Optimal. Leaf size=62 \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {1}{3};\frac {8}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 c^2 \left (a+b x^3\right )^{2/3}} \]
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Rubi [A] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {430, 429} \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {1}{3};\frac {8}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 c^2 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 429
Rule 430
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^3\right )^{8/3} \left (c+d x^3\right )^2} \, dx &=\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{8/3} \left (c+d x^3\right )^2} \, dx}{a^2 \left (a+b x^3\right )^{2/3}}\\ &=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {1}{3};\frac {8}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 c^2 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [B] time = 0.99, size = 550, normalized size = 8.87 \[ \frac {\frac {b d x^4 \left (\frac {b x^3}{a}+1\right )^{2/3} \left (5 a^2 d^2+21 a b c d-6 b^2 c^2\right ) F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{(a d-b c)^3}+\frac {4 c \left (x^4 \left (5 a^4 d^3+10 a^3 b d^3 x^3+a^2 b^2 d \left (24 c^2+24 c d x^3+5 d^2 x^6\right )+3 a b^3 c \left (-3 c^2+4 c d x^3+7 d^2 x^6\right )-6 b^4 c^2 x^3 \left (c+d x^3\right )\right ) \left (3 a d F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c F_1\left (\frac {4}{3};\frac {5}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )-4 a c x \left (15 a^4 d^3+5 a^3 b d^2 \left (4 d x^3-9 c\right )+a^2 b^2 d \left (45 c^2-21 c d x^3+5 d^2 x^6\right )+3 a b^3 c \left (-5 c^2+11 c d x^3+7 d^2 x^6\right )-6 b^4 c^2 x^3 \left (2 c+d x^3\right )\right ) F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}{\left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)^3 \left (4 a c F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-x^3 \left (3 a d F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c F_1\left (\frac {4}{3};\frac {5}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}}{60 a^2 c^2 \left (a+b x^3\right )^{2/3}} \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {8}{3}} {\left (d x^{3} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {8}{3}} \left (d \,x^{3}+c \right )^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {8}{3}} {\left (d x^{3} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{8/3}\,{\left (d\,x^3+c\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{3}\right )^{\frac {8}{3}} \left (c + d x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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