Optimal. Leaf size=331 \[ -\frac {b^{2/3} \left (20 a^2 d^2-24 a b c d+9 b^2 c^2\right ) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 d^3}+\frac {b^{2/3} \left (20 a^2 d^2-24 a b c d+9 b^2 c^2\right ) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} d^3}-\frac {(b c-a d)^{8/3} \log \left (c+d x^3\right )}{6 c^{2/3} d^3}+\frac {(b c-a d)^{8/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} d^3}-\frac {(b c-a d)^{8/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} d^3}-\frac {b x \left (a+b x^3\right )^{2/3} (6 b c-11 a d)}{18 d^2}+\frac {b x \left (a+b x^3\right )^{5/3}}{6 d} \]
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Rubi [C] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 0.19, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {430, 429} \[ \frac {a^2 x \left (a+b x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {8}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \left (\frac {b x^3}{a}+1\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Rule 429
Rule 430
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx &=\frac {\left (a^2 \left (a+b x^3\right )^{2/3}\right ) \int \frac {\left (1+\frac {b x^3}{a}\right )^{8/3}}{c+d x^3} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=\frac {a^2 x \left (a+b x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {8}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 1.23, size = 655, normalized size = 1.98 \[ \frac {3 b x^4 \sqrt [3]{\frac {b x^3}{a}+1} \sqrt [3]{b c-a d} \left (20 a^2 d^2-24 a b c d+9 b^2 c^2\right ) F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 \sqrt [3]{c} \left (9 a^3 d^2 \sqrt [3]{a+b x^3} \log \left (\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+\frac {x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+c^{2/3}\right )-2 a \sqrt [3]{a+b x^3} \left (9 a^2 d^2-7 a b c d+3 b^2 c^2\right ) \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt {3} a \sqrt [3]{a+b x^3} \left (9 a^2 d^2-7 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )-7 a^2 b c d \sqrt [3]{a+b x^3} \log \left (\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+\frac {x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+c^{2/3}\right )+42 a^2 b c^{2/3} d x \sqrt [3]{b c-a d}+9 b^3 c^{2/3} d x^7 \sqrt [3]{b c-a d}-18 b^3 c^{5/3} x^4 \sqrt [3]{b c-a d}+51 a b^2 c^{2/3} d x^4 \sqrt [3]{b c-a d}-18 a b^2 c^{5/3} x \sqrt [3]{b c-a d}+3 a b^2 c^2 \sqrt [3]{a+b x^3} \log \left (\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+\frac {x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+c^{2/3}\right )\right )}{108 c d^2 \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 15.64, size = 643, normalized size = 1.94 \[ -\frac {18 \, \sqrt {3} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (b c - a d\right )} x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} c \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}}}{3 \, {\left (b c - a d\right )} x}\right ) + 2 \, \sqrt {3} {\left (9 \, b^{2} c^{2} - 24 \, a b c d + 20 \, a^{2} d^{2}\right )} \left (-b^{2}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b x - 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {1}{3}}}{3 \, b x}\right ) - 18 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (\frac {c x \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )}}{x}\right ) - 2 \, {\left (9 \, b^{2} c^{2} - 24 \, a b c d + 20 \, a^{2} d^{2}\right )} \left (-b^{2}\right )^{\frac {1}{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} - 24 \, a b c d + 20 \, a^{2} d^{2}\right )} \left (-b^{2}\right )^{\frac {1}{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 ) + 9 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (-\frac {{\left (b c - a d\right )} x^{2} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} c x \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b c - a d\right )}}{x^{2}}\right ) - 3 \, {\left (3 \, b^{2} d^{2} x^{4} - 2 \, {\left (3 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, d^{3}} \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{d x^{3} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.70, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {8}{3}}}{d \,x^{3}+c}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{d x^{3} + c}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{8/3}}{d\,x^3+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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