Optimal. Leaf size=301 \[ -\frac {b^{5/3} \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 d^2}+\frac {b^{5/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2}-\frac {(b c-a d)^{2/3} (2 a d+3 b c) \log \left (c+d x^3\right )}{18 c^{5/3} d^2}+\frac {(b c-a d)^{2/3} (2 a d+3 b c) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{6 c^{5/3} d^2}-\frac {(b c-a d)^{2/3} (2 a d+3 b c) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} d^2}-\frac {x \left (a+b x^3\right )^{2/3} (b c-a d)}{3 c d \left (c+d x^3\right )} \]
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Rubi [C] time = 0.03, antiderivative size = 60, normalized size of antiderivative = 0.20, 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} \[ \frac {a x \left (a+b x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {5}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2 \left (\frac {b x^3}{a}+1\right )^{2/3}} \]
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 )^{5/3}}{\left (c+d x^3\right )^2} \, dx &=\frac {\left (a \left (a+b x^3\right )^{2/3}\right ) \int \frac {\left (1+\frac {b x^3}{a}\right )^{5/3}}{\left (c+d x^3\right )^2} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=\frac {a x \left (a+b x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {5}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2 \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.64, size = 450, normalized size = 1.50 \[ \frac {\frac {4 a^2 \left (\log \left (\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+\frac {x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+c^{2/3}\right )-2 \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )\right )}{\sqrt [3]{b c-a d}}+\frac {9 b^2 c^{2/3} x^4 \sqrt [3]{\frac {b x^3}{a}+1} F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{d \sqrt [3]{a+b x^3}}-\frac {12 c^{2/3} x \left (a+b x^3\right )^{2/3} (b c-a d)}{d \left (c+d x^3\right )}+\frac {2 a b c \left (\log \left (\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+\frac {x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+c^{2/3}\right )-2 \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )\right )}{d \sqrt [3]{b c-a d}}}{36 c^{5/3}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.34, size = 631, normalized size = 2.10 \[ -\frac {2 \, \sqrt {3} {\left ({\left (3 \, b c d + 2 \, a d^{2}\right )} x^{3} + 3 \, b c^{2} + 2 \, a c d\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 ) + 6 \, \sqrt {3} {\left (b c d x^{3} + b c^{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 ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b c d - a d^{2}\right )} x - 2 \, {\left ({\left (3 \, b c d + 2 \, a d^{2}\right )} x^{3} + 3 \, b c^{2} + 2 \, a c d\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 ) - 6 \, {\left (b c d x^{3} + b c^{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 ) + 3 \, {\left (b c d x^{3} + b c^{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 ) + {\left ({\left (3 \, b c d + 2 \, a d^{2}\right )} x^{3} + 3 \, b c^{2} + 2 \, a c d\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 )}{18 \, {\left (c d^{3} x^{3} + c^{2} d^{2}\right )}} \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{{\left (d x^{3} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {5}{3}}}{\left (d \,x^{3}+c \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{{\left (d x^{3} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{5/3}}{{\left (d\,x^3+c\right )}^2} \,d x \]
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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