Optimal. Leaf size=298 \[ \frac {d x \left (a+b x^3\right )^{m+1} \left (28 a^2 d^2-a b c d (15 m+92)+b^2 c^2 \left (9 m^2+60 m+118\right )\right )}{b^3 (3 m+4) (3 m+7) (3 m+10)}-\frac {x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \left (28 a^3 d^3-12 a^2 b c d^2 (3 m+10)+3 a b^2 c^2 d \left (9 m^2+51 m+70\right )-b^3 c^3 \left (27 m^3+189 m^2+414 m+280\right )\right ) \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )}{b^3 (3 m+4) (3 m+7) (3 m+10)}-\frac {d x \left (c+d x^3\right ) \left (a+b x^3\right )^{m+1} (7 a d-b c (3 m+16))}{b^2 (3 m+7) (3 m+10)}+\frac {d x \left (c+d x^3\right )^2 \left (a+b x^3\right )^{m+1}}{b (3 m+10)} \]
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Rubi [A] time = 0.30, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {416, 528, 388, 246, 245} \[ -\frac {x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \left (-12 a^2 b c d^2 (3 m+10)+28 a^3 d^3+3 a b^2 c^2 d \left (9 m^2+51 m+70\right )-b^3 c^3 \left (27 m^3+189 m^2+414 m+280\right )\right ) \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )}{b^3 (3 m+4) (3 m+7) (3 m+10)}+\frac {d x \left (a+b x^3\right )^{m+1} \left (28 a^2 d^2-a b c d (15 m+92)+b^2 c^2 \left (9 m^2+60 m+118\right )\right )}{b^3 (3 m+4) (3 m+7) (3 m+10)}-\frac {d x \left (c+d x^3\right ) \left (a+b x^3\right )^{m+1} (7 a d-b c (3 m+16))}{b^2 (3 m+7) (3 m+10)}+\frac {d x \left (c+d x^3\right )^2 \left (a+b x^3\right )^{m+1}}{b (3 m+10)} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 388
Rule 416
Rule 528
Rubi steps
\begin {align*} \int \left (a+b x^3\right )^m \left (c+d x^3\right )^3 \, dx &=\frac {d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )^2}{b (10+3 m)}+\frac {\int \left (a+b x^3\right )^m \left (c+d x^3\right ) \left (-c (a d-b c (10+3 m))-d (7 a d-b c (16+3 m)) x^3\right ) \, dx}{b (10+3 m)}\\ &=-\frac {d (7 a d-b c (16+3 m)) x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b^2 (7+3 m) (10+3 m)}+\frac {d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )^2}{b (10+3 m)}+\frac {\int \left (a+b x^3\right )^m \left (c \left (7 a^2 d^2-a b c d (23+6 m)+b^2 c^2 \left (70+51 m+9 m^2\right )\right )+d \left (28 a^2 d^2-a b c d (92+15 m)+b^2 c^2 \left (118+60 m+9 m^2\right )\right ) x^3\right ) \, dx}{b^2 (7+3 m) (10+3 m)}\\ &=\frac {d \left (28 a^2 d^2-a b c d (92+15 m)+b^2 c^2 \left (118+60 m+9 m^2\right )\right ) x \left (a+b x^3\right )^{1+m}}{b^3 (4+3 m) (7+3 m) (10+3 m)}-\frac {d (7 a d-b c (16+3 m)) x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b^2 (7+3 m) (10+3 m)}+\frac {d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )^2}{b (10+3 m)}-\frac {\left (28 a^3 d^3-12 a^2 b c d^2 (10+3 m)+3 a b^2 c^2 d \left (70+51 m+9 m^2\right )-b^3 c^3 \left (280+414 m+189 m^2+27 m^3\right )\right ) \int \left (a+b x^3\right )^m \, dx}{b^3 (4+3 m) (7+3 m) (10+3 m)}\\ &=\frac {d \left (28 a^2 d^2-a b c d (92+15 m)+b^2 c^2 \left (118+60 m+9 m^2\right )\right ) x \left (a+b x^3\right )^{1+m}}{b^3 (4+3 m) (7+3 m) (10+3 m)}-\frac {d (7 a d-b c (16+3 m)) x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b^2 (7+3 m) (10+3 m)}+\frac {d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )^2}{b (10+3 m)}-\frac {\left (\left (28 a^3 d^3-12 a^2 b c d^2 (10+3 m)+3 a b^2 c^2 d \left (70+51 m+9 m^2\right )-b^3 c^3 \left (280+414 m+189 m^2+27 m^3\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^3 (4+3 m) (7+3 m) (10+3 m)}\\ &=\frac {d \left (28 a^2 d^2-a b c d (92+15 m)+b^2 c^2 \left (118+60 m+9 m^2\right )\right ) x \left (a+b x^3\right )^{1+m}}{b^3 (4+3 m) (7+3 m) (10+3 m)}-\frac {d (7 a d-b c (16+3 m)) x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )}{b^2 (7+3 m) (10+3 m)}+\frac {d x \left (a+b x^3\right )^{1+m} \left (c+d x^3\right )^2}{b (10+3 m)}-\frac {\left (28 a^3 d^3-12 a^2 b c d^2 (10+3 m)+3 a b^2 c^2 d \left (70+51 m+9 m^2\right )-b^3 c^3 \left (280+414 m+189 m^2+27 m^3\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^3 (4+3 m) (7+3 m) (10+3 m)}\\ \end {align*}
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Mathematica [A] time = 5.08, size = 137, normalized size = 0.46 \[ \frac {1}{140} x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} \left (140 c^3 \, _2F_1\left (\frac {1}{3},-m;\frac {4}{3};-\frac {b x^3}{a}\right )+d x^3 \left (105 c^2 \, _2F_1\left (\frac {4}{3},-m;\frac {7}{3};-\frac {b x^3}{a}\right )+2 d x^3 \left (30 c \, _2F_1\left (\frac {7}{3},-m;\frac {10}{3};-\frac {b x^3}{a}\right )+7 d x^3 \, _2F_1\left (\frac {10}{3},-m;\frac {13}{3};-\frac {b x^3}{a}\right )\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.32, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{3} x^{9} + 3 \, c d^{2} x^{6} + 3 \, c^{2} d x^{3} + c^{3}\right )} {\left (b x^{3} + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (d x^{3} + c\right )}^{3} {\left (b x^{3} + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{3}+c \right )^{3} \left (b \,x^{3}+a \right )^{m}\, dx \]
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 \[ \int {\left (d x^{3} + c\right )}^{3} {\left (b x^{3} + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (b\,x^3+a\right )}^m\,{\left (d\,x^3+c\right )}^3 \,d x \]
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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