Optimal. Leaf size=50 \[ a x+\frac {b (f+g x) \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )}{g}+\frac {b e p \log (d (f+g x)+e)}{d g} \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2483, 2448, 263, 31} \[ a x+\frac {b (f+g x) \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )}{g}+\frac {b e p \log (d (f+g x)+e)}{d g} \]
Antiderivative was successfully verified.
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Rule 31
Rule 263
Rule 2448
Rule 2483
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right ) \, dx\\ &=a x+\frac {b \operatorname {Subst}\left (\int \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx,x,f+g x\right )}{g}\\ &=a x+\frac {b (f+g x) \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )}{g}+\frac {(b e p) \operatorname {Subst}\left (\int \frac {1}{\left (d+\frac {e}{x}\right ) x} \, dx,x,f+g x\right )}{g}\\ &=a x+\frac {b (f+g x) \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )}{g}+\frac {(b e p) \operatorname {Subst}\left (\int \frac {1}{e+d x} \, dx,x,f+g x\right )}{g}\\ &=a x+\frac {b (f+g x) \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )}{g}+\frac {b e p \log (e+d (f+g x))}{d g}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 70, normalized size = 1.40 \[ a x+b x \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )-b e g p \left (\frac {f \log (f+g x)}{e g^2}-\frac {(d f+e) \log (d f+d g x+e)}{d e g^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 76, normalized size = 1.52 \[ \frac {b d g p x \log \left (\frac {d g x + d f + e}{g x + f}\right ) - b d f p \log \left (g x + f\right ) + b d g x \log \relax (c) + a d g x + {\left (b d f + b e\right )} p \log \left (d g x + d f + e\right )}{d g} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 177, normalized size = 3.54 \[ \frac {{\left (d f g e^{\left (-2\right )} - {\left (d f + e\right )} g e^{\left (-2\right )}\right )} {\left (d p e^{2} \log \left (-d + \frac {d g x + d f + e}{g x + f}\right ) + d e^{2} \log \relax (c) - \frac {{\left (d g x + d f + e\right )} p e^{2} \log \left (-d + \frac {d g x + d f + e}{g x + f}\right )}{g x + f} + \frac {{\left (d g x + d f + e\right )} p e^{2} \log \left (\frac {d g x + d f + e}{g x + f}\right )}{g x + f}\right )} b}{d^{2} g^{2} - \frac {{\left (d g x + d f + e\right )} d g^{2}}{g x + f}} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 81, normalized size = 1.62 \[ -\frac {b f p \ln \left (g x +f \right )}{g}+\frac {b f p \ln \left (d g x +d f +e \right )}{g}+b x \ln \left (c \left (\frac {d g x +d f +e}{g x +f}\right )^{p}\right )+a x +\frac {b e p \ln \left (d g x +d f +e \right )}{d g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 70, normalized size = 1.40 \[ -b e g p {\left (\frac {f \log \left (g x + f\right )}{e g^{2}} - \frac {{\left (d f + e\right )} \log \left (d g x + d f + e\right )}{d e g^{2}}\right )} + b x \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 61, normalized size = 1.22 \[ a\,x+b\,x\,\ln \left (c\,{\left (d+\frac {e}{f+g\,x}\right )}^p\right )-\frac {b\,f\,p\,\ln \left (f+g\,x\right )}{g}+\frac {b\,p\,\ln \left (e+d\,f+d\,g\,x\right )\,\left (e+d\,f\right )}{d\,g} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.05, size = 114, normalized size = 2.28 \[ a x + b \left (\begin {cases} x \log {\left (c \left (\frac {e}{f}\right )^{p} \right )} & \text {for}\: d = 0 \wedge g = 0 \\x \log {\left (c \left (d + \frac {e}{f}\right )^{p} \right )} & \text {for}\: g = 0 \\- \frac {f p \log {\left (f + g x \right )}}{g} + p x \log {\relax (e )} - p x \log {\left (f + g x \right )} + p x + x \log {\relax (c )} & \text {for}\: d = 0 \\\frac {f p \log {\left (d + \frac {e}{f + g x} \right )}}{g} + p x \log {\left (d + \frac {e}{f + g x} \right )} + x \log {\relax (c )} + \frac {e p \log {\left (d f + d g x + e \right )}}{d g} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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