Optimal. Leaf size=81 \[ -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (d+e x)}+\frac {a p \log (a x+b)}{e (a d-b e)}-\frac {b p \log (d+e x)}{d (a d-b e)}-\frac {p \log (x)}{d e} \]
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Rubi [A] time = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2463, 514, 72} \[ -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (d+e x)}+\frac {a p \log (a x+b)}{e (a d-b e)}-\frac {b p \log (d+e x)}{d (a d-b e)}-\frac {p \log (x)}{d e} \]
Antiderivative was successfully verified.
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Rule 72
Rule 514
Rule 2463
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^2} \, dx &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (d+e x)}-\frac {(b p) \int \frac {1}{\left (a+\frac {b}{x}\right ) x^2 (d+e x)} \, dx}{e}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (d+e x)}-\frac {(b p) \int \frac {1}{x (b+a x) (d+e x)} \, dx}{e}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (d+e x)}-\frac {(b p) \int \left (\frac {1}{b d x}+\frac {a^2}{b (-a d+b e) (b+a x)}+\frac {e^2}{d (a d-b e) (d+e x)}\right ) \, dx}{e}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (d+e x)}-\frac {p \log (x)}{d e}+\frac {a p \log (b+a x)}{e (a d-b e)}-\frac {b p \log (d+e x)}{d (a d-b e)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 81, normalized size = 1.00 \[ -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (d+e x)}+\frac {a p \log (a x+b)}{e (a d-b e)}-\frac {b p \log (d+e x)}{d (a d-b e)}-\frac {p \log (x)}{d e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 148, normalized size = 1.83 \[ -\frac {{\left (a d^{2} - b d e\right )} p \log \left (\frac {a x + b}{x}\right ) - {\left (a d e p x + a d^{2} p\right )} \log \left (a x + b\right ) + {\left (b e^{2} p x + b d e p\right )} \log \left (e x + d\right ) + {\left (a d^{2} - b d e\right )} \log \relax (c) + {\left ({\left (a d e - b e^{2}\right )} p x + {\left (a d^{2} - b d e\right )} p\right )} \log \relax (x)}{a d^{3} e - b d^{2} e^{2} + {\left (a d^{2} e^{2} - b d e^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 192, normalized size = 2.37 \[ -\frac {a b^{2} d p \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right ) - b^{3} p e \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right ) - \frac {{\left (a x + b\right )} b^{2} d p \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right )}{x} + a b^{2} d \log \relax (c) - b^{3} e \log \relax (c) + \frac {{\left (a x + b\right )} b^{2} d p \log \left (\frac {a x + b}{x}\right )}{x}}{{\left (a^{2} d^{3} - 2 \, a b d^{2} e - \frac {{\left (a x + b\right )} a d^{3}}{x} + b^{2} d e^{2} + \frac {{\left (a x + b\right )} b d^{2} e}{x}\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{\left (e x +d \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 85, normalized size = 1.05 \[ \frac {b p {\left (\frac {a \log \left (a x + b\right )}{a b d - b^{2} e} - \frac {e \log \left (e x + d\right )}{a d^{2} - b d e} - \frac {\log \relax (x)}{b d}\right )}}{e} - \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{{\left (e x + d\right )} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 85, normalized size = 1.05 \[ -\frac {\ln \left (c\,{\left (\frac {b+a\,x}{x}\right )}^p\right )}{x\,e^2+d\,e}-\frac {p\,\ln \relax (x)}{d\,e}-\frac {a\,p\,\ln \left (b+a\,x\right )}{b\,e^2-a\,d\,e}-\frac {b\,p\,\ln \left (d+e\,x\right )}{a\,d^2-b\,d\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.62, size = 585, normalized size = 7.22 \[ \begin {cases} \frac {d p \log {\left (\frac {d}{e} + x \right )}}{d^{2} e + d e^{2} x} + \frac {e p x \log {\relax (b )}}{d^{2} e + d e^{2} x} - \frac {e p x \log {\relax (x )}}{d^{2} e + d e^{2} x} + \frac {e p x \log {\left (\frac {d}{e} + x \right )}}{d^{2} e + d e^{2} x} + \frac {e x \log {\relax (c )}}{d^{2} e + d e^{2} x} & \text {for}\: a = 0 \\- \frac {d p}{d^{2} e + d e^{2} x} + \frac {e p x \log {\left (\frac {b}{x} + \frac {b e}{d} \right )}}{d^{2} e + d e^{2} x} + \frac {e x \log {\relax (c )}}{d^{2} e + d e^{2} x} & \text {for}\: a = \frac {b e}{d} \\\frac {- \frac {a p \log {\left (a + \frac {b}{x} \right )}}{b} - \frac {p \log {\left (a + \frac {b}{x} \right )}}{x} + \frac {p}{x} - \frac {\log {\relax (c )}}{x}}{e^{2}} & \text {for}\: d = 0 \\\tilde {\infty } \left (p x \log {\left (a + \frac {b}{x} \right )} + x \log {\relax (c )} + \frac {b p \log {\left (a x + b \right )}}{a}\right ) & \text {for}\: d = - e x \\\frac {p x \log {\left (a + \frac {b}{x} \right )} + x \log {\relax (c )} + \frac {b p \log {\left (a x + b \right )}}{a}}{d^{2}} & \text {for}\: e = 0 \\\frac {a d p x \log {\left (a + \frac {b}{x} \right )}}{a d^{3} + a d^{2} e x - b d^{2} e - b d e^{2} x} + \frac {a d x \log {\relax (c )}}{a d^{3} + a d^{2} e x - b d^{2} e - b d e^{2} x} + \frac {b d p \log {\left (x + \frac {b}{a} \right )}}{a d^{3} + a d^{2} e x - b d^{2} e - b d e^{2} x} - \frac {b d p \log {\left (\frac {d}{e} + x \right )}}{a d^{3} + a d^{2} e x - b d^{2} e - b d e^{2} x} - \frac {b e p x \log {\left (a + \frac {b}{x} \right )}}{a d^{3} + a d^{2} e x - b d^{2} e - b d e^{2} x} + \frac {b e p x \log {\left (x + \frac {b}{a} \right )}}{a d^{3} + a d^{2} e x - b d^{2} e - b d e^{2} x} - \frac {b e p x \log {\left (\frac {d}{e} + x \right )}}{a d^{3} + a d^{2} e x - b d^{2} e - b d e^{2} x} - \frac {b e x \log {\relax (c )}}{a d^{3} + a d^{2} e x - b d^{2} e - b d e^{2} x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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