Optimal. Leaf size=298 \[ -\frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (-12 a^2 b c d^2 (3 m+10)+28 a^3 d^3+3 a b^2 c^2 d \left (9 m^2+51 m+70\right )-b^3 c^3 \left (27 m^3+189 m^2+414 m+280\right )\right ) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )}{b^3 (3 m+4) (3 m+7) (3 m+10)}+\frac{d x \left (a+b x^3\right )^{m+1} \left (28 a^2 d^2-a b c d (15 m+92)+b^2 c^2 \left (9 m^2+60 m+118\right )\right )}{b^3 (3 m+4) (3 m+7) (3 m+10)}-\frac{d x \left (c+d x^3\right ) \left (a+b x^3\right )^{m+1} (7 a d-b c (3 m+16))}{b^2 (3 m+7) (3 m+10)}+\frac{d x \left (c+d x^3\right )^2 \left (a+b x^3\right )^{m+1}}{b (3 m+10)} \]
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Rubi [A] time = 0.303879, antiderivative size = 298, normalized size of antiderivative = 1., 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 416
Rule 528
Rule 388
Rule 246
Rule 245
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*}
Mathematica [A] time = 5.06369, size = 137, normalized size = 0.46 \[ \frac{1}{140} x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (d x^3 \left (105 c^2 \, _2F_1\left (\frac{4}{3},-m;\frac{7}{3};-\frac{b x^3}{a}\right )+2 d x^3 \left (30 c \, _2F_1\left (\frac{7}{3},-m;\frac{10}{3};-\frac{b x^3}{a}\right )+7 d x^3 \, _2F_1\left (\frac{10}{3},-m;\frac{13}{3};-\frac{b x^3}{a}\right )\right )\right )+140 c^3 \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.41, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{3}+a \right ) ^{m} \left ( d{x}^{3}+c \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{3} + c\right )}^{3}{\left (b x^{3} + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{3} + c\right )}^{3}{\left (b x^{3} + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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