Optimal. Leaf size=133 \[ \frac {b p}{6 e (b d-a e) (d+e x)^2}+\frac {b^2 p}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2442, 46}
\begin {gather*} \frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3}+\frac {b^2 p}{3 e (d+e x) (b d-a e)^2}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}+\frac {b p}{6 e (d+e x)^2 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2442
Rubi steps
\begin {align*} \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx &=-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}+\frac {(b p) \int \frac {1}{(a+b x) (d+e x)^3} \, dx}{3 e}\\ &=-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}+\frac {(b p) \int \left (\frac {b^3}{(b d-a e)^3 (a+b x)}-\frac {e}{(b d-a e) (d+e x)^3}-\frac {b e}{(b d-a e)^2 (d+e x)^2}-\frac {b^2 e}{(b d-a e)^3 (d+e x)}\right ) \, dx}{3 e}\\ &=\frac {b p}{6 e (b d-a e) (d+e x)^2}+\frac {b^2 p}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 105, normalized size = 0.79 \begin {gather*} \frac {-2 \log \left (c (a+b x)^p\right )+\frac {b p (d+e x) \left ((b d-a e) (3 b d-a e+2 b e x)+2 b^2 (d+e x)^2 \log (a+b x)-2 b^2 (d+e x)^2 \log (d+e x)\right )}{(b d-a e)^3}}{6 e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.72, size = 873, normalized size = 6.56
method | result | size |
risch | \(-\frac {\ln \left (\left (b x +a \right )^{p}\right )}{3 e \left (e x +d \right )^{3}}+\frac {-i \pi \,a^{3} e^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+i \pi \,b^{3} d^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+2 a \,b^{2} e^{3} p \,x^{2}-2 b^{3} d \,e^{2} p \,x^{2}-a^{2} b \,e^{3} p x -5 b^{3} d^{2} e p x -a^{2} b d p \,e^{2}+4 a \,b^{2} d^{2} p e -3 b^{3} d^{3} p -3 i \pi a \,b^{2} d^{2} e \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}+3 i \pi \,a^{2} b d \,e^{2} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+3 i \pi \,a^{2} b d \,e^{2} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-3 i \pi a \,b^{2} d^{2} e \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-6 \ln \left (b x +a \right ) b^{3} d \,e^{2} p \,x^{2}+6 \ln \left (-e x -d \right ) b^{3} d \,e^{2} p \,x^{2}-6 \ln \left (b x +a \right ) b^{3} d^{2} e p x +6 \ln \left (-e x -d \right ) b^{3} d^{2} e p x +2 \ln \left (-e x -d \right ) b^{3} d^{3} p -2 \ln \left (b x +a \right ) b^{3} d^{3} p +6 a \,b^{2} d \,e^{2} p x -i \pi \,b^{3} d^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}+i \pi \,a^{3} e^{3} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}+2 \ln \left (c \right ) b^{3} d^{3}-2 \ln \left (c \right ) a^{3} e^{3}-i \pi \,a^{3} e^{3} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}+i \pi \,b^{3} d^{3} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-3 i \pi \,a^{2} b d \,e^{2} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+3 i \pi a \,b^{2} d^{2} e \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-6 \ln \left (c \right ) a \,b^{2} d^{2} e +3 i \pi a \,b^{2} d^{2} e \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-i \pi \,b^{3} d^{3} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+i \pi \,a^{3} e^{3} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-3 i \pi \,a^{2} b d \,e^{2} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-2 \ln \left (b x +a \right ) b^{3} e^{3} p \,x^{3}+2 \ln \left (-e x -d \right ) b^{3} e^{3} p \,x^{3}+6 \ln \left (c \right ) a^{2} b d \,e^{2}}{6 \left (e x +d \right )^{3} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (a e -b d \right ) e}\) | \(873\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 228, normalized size = 1.71 \begin {gather*} \frac {1}{6} \, {\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {2 \, b^{2} \log \left (x e + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac {2 \, b x e + 3 \, b d - a e}{b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x}\right )} b p e^{\left (-1\right )} - \frac {e^{\left (-1\right )} \log \left ({\left (b x + a\right )}^{p} c\right )}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 431 vs.
