Optimal. Leaf size=80 \[ \frac {b f p q \log (e+f x)}{h (f g-e h)}-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{h (g+h x)}-\frac {b f p q \log (g+h x)}{h (f g-e h)} \]
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Rubi [A]
time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2442, 36, 31,
2495} \begin {gather*} -\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{h (g+h x)}+\frac {b f p q \log (e+f x)}{h (f g-e h)}-\frac {b f p q \log (g+h x)}{h (f g-e h)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 2442
Rule 2495
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^2} \, dx &=\text {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(g+h x)^2} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{h (g+h x)}+\text {Subst}\left (\frac {(b f p q) \int \frac {1}{(e+f x) (g+h x)} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{h (g+h x)}-\text {Subst}\left (\frac {(b f p q) \int \frac {1}{g+h x} \, dx}{f g-e h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (b f^2 p q\right ) \int \frac {1}{e+f x} \, dx}{h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {b f p q \log (e+f x)}{h (f g-e h)}-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{h (g+h x)}-\frac {b f p q \log (g+h x)}{h (f g-e h)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 97, normalized size = 1.21 \begin {gather*} \frac {a f g-a e h-b f p q (g+h x) \log (e+f x)+b (f g-e h) \log \left (c \left (d (e+f x)^p\right )^q\right )+b f g p q \log (g+h x)+b f h p q x \log (g+h x)}{h (-f g+e h) (g+h x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 94, normalized size = 1.18 \begin {gather*} b f p q {\left (\frac {\log \left (f x + e\right )}{f g h - h^{2} e} - \frac {\log \left (h x + g\right )}{f g h - h^{2} e}\right )} - \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )}{h^{2} x + g h} - \frac {a}{h^{2} x + g h} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 119, normalized size = 1.49 \begin {gather*} -\frac {a f g - a h e - {\left (b f h p q x + b h p q e\right )} \log \left (f x + e\right ) + {\left (b f h p q x + b f g p q\right )} \log \left (h x + g\right ) + {\left (b f g - b h e\right )} \log \left (c\right ) + {\left (b f g q - b h q e\right )} \log \left (d\right )}{f g h^{2} x + f g^{2} h - {\left (h^{3} x + g h^{2}\right )} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.06, size = 129, normalized size = 1.61 \begin {gather*} \frac {b f h p q x \log \left (f x + e\right ) - b f h p q x \log \left (h x + g\right ) + b h p q e \log \left (f x + e\right ) - b f g p q \log \left (h x + g\right ) - b f g q \log \left (d\right ) + b h q e \log \left (d\right ) - b f g \log \left (c\right ) + b h e \log \left (c\right ) - a f g + a h e}{f g h^{2} x - h^{3} x e + f g^{2} h - g h^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.04, size = 89, normalized size = 1.11 \begin {gather*} -\frac {a}{x\,h^2+g\,h}-\frac {b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{h\,\left (g+h\,x\right )}+\frac {b\,f\,p\,q\,\mathrm {atan}\left (\frac {f\,g\,2{}\mathrm {i}+f\,h\,x\,2{}\mathrm {i}}{e\,h-f\,g}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{h\,\left (e\,h-f\,g\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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