Optimal. Leaf size=177 \[ \frac {(e+f x)^{m+1} \left (a+b \tan ^{-1}(c+d x)\right )}{f (m+1)}-\frac {i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {d (e+f x)}{d e-c f+i f}\right )}{2 f (m+1) (m+2) (d e+(-c+i) f)}+\frac {i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {d (e+f x)}{d e-(c+i) f}\right )}{2 f (m+1) (m+2) (d e-(c+i) f)} \]
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Rubi [A] time = 0.25, 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 = {5047, 4862, 712, 68} \[ -\frac {i b d (e+f x)^{m+2} \text {Hypergeometric2F1}\left (1,m+2,m+3,\frac {d (e+f x)}{-c f+d e+i f}\right )}{2 f (m+1) (m+2) (d e+(-c+i) f)}+\frac {i b d (e+f x)^{m+2} \text {Hypergeometric2F1}\left (1,m+2,m+3,\frac {d (e+f x)}{d e-(c+i) f}\right )}{2 f (m+1) (m+2) (d e-(c+i) f)}+\frac {(e+f x)^{m+1} \left (a+b \tan ^{-1}(c+d x)\right )}{f (m+1)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 712
Rule 4862
Rule 5047
Rubi steps
\begin {align*} \int (e+f x)^m \left (a+b \tan ^{-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 \tan ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^{1+m} \left (a+b \tan ^{-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 \tan ^{-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 \tan ^{-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 \tan ^{-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 = 0.39, size = 162, normalized size = 0.92 \[ \frac {(e+f x)^{m+1} \left (2 \left (a+b \tan ^{-1}(c+d x)\right )+\frac {b d (e+f x) \left ((d e-(c+i) f) \, _2F_1\left (1,m+2;m+3;\frac {d (e+f x)}{d e-(c-i) f}\right )+(-d e+(c-i) f) \, _2F_1\left (1,m+2;m+3;\frac {d (e+f x)}{d e-(c+i) f}\right )\right )}{(m+2) (-i c f+i d e+f) (d e-(c-i) f)}\right )}{2 f (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \arctan \left (d x + c\right ) + a\right )} {\left (f x + e\right )}^{m}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.96, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right )^{m} \left (a +b \arctan \left (d x +c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (e+f\,x\right )}^m\,\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right ) \,d x \]
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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