\(2 (126) = 252\).
time = 0.40, size = 431, normalized size = 3.24 \begin {gather*} \frac {3 \, b^{3} d^{3} p - {\left (2 \, a b^{2} p x^{2} - a^{2} b p x\right )} e^{3} + {\left (2 \, b^{3} d p x^{2} - 6 \, a b^{2} d p x + a^{2} b d p\right )} e^{2} + {\left (5 \, b^{3} d^{2} p x - 4 \, a b^{2} d^{2} p\right )} e + 2 \, {\left ({\left (b^{3} p x^{3} + a^{3} p\right )} e^{3} + 3 \, {\left (b^{3} d p x^{2} - a^{2} b d p\right )} e^{2} + 3 \, {\left (b^{3} d^{2} p x + a b^{2} d^{2} p\right )} e\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{3} p x^{3} e^{3} + 3 \, b^{3} d p x^{2} e^{2} + 3 \, b^{3} d^{2} p x e + b^{3} d^{3} p\right )} \log \left (x e + d\right ) - 2 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (c\right )}{6 \, {\left (b^{3} d^{6} e - a^{3} x^{3} e^{7} + 3 \, {\left (a^{2} b d x^{3} - a^{3} d x^{2}\right )} e^{6} - 3 \, {\left (a b^{2} d^{2} x^{3} - 3 \, a^{2} b d^{2} x^{2} + a^{3} d^{2} x\right )} e^{5} + {\left (b^{3} d^{3} x^{3} - 9 \, a b^{2} d^{3} x^{2} + 9 \, a^{2} b d^{3} x - a^{3} d^{3}\right )} e^{4} + 3 \, {\left (b^{3} d^{4} x^{2} - 3 \, a b^{2} d^{4} x + a^{2} b d^{4}\right )} e^{3} + 3 \, {\left (b^{3} d^{5} x - a b^{2} d^{5}\right )} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 4571 vs.
\(2 (109) = 218\).
time = 20.01, size = 4571, normalized size = 34.37 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs.
\(2 (126) = 252\).
time = 4.17, size = 495, normalized size = 3.72 \begin {gather*} \frac {2 \, b^{3} p x^{3} e^{3} \log \left (b x + a\right ) + 6 \, b^{3} d p x^{2} e^{2} \log \left (b x + a\right ) + 6 \, b^{3} d^{2} p x e \log \left (b x + a\right ) - 2 \, b^{3} p x^{3} e^{3} \log \left (x e + d\right ) - 6 \, b^{3} d p x^{2} e^{2} \log \left (x e + d\right ) - 6 \, b^{3} d^{2} p x e \log \left (x e + d\right ) + 2 \, b^{3} d p x^{2} e^{2} + 5 \, b^{3} d^{2} p x e + 6 \, a b^{2} d^{2} p e \log \left (b x + a\right ) - 2 \, b^{3} d^{3} p \log \left (x e + d\right ) + 3 \, b^{3} d^{3} p - 2 \, a b^{2} p x^{2} e^{3} - 6 \, a b^{2} d p x e^{2} - 4 \, a b^{2} d^{2} p e - 6 \, a^{2} b d p e^{2} \log \left (b x + a\right ) - 2 \, b^{3} d^{3} \log \left (c\right ) + 6 \, a b^{2} d^{2} e \log \left (c\right ) + a^{2} b p x e^{3} + a^{2} b d p e^{2} + 2 \, a^{3} p e^{3} \log \left (b x + a\right ) - 6 \, a^{2} b d e^{2} \log \left (c\right ) + 2 \, a^{3} e^{3} \log \left (c\right )}{6 \, {\left (b^{3} d^{3} x^{3} e^{4} + 3 \, b^{3} d^{4} x^{2} e^{3} + 3 \, b^{3} d^{5} x e^{2} + b^{3} d^{6} e - 3 \, a b^{2} d^{2} x^{3} e^{5} - 9 \, a b^{2} d^{3} x^{2} e^{4} - 9 \, a b^{2} d^{4} x e^{3} - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d x^{3} e^{6} + 9 \, a^{2} b d^{2} x^{2} e^{5} + 9 \, a^{2} b d^{3} x e^{4} + 3 \, a^{2} b d^{4} e^{3} - a^{3} x^{3} e^{7} - 3 \, a^{3} d x^{2} e^{6} - 3 \, a^{3} d^{2} x e^{5} - a^{3} d^{3} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.76, size = 145, normalized size = 1.09 \begin {gather*} \frac {b^2\,p\,x}{3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}-\frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{3\,e\,{\left (d+e\,x\right )}^3}-\frac {a\,b\,p}{6\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}+\frac {b^2\,d\,p}{2\,e\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}+\frac {b^3\,p\,\mathrm {atan}\left (\frac {a\,e\,1{}\mathrm {i}+b\,d\,1{}\mathrm {i}+b\,e\,x\,2{}\mathrm {i}}{a\,e-b\,d}\right )\,2{}\mathrm {i}}{3\,e\,{\left (a\,e-b\,d\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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