Optimal. Leaf size=162 \[ \frac {(e+f x)^{m+1} \left (a+b \tanh ^{-1}(c+d x)\right )}{f (m+1)}+\frac {b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {d (e+f x)}{d e-c f-f}\right )}{2 f (m+1) (m+2) (d e-(c+1) f)}-\frac {b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {d (e+f x)}{d e-c f+f}\right )}{2 f (m+1) (m+2) (-c f+d e+f)} \]
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Rubi [A] time = 0.26, antiderivative size = 162, 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 = {6111, 5926, 712, 68} \[ \frac {(e+f x)^{m+1} \left (a+b \tanh ^{-1}(c+d x)\right )}{f (m+1)}+\frac {b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {d (e+f x)}{d e-c f-f}\right )}{2 f (m+1) (m+2) (d e-(c+1) f)}-\frac {b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {d (e+f x)}{d e-c f+f}\right )}{2 f (m+1) (m+2) (-c f+d e+f)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 712
Rule 5926
Rule 6111
Rubi steps
\begin {align*} \int (e+f x)^m \left (a+b \tanh ^{-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 \tanh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^{1+m} \left (a+b \tanh ^{-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 \tanh ^{-1}(c+d x)\right )}{f (1+m)}-\frac {b \operatorname {Subst}\left (\int \left (\frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^{1+m}}{2 (1-x)}+\frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^{1+m}}{2 (1+x)}\right ) \, dx,x,c+d x\right )}{f (1+m)}\\ &=\frac {(e+f x)^{1+m} \left (a+b \tanh ^{-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} \, dx,x,c+d x\right )}{2 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} \, dx,x,c+d x\right )}{2 f (1+m)}\\ &=\frac {(e+f x)^{1+m} \left (a+b \tanh ^{-1}(c+d x)\right )}{f (1+m)}+\frac {b d (e+f x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {d (e+f x)}{d e-f-c f}\right )}{2 f (d e-(1+c) f) (1+m) (2+m)}-\frac {b d (e+f x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {d (e+f x)}{d e+f-c f}\right )}{2 f (d e+f-c f) (1+m) (2+m)}\\ \end {align*}
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Mathematica [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int (e+f x)^m \left (a+b \tanh ^{-1}(c+d x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )} {\left (f x + e\right )}^{m}, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )} {\left (f x + e\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.42, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right )^{m} \left (a +b \arctanh \left (d x +c \right )\right )\, dx \]
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 \[ -\frac {1}{2} \, b {\left (\frac {{\left (f x + e\right )} {\left (f x + e\right )}^{m} \log \left (-d x - c + 1\right )}{f {\left (m + 1\right )}} - \int \frac {{\left (d f x + d e + {\left (d f {\left (m + 1\right )} x + c f {\left (m + 1\right )} - f {\left (m + 1\right )}\right )} \log \left (d x + c + 1\right )\right )} {\left (f x + e\right )}^{m}}{d f {\left (m + 1\right )} x + c f {\left (m + 1\right )} - f {\left (m + 1\right )}}\,{d x}\right )} + \frac {{\left (f x + e\right )}^{m + 1} a}{f {\left (m + 1\right )}} \]
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 \[ \int {\left (e+f\,x\right )}^m\,\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atanh}{\left (c + d x \right )}\right ) \left (e + f x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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