Optimal. Leaf size=164 \[ \frac {d^3 q r \log (a+b x)}{3 b (b c-a d)^3}-\frac {d^3 q r \log (c+d x)}{3 b (b c-a d)^3}+\frac {d^2 q r}{3 b (a+b x) (b c-a d)^2}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {d q r}{6 b (a+b x)^2 (b c-a d)}-\frac {p r}{9 b (a+b x)^3} \]
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Rubi [A] time = 0.07, antiderivative size = 164, 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} \[ \frac {d^2 q r}{3 b (a+b x) (b c-a d)^2}+\frac {d^3 q r \log (a+b x)}{3 b (b c-a d)^3}-\frac {d^3 q r \log (c+d x)}{3 b (b c-a d)^3}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 b (a+b x)^3}-\frac {d q r}{6 b (a+b x)^2 (b c-a d)}-\frac {p r}{9 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 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)^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*}
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Mathematica [A] time = 0.36, size = 141, normalized size = 0.86 \[ \frac {r \left (\frac {d^3 q \log (a+b x)}{(b c-a d)^3}-\frac {d^3 q \log (c+d x)}{(b c-a d)^3}+\frac {\frac {6 d^2 q (a+b x)^2}{(b c-a d)^2}+\frac {3 d q (a+b x)}{a d-b c}-2 p}{6 (a+b x)^3}\right )-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^3}}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 580, normalized size = 3.54 \[ \frac {6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} q r x^{2} - 3 \, {\left (b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} q r x - 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} r \log \relax (f) - {\left (2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} p + 3 \, {\left (a b^{2} c^{2} d - 4 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} q\right )} r + 6 \, {\left (b^{3} d^{3} q r x^{3} + 3 \, a b^{2} d^{3} q r x^{2} + 3 \, a^{2} b d^{3} q r x + {\left (a^{3} d^{3} q - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} p\right )} r\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} d^{3} q r x^{3} + 3 \, a b^{2} d^{3} q r x^{2} + 3 \, a^{2} b d^{3} q r x + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} q r\right )} \log \left (d x + c\right ) - 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \relax (e)}{18 \, {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3} + {\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 469, normalized size = 2.86 \[ \frac {d^{3} q r \log \left (b x + a\right )}{3 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac {d^{3} q r \log \left (d x + c\right )}{3 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac {p r \log \left (b x + a\right )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} - \frac {q r \log \left (d x + c\right )}{3 \, {\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} + \frac {6 \, b^{2} d^{2} q r x^{2} - 3 \, b^{2} c d q r x + 15 \, a b d^{2} q r x - 2 \, b^{2} c^{2} p r + 4 \, a b c d p r - 2 \, a^{2} d^{2} p r - 3 \, a b c d q r + 9 \, a^{2} d^{2} q r - 6 \, b^{2} c^{2} r \log \relax (f) + 12 \, a b c d r \log \relax (f) - 6 \, a^{2} d^{2} r \log \relax (f) - 6 \, b^{2} c^{2} + 12 \, a b c d - 6 \, a^{2} d^{2}}{18 \, {\left (b^{6} c^{2} x^{3} - 2 \, a b^{5} c d x^{3} + a^{2} b^{4} d^{2} x^{3} + 3 \, a b^{5} c^{2} x^{2} - 6 \, a^{2} b^{4} c d x^{2} + 3 \, a^{3} b^{3} d^{2} x^{2} + 3 \, a^{2} b^{4} c^{2} x - 6 \, a^{3} b^{3} c d x + 3 \, a^{4} b^{2} d^{2} x + a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )}} \]
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 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (b x +a \right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 289, normalized size = 1.76 \[ \frac {{\left (3 \, {\left (\frac {2 \, d^{2} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac {2 \, b d x - b c + 3 \, a d}{a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x}\right )} d f q - \frac {2 \, b f p}{b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b}\right )} r}{18 \, b f} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{3 \, {\left (b x + a\right )}^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.87, size = 346, normalized size = 2.11 \[ \frac {\frac {x\,\left (5\,a\,b\,d^2\,q\,r-b^2\,c\,d\,q\,r\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {2\,a^2\,d^2\,p\,r+2\,b^2\,c^2\,p\,r-9\,a^2\,d^2\,q\,r-4\,a\,b\,c\,d\,p\,r+3\,a\,b\,c\,d\,q\,r}{6\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {b^2\,d^2\,q\,r\,x^2}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{3\,a^3\,b+9\,a^2\,b^2\,x+9\,a\,b^3\,x^2+3\,b^4\,x^3}-\frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (\frac {x}{3}+\frac {a}{3\,b}\right )}{{\left (a+b\,x\right )}^4}-\frac {2\,d^3\,q\,r\,\mathrm {atanh}\left (\frac {3\,a^3\,b\,d^3-3\,a^2\,b^2\,c\,d^2-3\,a\,b^3\,c^2\,d+3\,b^4\,c^3}{3\,b\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{3\,b\,{\left (a\,d-b\,c\right )}^3} \]
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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