Optimal. Leaf size=115 \[ -\frac {2 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )}{d g}+\frac {(d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}-\frac {2 b^2 e p^2 \text {Li}_2\left (\frac {e}{d (f+g x)}+1\right )}{d g} \]
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Rubi [A] time = 0.09, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2483, 2449, 2454, 2394, 2315} \[ -\frac {2 b^2 e p^2 \text {PolyLog}\left (2,\frac {e}{d (f+g x)}+1\right )}{d g}-\frac {2 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )}{d g}+\frac {(d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2394
Rule 2449
Rule 2454
Rule 2483
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^p\right )\right )^2 \, dx,x,f+g x\right )}{g}\\ &=\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(2 b e p) \operatorname {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{x} \, dx,x,f+g x\right )}{d g}\\ &=\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}-\frac {(2 b e p) \operatorname {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^p\right )}{x} \, dx,x,\frac {1}{f+g x}\right )}{d g}\\ &=-\frac {2 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {\left (2 b^2 e^2 p^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,\frac {1}{f+g x}\right )}{d g}\\ &=-\frac {2 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}-\frac {2 b^2 e p^2 \text {Li}_2\left (1+\frac {e}{d (f+g x)}\right )}{d g}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 219, normalized size = 1.90 \[ x \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2-\frac {b p \left (2 d f \log (f+g x) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )-2 (d f+e) \log (d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )+b d f p \left (\log (f+g x) \left (\log (f+g x)-2 \log \left (\frac {d f+d g x+e}{e}\right )\right )-2 \text {Li}_2\left (-\frac {d (f+g x)}{e}\right )\right )-b p (d f+e) \left (2 \text {Li}_2\left (\frac {e+d f+d g x}{e}\right )+\left (2 \log \left (-\frac {d (f+g x)}{e}\right )-\log (d f+d g x+e)\right ) \log (d (f+g x)+e)\right )\right )}{d g} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \log \left (c \left (\frac {d g x + d f + e}{g x + f}\right )^{p}\right )^{2} + 2 \, a b \log \left (c \left (\frac {d g x + d f + e}{g x + f}\right )^{p}\right ) + a^{2}, x\right ) \]
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 \[ \int {\left (b \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d +\frac {e}{g x +f}\right )^{p}\right )+a \right )^{2}\, dx \]
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 \[ -2 \, a b e g p {\left (\frac {f \log \left (g x + f\right )}{e g^{2}} - \frac {{\left (d f + e\right )} \log \left (d g x + d f + e\right )}{d e g^{2}}\right )} + 2 \, a b x \log \left (c {\left (d + \frac {e}{g x + f}\right )}^{p}\right ) + a^{2} x + b^{2} {\left (\frac {d g x \log \left ({\left (d g x + d f + e\right )}^{p}\right )^{2} + d g x \log \left ({\left (g x + f\right )}^{p}\right )^{2} - {\left (d f p^{2} + e p^{2}\right )} \log \left (d g x + d f + e\right )^{2} + 2 \, {\left (d f p^{2} + e p^{2}\right )} \log \left (d g x + d f + e\right ) \log \left (g x + f\right ) - 2 \, {\left (d f p \log \left (g x + f\right ) + d g x \log \left ({\left (g x + f\right )}^{p}\right ) - d g x \log \relax (c) - {\left (d f p + e p\right )} \log \left (d g x + d f + e\right )\right )} \log \left ({\left (d g x + d f + e\right )}^{p}\right ) + 2 \, {\left (d f p \log \left (g x + f\right ) - d g x \log \relax (c) - {\left (d f p + e p\right )} \log \left (d g x + d f + e\right )\right )} \log \left ({\left (g x + f\right )}^{p}\right )}{d g} - \int -\frac {d g^{2} x^{2} \log \relax (c)^{2} + {\left (d f^{2} + e f\right )} \log \relax (c)^{2} + {\left (2 \, e g p \log \relax (c) + {\left (2 \, d f g + e g\right )} \log \relax (c)^{2}\right )} x - 2 \, {\left (d f^{2} p^{2} + 2 \, e f p^{2} + {\left (d f g p^{2} + e g p^{2}\right )} x\right )} \log \left (g x + f\right )}{d g^{2} x^{2} + d f^{2} + e f + {\left (2 \, d f g + e g\right )} x}\,{d x}\right )} \]
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 \[ \int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{f+g\,x}\right )}^p\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \log {\left (c \left (d + \frac {e}{f + g x}\right )^{p} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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