Optimal. Leaf size=128 \[ -\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac {b p r \log (a+b x)}{h (b g-a h)}-\frac {b p r \log (g+h x)}{h (b g-a h)}+\frac {d q r \log (c+d x)}{h (d g-c h)}-\frac {d q r \log (g+h x)}{h (d g-c h)} \]
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Rubi [A] time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2495, 36, 31} \[ -\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac {b p r \log (a+b x)}{h (b g-a h)}-\frac {b p r \log (g+h x)}{h (b g-a h)}+\frac {d q r \log (c+d x)}{h (d g-c h)}-\frac {d q r \log (g+h x)}{h (d g-c h)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 2495
Rubi steps
\begin {align*} \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^2} \, dx &=-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac {(b p r) \int \frac {1}{(a+b x) (g+h x)} \, dx}{h}+\frac {(d q r) \int \frac {1}{(c+d x) (g+h x)} \, dx}{h}\\ &=-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}-\frac {(b p r) \int \frac {1}{g+h x} \, dx}{b g-a h}+\frac {\left (b^2 p r\right ) \int \frac {1}{a+b x} \, dx}{h (b g-a h)}-\frac {(d q r) \int \frac {1}{g+h x} \, dx}{d g-c h}+\frac {\left (d^2 q r\right ) \int \frac {1}{c+d x} \, dx}{h (d g-c h)}\\ &=\frac {b p r \log (a+b x)}{h (b g-a h)}+\frac {d q r \log (c+d x)}{h (d g-c h)}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}-\frac {b p r \log (g+h x)}{h (b g-a h)}-\frac {d q r \log (g+h x)}{h (d g-c h)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 93, normalized size = 0.73 \[ \frac {-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x}+\frac {b p r (\log (a+b x)-\log (g+h x))}{b g-a h}+\frac {d q r (\log (c+d x)-\log (g+h x))}{d g-c h}}{h} \]
Antiderivative was successfully verified.
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fricas [B] time = 109.51, size = 280, normalized size = 2.19 \[ -\frac {{\left (b d g^{2} + a c h^{2} - {\left (b c + a d\right )} g h\right )} r \log \relax (f) - {\left ({\left (b d g h - b c h^{2}\right )} p r x + {\left (a d g h - a c h^{2}\right )} p r\right )} \log \left (b x + a\right ) - {\left ({\left (b d g h - a d h^{2}\right )} q r x + {\left (b c g h - a c h^{2}\right )} q r\right )} \log \left (d x + c\right ) + {\left ({\left ({\left (b d g h - b c h^{2}\right )} p + {\left (b d g h - a d h^{2}\right )} q\right )} r x + {\left ({\left (b d g^{2} - b c g h\right )} p + {\left (b d g^{2} - a d g h\right )} q\right )} r\right )} \log \left (h x + g\right ) + {\left (b d g^{2} + a c h^{2} - {\left (b c + a d\right )} g h\right )} \log \relax (e)}{b d g^{3} h + a c g h^{3} - {\left (b c + a d\right )} g^{2} h^{2} + {\left (b d g^{2} h^{2} + a c h^{4} - {\left (b c + a d\right )} g h^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 190, normalized size = 1.48 \[ \frac {b^{2} p r \log \left ({\left | -b x - a \right |}\right )}{b^{2} g h - a b h^{2}} + \frac {d^{2} q r \log \left ({\left | d x + c \right |}\right )}{d^{2} g h - c d h^{2}} - \frac {p r \log \left (b x + a\right )}{h^{2} x + g h} - \frac {q r \log \left (d x + c\right )}{h^{2} x + g h} - \frac {{\left (b d g p r - b c h p r + b d g q r - a d h q r\right )} \log \left (h x + g\right )}{b d g^{2} h - b c g h^{2} - a d g h^{2} + a c h^{3}} - \frac {r \log \relax (f) + 1}{h^{2} x + g h} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (h x +g \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 123, normalized size = 0.96 \[ \frac {{\left (b f p {\left (\frac {\log \left (b x + a\right )}{b g - a h} - \frac {\log \left (h x + g\right )}{b g - a h}\right )} + d f q {\left (\frac {\log \left (d x + c\right )}{d g - c h} - \frac {\log \left (h x + g\right )}{d g - c h}\right )}\right )} r}{f h} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{{\left (h x + g\right )} h} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 152, normalized size = 1.19 \[ \frac {\ln \left (g+h\,x\right )\,\left (b\,c\,h\,p\,r-g\,\left (b\,d\,p\,r+b\,d\,q\,r\right )+a\,d\,h\,q\,r\right )}{a\,c\,h^3-a\,d\,g\,h^2-b\,c\,g\,h^2+b\,d\,g^2\,h}-\frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (x+\frac {g}{h}\right )}{{\left (g+h\,x\right )}^2}-\frac {b\,p\,r\,\ln \left (a+b\,x\right )}{a\,h^2-b\,g\,h}-\frac {d\,q\,r\,\ln \left (c+d\,x\right )}{c\,h^2-d\,g\,h} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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