Optimal. Leaf size=193 \[ -\frac {d^4 q r \log (a+b x)}{4 b (b c-a d)^4}+\frac {d^4 q r \log (c+d x)}{4 b (b c-a d)^4}-\frac {d^3 q r}{4 b (a+b x) (b c-a d)^3}+\frac {d^2 q r}{8 b (a+b x)^2 (b c-a d)^2}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {d q r}{12 b (a+b x)^3 (b c-a d)}-\frac {p r}{16 b (a+b x)^4} \]
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Rubi [A] time = 0.09, antiderivative size = 193, 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} \[ -\frac {d^3 q r}{4 b (a+b x) (b c-a d)^3}+\frac {d^2 q r}{8 b (a+b x)^2 (b c-a d)^2}-\frac {d^4 q r \log (a+b x)}{4 b (b c-a d)^4}+\frac {d^4 q r \log (c+d x)}{4 b (b c-a d)^4}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {d q r}{12 b (a+b x)^3 (b c-a d)}-\frac {p r}{16 b (a+b x)^4} \]
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)^5} \, dx &=-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\frac {1}{4} (p r) \int \frac {1}{(a+b x)^5} \, dx+\frac {(d q r) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{4 b}\\ &=-\frac {p r}{16 b (a+b x)^4}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}+\frac {(d q r) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{4 b}\\ &=-\frac {p r}{16 b (a+b x)^4}-\frac {d q r}{12 b (b c-a d) (a+b x)^3}+\frac {d^2 q r}{8 b (b c-a d)^2 (a+b x)^2}-\frac {d^3 q r}{4 b (b c-a d)^3 (a+b x)}-\frac {d^4 q r \log (a+b x)}{4 b (b c-a d)^4}+\frac {d^4 q r \log (c+d x)}{4 b (b c-a d)^4}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 164, normalized size = 0.85 \[ \frac {r \left (-\frac {d^4 q \log (a+b x)}{(b c-a d)^4}+\frac {d^4 q \log (c+d x)}{(b c-a d)^4}+\frac {-\frac {12 d^3 q (a+b x)^3}{(b c-a d)^3}+\frac {6 d^2 q (a+b x)^2}{(b c-a d)^2}+\frac {4 d q (a+b x)}{a d-b c}-3 p}{12 (a+b x)^4}\right )-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4}}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 861, normalized size = 4.46 \[ -\frac {12 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} q r x^{3} - 6 \, {\left (b^{4} c^{2} d^{2} - 8 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} q r x^{2} + 4 \, {\left (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 18 \, a^{2} b^{2} c d^{3} - 13 \, a^{3} b d^{4}\right )} q r x + 12 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} r \log \relax (f) + {\left (3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} p + 2 \, {\left (2 \, a b^{3} c^{3} d - 9 \, a^{2} b^{2} c^{2} d^{2} + 18 \, a^{3} b c d^{3} - 11 \, a^{4} d^{4}\right )} q\right )} r + 12 \, {\left (b^{4} d^{4} q r x^{4} + 4 \, a b^{3} d^{4} q r x^{3} + 6 \, a^{2} b^{2} d^{4} q r x^{2} + 4 \, a^{3} b d^{4} q r x + {\left (a^{4} d^{4} q + {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} p\right )} r\right )} \log \left (b x + a\right ) - 12 \, {\left (b^{4} d^{4} q r x^{4} + 4 \, a b^{3} d^{4} q r x^{3} + 6 \, a^{2} b^{2} d^{4} q r x^{2} + 4 \, a^{3} b d^{4} q r x - {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3}\right )} q r\right )} \log \left (d x + c\right ) + 12 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \relax (e)}{48 \, {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4} + {\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 748, normalized size = 3.88 \[ -\frac {d^{4} q r \log \left (b x + a\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} + \frac {d^{4} q r \log \left (d x + c\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {p r \log \left (b x + a\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {q r \log \left (d x + c\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {12 \, b^{3} d^{3} q r x^{3} - 6 \, b^{3} c d^{2} q r x^{2} + 42 \, a b^{2} d^{3} q r x^{2} + 4 \, b^{3} c^{2} d q r x - 20 \, a b^{2} c d^{2} q r x + 52 \, a^{2} b d^{3} q r x + 3 \, b^{3} c^{3} p r - 9 \, a b^{2} c^{2} d p r + 9 \, a^{2} b c d^{2} p r - 3 \, a^{3} d^{3} p r + 4 \, a b^{2} c^{2} d q r - 14 \, a^{2} b c d^{2} q r + 22 \, a^{3} d^{3} q r + 12 \, b^{3} c^{3} r \log \relax (f) - 36 \, a b^{2} c^{2} d r \log \relax (f) + 36 \, a^{2} b c d^{2} r \log \relax (f) - 12 \, a^{3} d^{3} r \log \relax (f) + 12 \, b^{3} c^{3} - 36 \, a b^{2} c^{2} d + 36 \, a^{2} b c d^{2} - 12 \, a^{3} d^{3}}{48 \, {\left (b^{8} c^{3} x^{4} - 3 \, a b^{7} c^{2} d x^{4} + 3 \, a^{2} b^{6} c d^{2} x^{4} - a^{3} b^{5} d^{3} x^{4} + 4 \, a b^{7} c^{3} x^{3} - 12 \, a^{2} b^{6} c^{2} d x^{3} + 12 \, a^{3} b^{5} c d^{2} x^{3} - 4 \, a^{4} b^{4} d^{3} x^{3} + 6 \, a^{2} b^{6} c^{3} x^{2} - 18 \, a^{3} b^{5} c^{2} d x^{2} + 18 \, a^{4} b^{4} c d^{2} x^{2} - 6 \, a^{5} b^{3} d^{3} x^{2} + 4 \, a^{3} b^{5} c^{3} x - 12 \, a^{4} b^{4} c^{2} d x + 12 \, a^{5} b^{3} c d^{2} x - 4 \, a^{6} b^{2} d^{3} x + a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )}} \]
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 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (b x +a \right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.85, size = 459, normalized size = 2.38 \[ -\frac {{\left (2 \, {\left (\frac {6 \, d^{3} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x}\right )} d f q + \frac {3 \, b f p}{b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b}\right )} r}{48 \, b f} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{4 \, {\left (b x + a\right )}^{4} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.71, size = 526, normalized size = 2.73 \[ \frac {\frac {3\,b^3\,c^3\,p\,r-3\,a^3\,d^3\,p\,r+22\,a^3\,d^3\,q\,r-9\,a\,b^2\,c^2\,d\,p\,r+9\,a^2\,b\,c\,d^2\,p\,r+4\,a\,b^2\,c^2\,d\,q\,r-14\,a^2\,b\,c\,d^2\,q\,r}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x\,\left (13\,q\,r\,a^2\,b\,d^3-5\,q\,r\,a\,b^2\,c\,d^2+q\,r\,b^3\,c^2\,d\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d\,x^2\,\left (7\,a\,b^2\,d^2\,q\,r-b^3\,c\,d\,q\,r\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b^3\,d^3\,q\,r\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b+16\,a^3\,b^2\,x+24\,a^2\,b^3\,x^2+16\,a\,b^4\,x^3+4\,b^5\,x^4}-\frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (\frac {x}{4}+\frac {a}{4\,b}\right )}{{\left (a+b\,x\right )}^5}+\frac {d^4\,q\,r\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4+8\,a^3\,b^2\,c\,d^3-8\,a\,b^4\,c^3\,d+4\,b^5\,c^4}{4\,b\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,{\left (a\,d-b\,c\right )}^4} \]
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 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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