Optimal. Leaf size=73 \[ \frac{(d x)^{m+1} \left (a+b \tan ^{-1}(c x)\right )}{d (m+1)}-\frac{b c (d x)^{m+2} \text{Hypergeometric2F1}\left (1,\frac{m+2}{2},\frac{m+4}{2},-c^2 x^2\right )}{d^2 (m+1) (m+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0311376, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4852, 364} \[ \frac{(d x)^{m+1} \left (a+b \tan ^{-1}(c x)\right )}{d (m+1)}-\frac{b c (d x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right )}{d^2 (m+1) (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4852
Rule 364
Rubi steps
\begin{align*} \int (d x)^m \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{(d x)^{1+m} \left (a+b \tan ^{-1}(c x)\right )}{d (1+m)}-\frac{(b c) \int \frac{(d x)^{1+m}}{1+c^2 x^2} \, dx}{d (1+m)}\\ &=\frac{(d x)^{1+m} \left (a+b \tan ^{-1}(c x)\right )}{d (1+m)}-\frac{b c (d x)^{2+m} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-c^2 x^2\right )}{d^2 (1+m) (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0313844, size = 60, normalized size = 0.82 \[ -\frac{x (d x)^m \left (b c x \text{Hypergeometric2F1}\left (1,\frac{m}{2}+1,\frac{m}{2}+2,-c^2 x^2\right )-(m+2) \left (a+b \tan ^{-1}(c x)\right )\right )}{(m+1) (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.434, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b\arctan \left ( cx \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \arctan \left (c x\right ) + a\right )} \left (d x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \left (a + b \operatorname{atan}{\left (c x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \left (c x\right ) + a\right )} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]