Optimal. Leaf size=177 \[ \frac {(e+f x)^{m+1} \left (a+b \cot ^{-1}(c+d x)\right )}{f (m+1)}+\frac {i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {d (e+f x)}{d e-c f+i f}\right )}{2 f (m+1) (m+2) (d e+(-c+i) f)}-\frac {i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {d (e+f x)}{d e-(c+i) f}\right )}{2 f (m+1) (m+2) (d e-(c+i) f)} \]
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Rubi [A] time = 0.24, antiderivative size = 177, normalized size of antiderivative = 1.00, 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.
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Rule 68
Rule 712
Rule 4863
Rule 5048
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*}
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Mathematica [A] time = 0.36, 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.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )} {\left (f x + e\right )}^{m}, x\right ) \]
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 \[ \int {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )} {\left (f x + e\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.11, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right )^{m} \left (a +b \,\mathrm {arccot}\left (d x +c \right )\right )\, dx \]
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 \[ \frac {\frac {1}{2} \, {\left (3 \, {\left (f x \arctan \left (1, d x + c\right ) + e \arctan \left (1, d x + c\right )\right )} {\left (f x + e\right )}^{m} + 2 \, {\left (f m + f\right )} \int \frac {{\left ({\left (c^{2} \arctan \left (1, d x + c\right ) + \arctan \left (1, d x + c\right )\right )} f m + {\left (d^{2} f m \arctan \left (1, d x + c\right ) + d^{2} f \arctan \left (1, d x + c\right )\right )} x^{2} + 3 \, d e + {\left (c^{2} \arctan \left (1, d x + c\right ) + \arctan \left (1, d x + c\right )\right )} f + {\left (2 \, c d f m \arctan \left (1, d x + c\right ) + {\left (2 \, c \arctan \left (1, d x + c\right ) + 3\right )} d f\right )} x\right )} {\left (f x + e\right )}^{m}}{2 \, {\left ({\left (c^{2} + 1\right )} f m + {\left (d^{2} f m + d^{2} f\right )} x^{2} + {\left (c^{2} + 1\right )} f + 2 \, {\left (c d f m + c d f\right )} x\right )}}\,{d x}\right )} b}{2 \, {\left (f m + f\right )}} + \frac {{\left (f x + e\right )}^{m + 1} a}{f {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e+f\,x\right )}^m\,\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right ) \,d x \]
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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