Optimal. Leaf size=541 \[ -\frac {b (2 b c-a d) \left (18 b^2 c^2-18 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^4}+\frac {b \left (18 b^2 c^2-10 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{5/3}}{18 c^2 d^3}-\frac {(b c-a d) x \left (a+b x^3\right )^{11/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (12 b c+5 a d) x \left (a+b x^3\right )^{8/3}}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {b^{8/3} \left (54 b^2 c^2-126 a b c d+77 a^2 d^2\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} d^5}-\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} d^5}-\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^5}+\frac {(b c-a d)^{8/3} \left (54 b^2 c^2+18 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^5}-\frac {b^{8/3} \left (54 b^2 c^2-126 a b c d+77 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 d^5} \]
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Rubi [A]
time = 0.50, antiderivative size = 541, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {424, 540, 542,
544, 245, 384} \begin {gather*} -\frac {(b c-a d)^{8/3} \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \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 )}{9 \sqrt {3} c^{8/3} d^5}+\frac {b^{8/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) \left (77 a^2 d^2-126 a b c d+54 b^2 c^2\right )}{9 \sqrt {3} d^5}-\frac {b x \left (a+b x^3\right )^{2/3} (2 b c-a d) \left (-5 a^2 d^2-18 a b c d+18 b^2 c^2\right )}{18 c^2 d^4}+\frac {b x \left (a+b x^3\right )^{5/3} \left (-5 a^2 d^2-10 a b c d+18 b^2 c^2\right )}{18 c^2 d^3}-\frac {(b c-a d)^{8/3} \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^5}+\frac {(b c-a d)^{8/3} \left (5 a^2 d^2+18 a b c d+54 b^2 c^2\right ) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^5}-\frac {b^{8/3} \left (77 a^2 d^2-126 a b c d+54 b^2 c^2\right ) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 d^5}-\frac {x \left (a+b x^3\right )^{8/3} (b c-a d) (5 a d+12 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac {x \left (a+b x^3\right )^{11/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 384
Rule 424
Rule 540
Rule 542
Rule 544
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx &=\frac {\left (a^4 \left (a+b x^3\right )^{2/3}\right ) \int \frac {\left (1+\frac {b x^3}{a}\right )^{14/3}}{\left (c+d x^3\right )^3} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=\frac {a^4 x \left (a+b x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {14}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3 \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 = 11.57, size = 1171, normalized size = 2.16 \begin {gather*} \frac {1}{108} \left (\frac {6 x \left (a+b x^3\right )^{2/3} \left (-2 b^3 (9 b c-13 a d)+3 b^4 d x^3+\frac {3 (b c-a d)^4}{c \left (c+d x^3\right )^2}-\frac {(b c-a d)^3 (21 b c+5 a d)}{c^2 \left (c+d x^3\right )}\right )}{d^4}+\frac {162 b^5 c 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^4 \sqrt [3]{a+b x^3}}-\frac {378 a b^4 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^3 \sqrt [3]{a+b x^3}}+\frac {231 a^2 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 )}{c d^2 \sqrt [3]{a+b x^3}}+\frac {10 a^5 \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^{8/3} \sqrt [3]{b c-a d}}+\frac {36 a b^4 c^{4/3} \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 )}{d^4 \sqrt [3]{b c-a d}}-\frac {72 a^2 b^3 \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^3 \sqrt [3]{b c-a d}}+\frac {30 a^3 b^2 \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^2 \sqrt [3]{b c-a d}}+\frac {6 a^4 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^{5/3} 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 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {14}{3}}}{\left (d \,x^{3}+c \right )^{3}}\, dx\]
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 [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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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 )}^{14/3}}{{\left (d\,x^3+c\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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