\right ) \log \left (\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}+b^{2/3} x^2\right )}{54 b^{5/3} d^3}+\frac {i \left (\sqrt {3} c^{5/3} \sqrt [3]{b c-a d}+i c^{5/3} \sqrt [3]{b c-a d}\right ) \log \left (2 x \sqrt [3]{b c-a d}+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{6 d^3}-\frac {\sqrt {-1-i \sqrt {3}} c^{5/3} \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac {3 x \sqrt [3]{b c-a d}}{\sqrt {3} x \sqrt [3]{b c-a d}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3}-3 i \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{\sqrt {6} d^3}+\frac {\left (c^{5/3} \sqrt [3]{b c-a d}-i \sqrt {3} c^{5/3} \sqrt [3]{b c-a d}\right ) \log \left (\left (\sqrt {3}+i\right ) c^{2/3} \left (a+b x^3\right )^{2/3}+\sqrt [3]{c} \left (-\sqrt {3} x+i x\right ) \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}-2 i x^2 (b c-a d)^{2/3}\right )}{12 d^3}+\frac {\sqrt [3]{a+b x^3} \left (a d x^2-6 b c x^2+3 b d x^5\right )}{18 b d^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.95, size = 494, normalized size = 1.47 \begin {gather*} \frac {18 \, \sqrt {3} {\left (b c^{3} - a c^{2} d\right )}^{\frac {1}{3}} b^{3} c \arctan \left (-\frac {\sqrt {3} {\left (b c^{2} - a c d\right )} x + 2 \, \sqrt {3} {\left (b c^{3} - a c^{2} d\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{3 \, {\left (b c^{2} - a c d\right )} x}\right ) + 18 \, {\left (b c^{3} - a c^{2} d\right )}^{\frac {1}{3}} b^{3} c \log \left (\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} c - {\left (b c^{3} - a c^{2} d\right )}^{\frac {1}{3}} x}{x}\right ) - 9 \, {\left (b c^{3} - a c^{2} d\right )}^{\frac {1}{3}} b^{3} c \log \left (\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} c^{2} + {\left (b c^{3} - a c^{2} d\right )}^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} c x + {\left (b c^{3} - a c^{2} d\right )}^{\frac {2}{3}} x^{2}}{x^{2}}\right ) + 2 \, \sqrt {3} {\left (9 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} {\left (b^{2}\right )}^{\frac {1}{6}} \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 )} {\left (b^{2}\right )}^{\frac {1}{6}}}{3 \, b^{2} x}\right ) - 2 \, {\left (9 \, b^{2} c^{2} - 3 \, a b c d - a^{2} d^{2}\right )} {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2}\right )}^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + {\left (9 \, b^{2} c^{2} - 3 \, a b c d - a^{2} d^{2}\right )} {\left (b^{2}\right )}^{\frac {2}{3}} \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} d^{2} x^{5} - {\left (6 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{54 \, b^{3} d^{3}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{7}}{d x^{3} + c}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{7}}{d \,x^{3}+c}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{7}}{d x^{3} + c}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^7\,{\left (b\,x^3+a\right )}^{1/3}}{d\,x^3+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7} \sqrt [3]{a + b x^{3}}}{c + d x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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