Optimal. Leaf size=75 \[ \frac{(d x)^{m+1} \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{d (m+1)}-\frac{3 b c (d x)^{m+4} \text{Hypergeometric2F1}\left (1,\frac{m+4}{6},\frac{m+10}{6},-c^2 x^6\right )}{d^4 (m+1) (m+4)} \]
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Rubi [A] time = 0.0424709, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5033, 16, 364} \[ \frac{(d x)^{m+1} \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{d (m+1)}-\frac{3 b c (d x)^{m+4} \, _2F_1\left (1,\frac{m+4}{6};\frac{m+10}{6};-c^2 x^6\right )}{d^4 (m+1) (m+4)} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 16
Rule 364
Rubi steps
\begin{align*} \int (d x)^m \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{(d x)^{1+m} \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{d (1+m)}-\frac{(3 b c) \int \frac{x^2 (d x)^{1+m}}{1+c^2 x^6} \, dx}{d (1+m)}\\ &=\frac{(d x)^{1+m} \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{d (1+m)}-\frac{(3 b c) \int \frac{(d x)^{3+m}}{1+c^2 x^6} \, dx}{d^3 (1+m)}\\ &=\frac{(d x)^{1+m} \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{d (1+m)}-\frac{3 b c (d x)^{4+m} \, _2F_1\left (1,\frac{4+m}{6};\frac{10+m}{6};-c^2 x^6\right )}{d^4 (1+m) (4+m)}\\ \end{align*}
Mathematica [A] time = 0.0611742, size = 65, normalized size = 0.87 \[ -\frac{x (d x)^m \left (3 b c x^3 \text{Hypergeometric2F1}\left (1,\frac{m+4}{6},\frac{m+10}{6},-c^2 x^6\right )-(m+4) \left (a+b \tan ^{-1}\left (c x^3\right )\right )\right )}{(m+1) (m+4)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.239, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b\arctan \left ( c{x}^{3} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (b \arctan \left (c x^{3}\right ) + a\right )} \left (d x\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 (b \arctan \left (c x^{3}\right ) + a\right )} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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