Optimal. Leaf size=147 \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \left (2 a^2 d^2+a b c d+2 b^2 c^2\right ) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{5 a^2 b^2 \left (a+b x^3\right )^{2/3}}+\frac {2 x \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right )}{5 \left (a+b x^3\right )^{2/3}}+\frac {x \left (c+d x^3\right ) (b c-a d)}{5 a b \left (a+b x^3\right )^{5/3}} \]
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Rubi [A] time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {413, 385, 246, 245} \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \left (2 a^2 d^2+a b c d+2 b^2 c^2\right ) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{5 a^2 b^2 \left (a+b x^3\right )^{2/3}}+\frac {2 x \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right )}{5 \left (a+b x^3\right )^{2/3}}+\frac {x \left (c+d x^3\right ) (b c-a d)}{5 a b \left (a+b x^3\right )^{5/3}} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 385
Rule 413
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{8/3}} \, dx &=\frac {(b c-a d) x \left (c+d x^3\right )}{5 a b \left (a+b x^3\right )^{5/3}}+\frac {\int \frac {c (4 b c+a d)+d (b c+4 a d) x^3}{\left (a+b x^3\right )^{5/3}} \, dx}{5 a b}\\ &=\frac {2 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) x}{5 \left (a+b x^3\right )^{2/3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{5 a b \left (a+b x^3\right )^{5/3}}+\frac {\left (2 b^2 c^2+a b c d+2 a^2 d^2\right ) \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx}{5 a^2 b^2}\\ &=\frac {2 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) x}{5 \left (a+b x^3\right )^{2/3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{5 a b \left (a+b x^3\right )^{5/3}}+\frac {\left (\left (2 b^2 c^2+a b c d+2 a^2 d^2\right ) \left (1+\frac {b x^3}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{5 a^2 b^2 \left (a+b x^3\right )^{2/3}}\\ &=\frac {2 \left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) x}{5 \left (a+b x^3\right )^{2/3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{5 a b \left (a+b x^3\right )^{5/3}}+\frac {\left (2 b^2 c^2+a b c d+2 a^2 d^2\right ) x \left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )}{5 a^2 b^2 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 5.16, size = 128, normalized size = 0.87 \[ \frac {x \left (\left (a+b x^3\right ) \left (\frac {b x^3}{a}+1\right )^{2/3} \left (2 a^2 d^2+a b c d+2 b^2 c^2\right ) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+\left (a+b x^3\right ) \left (-3 a^2 d^2+a b c d+2 b^2 c^2\right )+a (b c-a d)^2\right )}{5 a^2 b^2 \left (a+b x^3\right )^{5/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{2} x^{6} + 2 \, c d x^{3} + c^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {8}{3}}}\,{d x} \]
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 \[ \int \frac {\left (d \,x^{3}+c \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {8}{3}}}\, dx \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {8}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,x^3+c\right )}^2}{{\left (b\,x^3+a\right )}^{8/3}} \,d x \]
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 \[ \int \frac {\left (c + d x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac {8}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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