Optimal. Leaf size=167 \[ \frac {x \left (c+d x^3\right )^{q+1} \left (a^2 d^2 \left (9 q^2+33 q+28\right )-2 a b c d (3 q+7)+4 b^2 c^2\right ) \, _2F_1\left (1,q+\frac {4}{3};\frac {4}{3};-\frac {d x^3}{c}\right )}{c d^2 (3 q+4) (3 q+7)}-\frac {b x \left (c+d x^3\right )^{q+1} (4 b c-a d (3 q+10))}{d^2 (3 q+4) (3 q+7)}+\frac {b x \left (a+b x^3\right ) \left (c+d x^3\right )^{q+1}}{d (3 q+7)} \]
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Rubi [A] time = 0.13, antiderivative size = 176, normalized size of antiderivative = 1.05, 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} \[ \frac {x \left (c+d x^3\right )^q \left (\frac {d x^3}{c}+1\right )^{-q} \left (a^2 d^2 \left (9 q^2+33 q+28\right )-2 a b c d (3 q+7)+4 b^2 c^2\right ) \, _2F_1\left (\frac {1}{3},-q;\frac {4}{3};-\frac {d x^3}{c}\right )}{d^2 (3 q+4) (3 q+7)}-\frac {b x \left (c+d x^3\right )^{q+1} (4 b c-a d (3 q+10))}{d^2 (3 q+4) (3 q+7)}+\frac {b x \left (a+b x^3\right ) \left (c+d x^3\right )^{q+1}}{d (3 q+7)} \]
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 )^2 \left (c+d x^3\right )^q \, dx &=\frac {b x \left (a+b x^3\right ) \left (c+d x^3\right )^{1+q}}{d (7+3 q)}+\frac {\int \left (c+d x^3\right )^q \left (-a (b c-a d (7+3 q))-b (4 b c-a d (10+3 q)) x^3\right ) \, dx}{d (7+3 q)}\\ &=-\frac {b (4 b c-a d (10+3 q)) x \left (c+d x^3\right )^{1+q}}{d^2 (4+3 q) (7+3 q)}+\frac {b x \left (a+b x^3\right ) \left (c+d x^3\right )^{1+q}}{d (7+3 q)}+\frac {\left (4 b^2 c^2-2 a b c d (7+3 q)+a^2 d^2 \left (28+33 q+9 q^2\right )\right ) \int \left (c+d x^3\right )^q \, dx}{d^2 (4+3 q) (7+3 q)}\\ &=-\frac {b (4 b c-a d (10+3 q)) x \left (c+d x^3\right )^{1+q}}{d^2 (4+3 q) (7+3 q)}+\frac {b x \left (a+b x^3\right ) \left (c+d x^3\right )^{1+q}}{d (7+3 q)}+\frac {\left (\left (4 b^2 c^2-2 a b c d (7+3 q)+a^2 d^2 \left (28+33 q+9 q^2\right )\right ) \left (c+d x^3\right )^q \left (1+\frac {d x^3}{c}\right )^{-q}\right ) \int \left (1+\frac {d x^3}{c}\right )^q \, dx}{d^2 (4+3 q) (7+3 q)}\\ &=-\frac {b (4 b c-a d (10+3 q)) x \left (c+d x^3\right )^{1+q}}{d^2 (4+3 q) (7+3 q)}+\frac {b x \left (a+b x^3\right ) \left (c+d x^3\right )^{1+q}}{d (7+3 q)}+\frac {\left (4 b^2 c^2-2 a b c d (7+3 q)+a^2 d^2 \left (28+33 q+9 q^2\right )\right ) x \left (c+d x^3\right )^q \left (1+\frac {d x^3}{c}\right )^{-q} \, _2F_1\left (\frac {1}{3},-q;\frac {4}{3};-\frac {d x^3}{c}\right )}{d^2 (4+3 q) (7+3 q)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 106, normalized size = 0.63 \[ \frac {1}{14} x \left (c+d x^3\right )^q \left (\frac {d x^3}{c}+1\right )^{-q} \left (14 a^2 \, _2F_1\left (\frac {1}{3},-q;\frac {4}{3};-\frac {d x^3}{c}\right )+b x^3 \left (7 a \, _2F_1\left (\frac {4}{3},-q;\frac {7}{3};-\frac {d x^3}{c}\right )+2 b x^3 \, _2F_1\left (\frac {7}{3},-q;\frac {10}{3};-\frac {d x^3}{c}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} {\left (d x^{3} + c\right )}^{q}, x\right ) \]
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 \[ \int {\left (b x^{3} + a\right )}^{2} {\left (d x^{3} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{3}+a \right )^{2} \left (d \,x^{3}+c \right )^{q}\, dx \]
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 \[ \int {\left (b x^{3} + a\right )}^{2} {\left (d x^{3} + c\right )}^{q}\,{d x} \]
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 \[ \int {\left (b\,x^3+a\right )}^2\,{\left (d\,x^3+c\right )}^q \,d x \]
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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