Optimal. Leaf size=177 \[ \frac{(e+f x)^{m+1} \left (a+b \cot ^{-1}(c+d x)\right )}{f (m+1)}+\frac{i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-c f+i f}\right )}{2 f (m+1) (m+2) (d e+(-c+i) f)}-\frac{i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right )}{2 f (m+1) (m+2) (d e-(c+i) f)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.243636, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5048, 4863, 712, 68} \[ \frac{(e+f x)^{m+1} \left (a+b \cot ^{-1}(c+d x)\right )}{f (m+1)}+\frac{i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-c f+i f}\right )}{2 f (m+1) (m+2) (d e+(-c+i) f)}-\frac{i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right )}{2 f (m+1) (m+2) (d e-(c+i) f)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5048
Rule 4863
Rule 712
Rule 68
Rubi steps
\begin{align*} \int (e+f x)^m \left (a+b \cot ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^m \left (a+b \cot ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{(e+f x)^{1+m} \left (a+b \cot ^{-1}(c+d x)\right )}{f (1+m)}+\frac{b \operatorname{Subst}\left (\int \frac{\left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^{1+m}}{1+x^2} \, dx,x,c+d x\right )}{f (1+m)}\\ &=\frac{(e+f x)^{1+m} \left (a+b \cot ^{-1}(c+d x)\right )}{f (1+m)}+\frac{b \operatorname{Subst}\left (\int \left (\frac{i \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^{1+m}}{2 (i-x)}+\frac{i \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^{1+m}}{2 (i+x)}\right ) \, dx,x,c+d x\right )}{f (1+m)}\\ &=\frac{(e+f x)^{1+m} \left (a+b \cot ^{-1}(c+d x)\right )}{f (1+m)}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^{1+m}}{i-x} \, dx,x,c+d x\right )}{2 f (1+m)}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^{1+m}}{i+x} \, dx,x,c+d x\right )}{2 f (1+m)}\\ &=\frac{(e+f x)^{1+m} \left (a+b \cot ^{-1}(c+d x)\right )}{f (1+m)}+\frac{i b d (e+f x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac{d (e+f x)}{d e+i f-c f}\right )}{2 f (d e+(i-c) f) (1+m) (2+m)}-\frac{i b d (e+f x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac{d (e+f x)}{d e-(i+c) f}\right )}{2 f (d e-(i+c) f) (1+m) (2+m)}\\ \end{align*}
Mathematica [A] time = 0.326997, size = 162, normalized size = 0.92 \[ \frac{(e+f x)^{m+1} \left (2 \left (a+b \cot ^{-1}(c+d x)\right )+\frac{b d (e+f x) \left ((d e-(c+i) f) \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-(c-i) f}\right )+(-d e+(c-i) f) \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right )\right )}{(m+2) (i c f-i d e+f) (d e-(c+i) f)}\right )}{2 f (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 1.395, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) ^{m} \left ( a+b{\rm arccot} \left (dx+c\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 \operatorname{arccot}\left (d x + c\right ) + a\right )}{\left (f x + e\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arccot}\left (d x + c\right ) + a\right )}{\left (f x + e\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]