Optimal. Leaf size=64 \[ -\frac {\sqrt [3]{\frac {b x^3}{a}+1} F_1\left (-\frac {4}{3};\frac {1}{3},1;-\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 c x^4 \sqrt [3]{a+b x^3}} \]
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Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac {\sqrt [3]{\frac {b x^3}{a}+1} F_1\left (-\frac {4}{3};\frac {1}{3},1;-\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 c x^4 \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx &=\frac {\sqrt [3]{1+\frac {b x^3}{a}} \int \frac {1}{x^5 \sqrt [3]{1+\frac {b x^3}{a}} \left (c+d x^3\right )} \, dx}{\sqrt [3]{a+b x^3}}\\ &=-\frac {\sqrt [3]{1+\frac {b x^3}{a}} F_1\left (-\frac {4}{3};\frac {1}{3},1;-\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 c x^4 \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [B] time = 0.22, size = 183, normalized size = 2.86 \[ \frac {5 x^6 \sqrt [3]{\frac {b x^3}{a}+1} \left (2 a^2 d^2-2 a b c d-b^2 c^2\right ) F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-2 b d x^9 \sqrt [3]{\frac {b x^3}{a}+1} (2 a d+b c) F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+5 c \left (a+b x^3\right ) \left (-a c+4 a d x^3+2 b c x^3\right )}{20 a^2 c^3 x^4 \sqrt [3]{a+b x^3}} \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (d \,x^{3}+c \right ) x^{5}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^5\,{\left (b\,x^3+a\right )}^{1/3}\,\left (d\,x^3+c\right )} \,d x \]
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 \[ \int \frac {1}{x^{5} \sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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