Optimal. Leaf size=217 \[ \frac {(3 b c-2 a d) \log \left (c+d x^3\right )}{18 c^{5/3} (b c-a d)^{4/3}}-\frac {(3 b c-2 a d) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{6 c^{5/3} (b c-a d)^{4/3}}+\frac {(3 b c-2 a d) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^{4/3}}-\frac {d x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right ) (b c-a d)} \]
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Rubi [A] time = 0.20, antiderivative size = 276, normalized size of antiderivative = 1.27, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {382, 377, 200, 31, 634, 617, 204, 628} \[ -\frac {(3 b c-2 a d) \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} (b c-a d)^{4/3}}+\frac {(3 b c-2 a d) \log \left (\frac {x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{18 c^{5/3} (b c-a d)^{4/3}}+\frac {(3 b c-2 a d) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^{4/3}}-\frac {d x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 377
Rule 382
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )^2} \, dx &=-\frac {d x \left (a+b x^3\right )^{2/3}}{3 c (b c-a d) \left (c+d x^3\right )}+\frac {(3 b c-2 a d) \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{3 c (b c-a d)}\\ &=-\frac {d x \left (a+b x^3\right )^{2/3}}{3 c (b c-a d) \left (c+d x^3\right )}+\frac {(3 b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{c-(b c-a d) x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 c (b c-a d)}\\ &=-\frac {d x \left (a+b x^3\right )^{2/3}}{3 c (b c-a d) \left (c+d x^3\right )}+\frac {(3 b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{c}-\sqrt [3]{b c-a d} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} (b c-a d)}+\frac {(3 b c-2 a d) \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{c}+\sqrt [3]{b c-a d} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} (b c-a d)}\\ &=-\frac {d x \left (a+b x^3\right )^{2/3}}{3 c (b c-a d) \left (c+d x^3\right )}-\frac {(3 b c-2 a d) \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} (b c-a d)^{4/3}}+\frac {(3 b c-2 a d) \operatorname {Subst}\left (\int \frac {\sqrt [3]{c} \sqrt [3]{b c-a d}+2 (b c-a d)^{2/3} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{18 c^{5/3} (b c-a d)^{4/3}}+\frac {(3 b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{6 c^{4/3} (b c-a d)}\\ &=-\frac {d x \left (a+b x^3\right )^{2/3}}{3 c (b c-a d) \left (c+d x^3\right )}-\frac {(3 b c-2 a d) \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} (b c-a d)^{4/3}}+\frac {(3 b c-2 a d) \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{18 c^{5/3} (b c-a d)^{4/3}}-\frac {(3 b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{3 c^{5/3} (b c-a d)^{4/3}}\\ &=-\frac {d x \left (a+b x^3\right )^{2/3}}{3 c (b c-a d) \left (c+d x^3\right )}+\frac {(3 b c-2 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} (b c-a d)^{4/3}}-\frac {(3 b c-2 a d) \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} (b c-a d)^{4/3}}+\frac {(3 b c-2 a d) \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{18 c^{5/3} (b c-a d)^{4/3}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 99, normalized size = 0.46 \[ \frac {x \left (\left (c+d x^3\right ) (3 b c-2 a d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-c d \left (a+b x^3\right )\right )}{3 c^2 \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{1/3}\,{\left (d\,x^3+c\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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