Optimal. Leaf size=377 \[ -\frac {d x}{6 c (b c-a d) \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2}+\frac {b (6 b c+a d) x}{6 a c (b c-a d)^2 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {d \left (18 b^2 c^2+15 a b c d-5 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{18 a c^2 (b c-a d)^3 \left (c+d x^3\right )}-\frac {d \left (27 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{10/3}}-\frac {d \left (27 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{10/3}}+\frac {d \left (27 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{10/3}} \]
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Rubi [A]
time = 0.25, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {425, 541, 12,
384} \begin {gather*} -\frac {d \left (5 a^2 d^2-18 a b c d+27 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} (b c-a d)^{10/3}}+\frac {d x \left (a+b x^3\right )^{2/3} \left (-5 a^2 d^2+15 a b c d+18 b^2 c^2\right )}{18 a c^2 \left (c+d x^3\right ) (b c-a d)^3}-\frac {d \left (5 a^2 d^2-18 a b c d+27 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{10/3}}+\frac {d \left (5 a^2 d^2-18 a b c d+27 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} (b c-a d)^{10/3}}+\frac {b x (a d+6 b c)}{6 a c \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)^2}-\frac {d x}{6 c \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 384
Rule 425
Rule 541
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^3} \, dx &=\frac {\sqrt [3]{1+\frac {b x^3}{a}} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{4/3} \left (c+d x^3\right )^3} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=-\frac {65 c^2 \left (a+b x^3\right )^2 \left (14000 a^2 c^5+21896 a b c^5 x^3+48104 a^2 c^4 d x^3+8391 b^2 c^5 x^6+70802 a b c^4 d x^6+60807 a^2 c^3 d^2 x^6+24417 b^2 c^4 d x^9+81534 a b c^3 d^2 x^9+33657 a^2 c^2 d^3 x^9+23409 b^2 c^3 d^2 x^{12}+38652 a b c^2 d^3 x^{12}+7155 a^2 c d^4 x^{12}+7425 b^2 c^2 d^3 x^{15}+5940 a b c d^4 x^{15}+243 a^2 d^5 x^{15}-28 \left (c+d x^3\right )^2 \left (27 b^2 c^2 x^6 \left (7 c+6 d x^3\right )+9 a b c x^3 \left (73 c^2+104 c d x^3+33 d^2 x^6\right )+a^2 \left (500 c^3+843 c^2 d x^3+375 c d^2 x^6+27 d^3 x^9\right )\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )\right )-486 (b c-a d)^4 x^{12} \left (c+d x^3\right )^3 \, _4F_3\left (2,2,2,\frac {7}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{16380 c^5 (b c-a d)^3 x^8 \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 7.99, size = 471, normalized size = 1.25 \begin {gather*} \frac {-\frac {6 c^{2/3} x \left (18 b^3 c^2 \left (c+d x^3\right )^2+3 a b^2 c d^2 x^3 \left (6 c+5 d x^3\right )-a^3 d^3 \left (8 c+5 d x^3\right )+a^2 b d^2 \left (18 c^2+7 c d x^3-5 d^2 x^6\right )\right )}{a (-b c+a d)^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2}+\frac {2 i \left (3 i+\sqrt {3}\right ) d \left (27 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {i+\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d} x}}{\sqrt {3}}\right )}{(b c-a d)^{10/3}}-\frac {2 i \left (-i+\sqrt {3}\right ) d \left (27 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{(b c-a d)^{10/3}}+\frac {\left (1+i \sqrt {3}\right ) d \left (27 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{(b c-a d)^{10/3}}}{108 c^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )^{3}}\, dx\]
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 {1}{{\left (b\,x^3+a\right )}^{4/3}\,{\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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