Optimal. Leaf size=351 \[ \frac {b (2 b c-a d) x \left (a+b x^3\right )^{2/3}}{3 c d^2}-\frac {(b c-a d) x \left (a+b x^3\right )^{5/3}}{3 c d \left (c+d x^3\right )}-\frac {2 b^{5/3} (3 b c-4 a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} d^3}+\frac {2 (b c-a d)^{5/3} (3 b c+a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} d^3}+\frac {(b c-a d)^{5/3} (3 b c+a d) \log \left (c+d x^3\right )}{9 c^{5/3} d^3}-\frac {(b c-a d)^{5/3} (3 b c+a d) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} d^3}+\frac {b^{5/3} (3 b c-4 a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 d^3} \]
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Rubi [A]
time = 0.46, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {424, 542, 544,
245, 384} \begin {gather*} -\frac {2 b^{5/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) (3 b c-4 a d)}{3 \sqrt {3} d^3}+\frac {2 (b c-a d)^{5/3} (a d+3 b c) \text {ArcTan}\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^3}+\frac {b^{5/3} (3 b c-4 a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 d^3}+\frac {(b c-a d)^{5/3} (a d+3 b c) \log \left (c+d x^3\right )}{9 c^{5/3} d^3}-\frac {(b c-a d)^{5/3} (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 )}{3 c^{5/3} d^3}+\frac {b x \left (a+b x^3\right )^{2/3} (2 b c-a d)}{3 c d^2}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{3 c d \left (c+d x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 384
Rule 424
Rule 542
Rule 544
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, 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}}{\left (c+d x^3\right )^2} \, 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},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] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 10.70, size = 698, normalized size = 1.99 \begin {gather*} \frac {1}{18} \left (\frac {6 x \left (a+b x^3\right )^{2/3} \left (b^2+\frac {(b c-a d)^2}{c \left (c+d x^3\right )}\right )}{d^2}-\frac {9 b^3 x^4 \sqrt [3]{1+\frac {b x^3}{a}} 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^2 \sqrt [3]{a+b x^3}}+\frac {12 a b^2 x^4 \sqrt [3]{1+\frac {b x^3}{a}} F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c d \sqrt [3]{a+b x^3}}+\frac {2 a^3 \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{c^{5/3} \sqrt [3]{b c-a d}}-\frac {2 a b^2 \sqrt [3]{c} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{d^2 \sqrt [3]{b c-a d}}+\frac {2 a^2 b \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{c^{2/3} d \sqrt [3]{b c-a d}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {8}{3}}}{\left (d \,x^{3}+c \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 819 vs.
\(2 (291) = 582\).
time = 11.01, size = 819, normalized size = 2.33 \begin {gather*} \frac {2 \, \sqrt {3} {\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{3}\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 (3 \, b^{2} c^{3} - 4 \, a b c^{2} d + {\left (3 \, b^{2} c^{2} d - 4 \, a b c d^{2}\right )} x^{3}\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 ) - 2 \, {\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{3}\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 (3 \, b^{2} c^{3} - 4 \, a b c^{2} d + {\left (3 \, b^{2} c^{2} d - 4 \, a b c d^{2}\right )} x^{3}\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 (3 \, b^{2} c^{3} - 4 \, a b c^{2} d + {\left (3 \, b^{2} c^{2} d - 4 \, a b c d^{2}\right )} x^{3}\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 (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{3}\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 (b^{2} c d^{2} x^{4} + {\left (2 \, b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{9 \, {\left (c d^{4} x^{3} + c^{2} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} \text {could not integrate} \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 {{\left (b\,x^3+a\right )}^{8/3}}{{\left (d\,x^3+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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