Optimal. Leaf size=135 \[ -\frac {d^2 q r \log (a+b x)}{2 b (b c-a d)^2}+\frac {d^2 q r \log (c+d x)}{2 b (b c-a d)^2}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}-\frac {d q r}{2 b (a+b x) (b c-a d)}-\frac {p r}{4 b (a+b x)^2} \]
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Rubi [A] time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2495, 32, 44} \[ -\frac {d^2 q r \log (a+b x)}{2 b (b c-a d)^2}+\frac {d^2 q r \log (c+d x)}{2 b (b c-a d)^2}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}-\frac {d q r}{2 b (a+b x) (b c-a d)}-\frac {p r}{4 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 32
Rule 44
Rule 2495
Rubi steps
\begin {align*} \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3} \, dx &=-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}+\frac {1}{2} (p r) \int \frac {1}{(a+b x)^3} \, dx+\frac {(d q r) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{2 b}\\ &=-\frac {p r}{4 b (a+b x)^2}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}+\frac {(d q r) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{2 b}\\ &=-\frac {p r}{4 b (a+b x)^2}-\frac {d q r}{2 b (b c-a d) (a+b x)}-\frac {d^2 q r \log (a+b x)}{2 b (b c-a d)^2}+\frac {d^2 q r \log (c+d x)}{2 b (b c-a d)^2}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b (a+b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 116, normalized size = 0.86 \[ \frac {r \left (-\frac {d^2 q \log (a+b x)}{(b c-a d)^2}+\frac {d^2 q \log (c+d x)}{(b c-a d)^2}-\frac {p-\frac {2 d q (a+b x)}{a d-b c}}{2 (a+b x)^2}\right )-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2}}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 323, normalized size = 2.39 \[ -\frac {2 \, {\left (b^{2} c d - a b d^{2}\right )} q r x + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} r \log \relax (f) + {\left ({\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} p + 2 \, {\left (a b c d - a^{2} d^{2}\right )} q\right )} r + 2 \, {\left (b^{2} d^{2} q r x^{2} + 2 \, a b d^{2} q r x + {\left (a^{2} d^{2} q + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} p\right )} r\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{2} d^{2} q r x^{2} + 2 \, a b d^{2} q r x - {\left (b^{2} c^{2} - 2 \, a b c d\right )} q r\right )} \log \left (d x + c\right ) + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \relax (e)}{4 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2} + {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 246, normalized size = 1.82 \[ -\frac {d^{2} q r \log \left (b x + a\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} + \frac {d^{2} q r \log \left (d x + c\right )}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}} - \frac {p r \log \left (b x + a\right )}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} - \frac {q r \log \left (d x + c\right )}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} - \frac {2 \, b d q r x + b c p r - a d p r + 2 \, a d q r + 2 \, b c r \log \relax (f) - 2 \, a d r \log \relax (f) + 2 \, b c - 2 \, a d}{4 \, {\left (b^{4} c x^{2} - a b^{3} d x^{2} + 2 \, a b^{3} c x - 2 \, a^{2} b^{2} d x + a^{2} b^{2} c - a^{3} b d\right )}} \]
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 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (b x +a \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 165, normalized size = 1.22 \[ -\frac {{\left (2 \, d f q {\left (\frac {d \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac {d \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac {1}{a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x}\right )} + \frac {b f p}{b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b}\right )} r}{4 \, b f} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{2 \, {\left (b x + a\right )}^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.39, size = 182, normalized size = 1.35 \[ \frac {\frac {b\,c\,p\,r-a\,d\,p\,r+2\,a\,d\,q\,r}{2\,\left (a\,d-b\,c\right )}+\frac {b\,d\,q\,r\,x}{a\,d-b\,c}}{2\,a^2\,b+4\,a\,b^2\,x+2\,b^3\,x^2}-\frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )}{{\left (a+b\,x\right )}^3}+\frac {d^2\,q\,r\,\mathrm {atanh}\left (\frac {2\,b^3\,c^2-2\,a^2\,b\,d^2}{2\,b\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{b\,{\left (a\,d-b\,c\right )}^2} \]
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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