Optimal. Leaf size=215 \[ \frac {\sqrt [3]{a+b x^3} (4 a d+3 b c)}{4 a^2 c^2 x}+\frac {d^2 \log \left (c+d x^3\right )}{6 c^{7/3} (b c-a d)^{2/3}}-\frac {d^2 \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{7/3} (b c-a d)^{2/3}}-\frac {d^2 \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{7/3} (b c-a d)^{2/3}}-\frac {\sqrt [3]{a+b x^3}}{4 a c x^4} \]
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Rubi [A] time = 0.30, antiderivative size = 269, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {494, 461, 292, 31, 634, 617, 204, 628} \[ \frac {\sqrt [3]{a+b x^3} (a d+b c)}{a^2 c^2 x}-\frac {\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}-\frac {d^2 \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{3 c^{7/3} (b c-a d)^{2/3}}+\frac {d^2 \log \left (\frac {x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{6 c^{7/3} (b c-a d)^{2/3}}-\frac {d^2 \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{7/3} (b c-a d)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 461
Rule 494
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-b x^3\right )^2}{x^5 \left (c-(b c-a d) x^3\right )} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{c x^5}+\frac {-b c-a d}{c^2 x^2}+\frac {a^2 d^2 x}{c^2 \left (c-(b c-a d) x^3\right )}\right ) \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{a^2}\\ &=\frac {(b c+a d) \sqrt [3]{a+b x^3}}{a^2 c^2 x}-\frac {\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}+\frac {d^2 \operatorname {Subst}\left (\int \frac {x}{c-(b c-a d) x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{c^2}\\ &=\frac {(b c+a d) \sqrt [3]{a+b x^3}}{a^2 c^2 x}-\frac {\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}+\frac {d^2 \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 )}{3 c^{7/3} \sqrt [3]{b c-a d}}-\frac {d^2 \operatorname {Subst}\left (\int \frac {\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 )}{3 c^{7/3} \sqrt [3]{b c-a d}}\\ &=\frac {(b c+a d) \sqrt [3]{a+b x^3}}{a^2 c^2 x}-\frac {\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}-\frac {d^2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{7/3} (b c-a d)^{2/3}}+\frac {d^2 \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 )}{6 c^{7/3} (b c-a d)^{2/3}}-\frac {d^2 \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 )}{2 c^2 \sqrt [3]{b c-a d}}\\ &=\frac {(b c+a d) \sqrt [3]{a+b x^3}}{a^2 c^2 x}-\frac {\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}-\frac {d^2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{7/3} (b c-a d)^{2/3}}+\frac {d^2 \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 )}{6 c^{7/3} (b c-a d)^{2/3}}+\frac {d^2 \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 )}{c^{7/3} (b c-a d)^{2/3}}\\ &=\frac {(b c+a d) \sqrt [3]{a+b x^3}}{a^2 c^2 x}-\frac {\left (a+b x^3\right )^{4/3}}{4 a^2 c x^4}-\frac {d^2 \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 )}{\sqrt {3} c^{7/3} (b c-a d)^{2/3}}-\frac {d^2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{7/3} (b c-a d)^{2/3}}+\frac {d^2 \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 )}{6 c^{7/3} (b c-a d)^{2/3}}\\ \end {align*}
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Mathematica [C] time = 1.89, size = 267, normalized size = 1.24 \[ -\frac {-81 x^3 \left (c+d x^3\right )^2 (b c-a d) \, _4F_3\left (\frac {2}{3},2,2,2;1,1,\frac {8}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+216 d x^6 \left (c+d x^3\right ) (a d-b c) \, _3F_2\left (\frac {2}{3},2,2;1,\frac {8}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-5 \left (2 c \left (a+b x^3\right ) \left (c^2+10 c d x^3+9 d^2 x^6\right )+\left (a \left (-8 c^3+17 c^2 d x^3+46 c d^2 x^6+9 d^3 x^9\right )+3 b c x^3 \left (-3 c^2+2 c d x^3+9 d^2 x^6\right )\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )\right )}{120 c^4 x^4 \left (a+b x^3\right )^{5/3}} \]
Warning: Unable to verify antiderivative.
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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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x^{3} + c\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right ) x^{5}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x^{3} + c\right )} x^{5}}\,{d x} \]
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 \[ \int \frac {1}{x^5\,{\left (b\,x^3+a\right )}^{2/3}\,\left (d\,x^3+c\right )} \,d x \]
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 \[ \int \frac {1}{x^{5} \left (a + b x^{3}\right )^{\frac {2}{3}} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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