Optimal. Leaf size=267 \[ \frac {a (6 b c-5 a d) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{4/3}}-\frac {a (6 b c-5 a d) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{4/3}}+\frac {a (6 b c-5 a d) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{4/3}}+\frac {x \left (a+b x^3\right )^{2/3} (6 b c-5 a d)}{18 c^2 \left (c+d x^3\right ) (b c-a d)}-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2 (b c-a d)} \]
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Rubi [A] time = 0.24, antiderivative size = 326, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {382, 378, 377, 200, 31, 634, 617, 204, 628} \[ \frac {x \left (a+b x^3\right )^{2/3} (6 b c-5 a d)}{18 c^2 \left (c+d x^3\right ) (b c-a d)}-\frac {a (6 b c-5 a d) \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} (b c-a d)^{4/3}}+\frac {a (6 b c-5 a d) \log \left (\frac {x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{54 c^{8/3} (b c-a d)^{4/3}}+\frac {a (6 b c-5 a d) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{4/3}}-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 377
Rule 378
Rule 382
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^3} \, dx &=-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac {(6 b c-5 a d) \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^2} \, dx}{6 c (b c-a d)}\\ &=-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac {(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}+\frac {(a (6 b c-5 a d)) \int \frac {1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{9 c^2 (b c-a d)}\\ &=-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac {(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}+\frac {(a (6 b c-5 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 )}{9 c^2 (b c-a d)}\\ &=-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac {(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}+\frac {(a (6 b c-5 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 )}{27 c^{8/3} (b c-a d)}+\frac {(a (6 b c-5 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 )}{27 c^{8/3} (b c-a d)}\\ &=-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac {(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}-\frac {a (6 b c-5 a d) \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} (b c-a d)^{4/3}}+\frac {(a (6 b c-5 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 )}{54 c^{8/3} (b c-a d)^{4/3}}+\frac {(a (6 b c-5 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 )}{18 c^{7/3} (b c-a d)}\\ &=-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac {(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}-\frac {a (6 b c-5 a d) \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} (b c-a d)^{4/3}}+\frac {a (6 b c-5 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 )}{54 c^{8/3} (b c-a d)^{4/3}}-\frac {(a (6 b c-5 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 )}{9 c^{8/3} (b c-a d)^{4/3}}\\ &=-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac {(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}+\frac {a (6 b c-5 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 )}{9 \sqrt {3} c^{8/3} (b c-a d)^{4/3}}-\frac {a (6 b c-5 a d) \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} (b c-a d)^{4/3}}+\frac {a (6 b c-5 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 )}{54 c^{8/3} (b c-a d)^{4/3}}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 153, normalized size = 0.57 \[ \frac {x \left (c \left (-a^2 d \left (8 c+5 d x^3\right )+a b \left (6 c^2-5 c d x^3-5 d^2 x^6\right )+3 b^2 c x^3 \left (2 c+d x^3\right )\right )-2 a \left (c+d x^3\right )^2 (5 a d-6 b c) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )\right )}{18 c^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2 (b c-a d)} \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (d \,x^{3}+c \right )^{3}}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{2/3}}{{\left (d\,x^3+c\right )}^3} \,d x \]
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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