Optimal. Leaf size=176 \[ \frac {x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \left (4 a^2 d^2-2 a b c d (3 m+7)+b^2 c^2 \left (9 m^2+33 m+28\right )\right ) \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )}{b^2 (3 m+4) (3 m+7)}-\frac {d x \left (a+b x^3\right )^{m+1} (4 a d-b c (3 m+10))}{b^2 (3 m+4) (3 m+7)}+\frac {d x \left (c+d x^3\right ) \left (a+b x^3\right )^{m+1}}{b (3 m+7)} \]
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Rubi [A] time = 0.13, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {416, 388, 246, 245} \[ \frac {x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \left (4 a^2 d^2-2 a b c d (3 m+7)+b^2 c^2 \left (9 m^2+33 m+28\right )\right ) \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )}{b^2 (3 m+4) (3 m+7)}-\frac {d x \left (a+b x^3\right )^{m+1} (4 a d-b c (3 m+10))}{b^2 (3 m+4) (3 m+7)}+\frac {d x \left (c+d x^3\right ) \left (a+b x^3\right )^{m+1}}{b (3 m+7)} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 388
Rule 416
Rubi steps
\begin {align*} \int \left (a+b x^3\right )^m \left (c+d x^3\right )^2 \, dx &=\frac {d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b (7+3 m)}+\frac {\int \left (a+b x^3\right )^m \left (-c (a d-b c (7+3 m))-d (4 a d-b c (10+3 m)) x^3\right ) \, dx}{b (7+3 m)}\\ &=-\frac {d (4 a d-b c (10+3 m)) x \left (a+b x^3\right )^{1+m}}{b^2 (4+3 m) (7+3 m)}+\frac {d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b (7+3 m)}+\frac {\left (4 a^2 d^2-2 a b c d (7+3 m)+b^2 c^2 \left (28+33 m+9 m^2\right )\right ) \int \left (a+b x^3\right )^m \, dx}{b^2 (4+3 m) (7+3 m)}\\ &=-\frac {d (4 a d-b c (10+3 m)) x \left (a+b x^3\right )^{1+m}}{b^2 (4+3 m) (7+3 m)}+\frac {d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b (7+3 m)}+\frac {\left (\left (4 a^2 d^2-2 a b c d (7+3 m)+b^2 c^2 \left (28+33 m+9 m^2\right )\right ) \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m}\right ) \int \left (1+\frac {b x^3}{a}\right )^m \, dx}{b^2 (4+3 m) (7+3 m)}\\ &=-\frac {d (4 a d-b c (10+3 m)) x \left (a+b x^3\right )^{1+m}}{b^2 (4+3 m) (7+3 m)}+\frac {d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b (7+3 m)}+\frac {\left (4 a^2 d^2-2 a b c d (7+3 m)+b^2 c^2 \left (28+33 m+9 m^2\right )\right ) x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )}{b^2 (4+3 m) (7+3 m)}\\ \end {align*}
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Mathematica [A] time = 5.04, size = 106, normalized size = 0.60 \[ \frac {1}{14} x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \left (14 c^2 \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )+d x^3 \left (7 c \, _2F_1\left (\frac {4}{3},-m;\frac {7}{3};-\frac {b x^3}{a}\right )+2 d x^3 \, _2F_1\left (\frac {7}{3},-m;\frac {10}{3};-\frac {b x^3}{a}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{2} x^{6} + 2 \, c d x^{3} + c^{2}\right )} {\left (b x^{3} + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (d x^{3} + c\right )}^{2} {\left (b x^{3} + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{3}+c \right )^{2} \left (b \,x^{3}+a \right )^{m}\, dx \]
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 \[ \int {\left (d x^{3} + c\right )}^{2} {\left (b x^{3} + a\right )}^{m}\,{d x} \]
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 \[ \int {\left (b\,x^3+a\right )}^m\,{\left (d\,x^3+c\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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