Optimal. Leaf size=164 \[ \frac{d^2 q r}{3 b (a+b x) (b c-a d)^2}+\frac{d^3 q r \log (a+b x)}{3 b (b c-a d)^3}-\frac{d^3 q r \log (c+d x)}{3 b (b c-a d)^3}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac{d q r}{6 b (a+b x)^2 (b c-a d)}-\frac{p r}{9 b (a+b x)^3} \]
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Rubi [A] time = 0.0693974, antiderivative size = 164, normalized size of antiderivative = 1., 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} \[ \frac{d^2 q r}{3 b (a+b x) (b c-a d)^2}+\frac{d^3 q r \log (a+b x)}{3 b (b c-a d)^3}-\frac{d^3 q r \log (c+d x)}{3 b (b c-a d)^3}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac{d q r}{6 b (a+b x)^2 (b c-a d)}-\frac{p r}{9 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 2495
Rule 32
Rule 44
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)^4} \, dx &=-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}+\frac{1}{3} (p r) \int \frac{1}{(a+b x)^4} \, dx+\frac{(d q r) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{3 b}\\ &=-\frac{p r}{9 b (a+b x)^3}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}+\frac{(d q r) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{3 b}\\ &=-\frac{p r}{9 b (a+b x)^3}-\frac{d q r}{6 b (b c-a d) (a+b x)^2}+\frac{d^2 q r}{3 b (b c-a d)^2 (a+b x)}+\frac{d^3 q r \log (a+b x)}{3 b (b c-a d)^3}-\frac{d^3 q r \log (c+d x)}{3 b (b c-a d)^3}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.391948, size = 141, normalized size = 0.86 \[ \frac{r \left (\frac{\frac{6 d^2 q (a+b x)^2}{(b c-a d)^2}+\frac{3 d q (a+b x)}{a d-b c}-2 p}{6 (a+b x)^3}+\frac{d^3 q \log (a+b x)}{(b c-a d)^3}-\frac{d^3 q \log (c+d x)}{(b c-a d)^3}\right )-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3}}{3 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.425, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) }{ \left ( bx+a \right ) ^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.27755, size = 390, normalized size = 2.38 \begin{align*} \frac{{\left (3 \,{\left (\frac{2 \, d^{2} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{2 \, d^{2} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{2 \, b d x - b c + 3 \, a d}{a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x}\right )} d f q - \frac{2 \, b f p}{b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b}\right )} r}{18 \, b f} - \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{3 \,{\left (b x + a\right )}^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.854387, size = 1177, normalized size = 7.18 \begin{align*} \frac{6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} q r x^{2} - 3 \,{\left (b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} q r x - 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} r \log \left (f\right ) -{\left (2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} p + 3 \,{\left (a b^{2} c^{2} d - 4 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} q\right )} r + 6 \,{\left (b^{3} d^{3} q r x^{3} + 3 \, a b^{2} d^{3} q r x^{2} + 3 \, a^{2} b d^{3} q r x +{\left (a^{3} d^{3} q -{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} p\right )} r\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} d^{3} q r x^{3} + 3 \, a b^{2} d^{3} q r x^{2} + 3 \, a^{2} b d^{3} q r x +{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} q r\right )} \log \left (d x + c\right ) - 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (e\right )}{18 \,{\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3} +{\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{3} + 3 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21093, size = 633, normalized size = 3.86 \begin{align*} \frac{d^{3} q r \log \left (b x + a\right )}{3 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac{d^{3} q r \log \left (d x + c\right )}{3 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac{p r \log \left (b x + a\right )}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac{q r \log \left (d x + c\right )}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac{6 \, b^{2} d^{2} q r x^{2} - 3 \, b^{2} c d q r x + 15 \, a b d^{2} q r x - 2 \, b^{2} c^{2} p r + 4 \, a b c d p r - 2 \, a^{2} d^{2} p r - 3 \, a b c d q r + 9 \, a^{2} d^{2} q r - 6 \, b^{2} c^{2} r \log \left (f\right ) + 12 \, a b c d r \log \left (f\right ) - 6 \, a^{2} d^{2} r \log \left (f\right ) - 6 \, b^{2} c^{2} + 12 \, a b c d - 6 \, a^{2} d^{2}}{18 \,{\left (b^{6} c^{2} x^{3} - 2 \, a b^{5} c d x^{3} + a^{2} b^{4} d^{2} x^{3} + 3 \, a b^{5} c^{2} x^{2} - 6 \, a^{2} b^{4} c d x^{2} + 3 \, a^{3} b^{3} d^{2} x^{2} + 3 \, a^{2} b^{4} c^{2} x - 6 \, a^{3} b^{3} c d x + 3 \, a^{4} b^{2} d^{2} x + a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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