Integrand size = 30, antiderivative size = 250 \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=-\frac {b f}{2 d (f h-e i)^2 (h+i x)}-\frac {b f^2 \log (e+f x)}{2 d (f h-e i)^3}+\frac {a+b \log (c (e+f x))}{2 d (f h-e i) (h+i x)^2}-\frac {f i (e+f x) (a+b \log (c (e+f x)))}{d (f h-e i)^3 (h+i x)}+\frac {3 b f^2 \log (h+i x)}{2 d (f h-e i)^3}-\frac {f^2 (a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {b f^2 \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3} \]
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Time = 0.35 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2458, 12, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=-\frac {f^2 \log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^3}-\frac {f i (e+f x) (a+b \log (c (e+f x)))}{d (h+i x) (f h-e i)^3}+\frac {a+b \log (c (e+f x))}{2 d (h+i x)^2 (f h-e i)}+\frac {b f^2 \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}-\frac {b f^2 \log (e+f x)}{2 d (f h-e i)^3}+\frac {3 b f^2 \log (h+i x)}{2 d (f h-e i)^3}-\frac {b f}{2 d (h+i x) (f h-e i)^2} \]
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Rule 12
Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2458
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{d x \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3} \, dx,x,e+f x\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{x \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3} \, dx,x,e+f x\right )}{d f} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \log (c x)}{x \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^2} \, dx,x,e+f x\right )}{d (f h-e i)}-\frac {i \text {Subst}\left (\int \frac {a+b \log (c x)}{\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^3} \, dx,x,e+f x\right )}{d f (f h-e i)} \\ & = \frac {a+b \log (c (e+f x))}{2 d (f h-e i) (h+i x)^2}+\frac {f \text {Subst}\left (\int \frac {a+b \log (c x)}{x \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )} \, dx,x,e+f x\right )}{d (f h-e i)^2}-\frac {i \text {Subst}\left (\int \frac {a+b \log (c x)}{\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^2} \, dx,x,e+f x\right )}{d (f h-e i)^2}-\frac {b \text {Subst}\left (\int \frac {1}{x \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^2} \, dx,x,e+f x\right )}{2 d (f h-e i)} \\ & = \frac {a+b \log (c (e+f x))}{2 d (f h-e i) (h+i x)^2}-\frac {f i (e+f x) (a+b \log (c (e+f x)))}{d (f h-e i)^3 (h+i x)}-\frac {f^2 (a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {\left (b f^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {f h-e i}{i x}\right )}{x} \, dx,x,e+f x\right )}{d (f h-e i)^3}+\frac {(b f i) \text {Subst}\left (\int \frac {1}{\frac {f h-e i}{f}+\frac {i x}{f}} \, dx,x,e+f x\right )}{d (f h-e i)^3}-\frac {b \text {Subst}\left (\int \left (\frac {f^2}{(f h-e i)^2 x}-\frac {f^2 i}{(f h-e i) (f h-e i+i x)^2}-\frac {f^2 i}{(f h-e i)^2 (f h-e i+i x)}\right ) \, dx,x,e+f x\right )}{2 d (f h-e i)} \\ & = -\frac {b f}{2 d (f h-e i)^2 (h+i x)}-\frac {b f^2 \log (e+f x)}{2 d (f h-e i)^3}+\frac {a+b \log (c (e+f x))}{2 d (f h-e i) (h+i x)^2}-\frac {f i (e+f x) (a+b \log (c (e+f x)))}{d (f h-e i)^3 (h+i x)}+\frac {3 b f^2 \log (h+i x)}{2 d (f h-e i)^3}-\frac {f^2 (a+b \log (c (e+f x))) \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3}+\frac {b f^2 \text {Li}_2\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\frac {\frac {(f h-e i)^2 (a+b \log (c (e+f x)))}{(h+i x)^2}+\frac {2 f (f h-e i) (a+b \log (c (e+f x)))}{h+i x}+\frac {f^2 (a+b \log (c (e+f x)))^2}{b}+2 b f^2 (-\log (e+f x)+\log (h+i x))-\frac {b f (f h-e i+f (h+i x) \log (e+f x)-f (h+i x) \log (h+i x))}{h+i x}-2 f^2 (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )-2 b f^2 \operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )}{2 d (f h-e i)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(241)=482\).
Time = 1.20 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.04
method | result | size |
parts | \(\frac {a \left (-\frac {1}{2 \left (e i -f h \right ) \left (i x +h \right )^{2}}+\frac {f^{2} \ln \left (i x +h \right )}{\left (e i -f h \right )^{3}}+\frac {f}{\left (e i -f h \right )^{2} \left (i x +h \right )}-\frac {f^{2} \ln \left (f x +e \right )}{\left (e i -f h \right )^{3}}\right )}{d}+\frac {b \left (-\frac {c \,f^{3} \ln \left (c f x +c e \right )^{2}}{2 \left (e i -f h \right )^{3}}+\frac {c^{3} f^{3} i \left (-\frac {\frac {\ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{i}+\frac {c \left (e i -f h \right )}{i \left (-c e i +h c f +i \left (c f x +c e \right )\right )}}{2 c^{2} \left (e i -f h \right )^{2}}+\frac {\ln \left (c f x +c e \right ) \left (-2 c e i +2 h c f +i \left (c f x +c e \right )\right ) \left (c f x +c e \right )}{2 \left (-c e i +h c f +i \left (c f x +c e \right )\right )^{2} c^{2} \left (e i -f h \right )^{2}}\right )}{e i -f h}+\frac {c \,f^{3} i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{\left (e i -f h \right )^{3}}-\frac {c^{2} f^{3} i \left (\frac {\ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}-\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (-c e i +h c f +i \left (c f x +c e \right )\right )}\right )}{\left (e i -f h \right )^{2}}\right )}{d c f}\) | \(510\) |
derivativedivides | \(\frac {-\frac {c^{4} f^{3} a \left (\frac {\ln \left (c f x +c e \right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )}-\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{2 c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{d}-\frac {c^{4} f^{3} b \left (\frac {\ln \left (c f x +c e \right )^{2}}{2 c^{3} \left (e i -f h \right )^{3}}+\frac {i \left (\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}+\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}\right )}{c^{2} \left (e i -f h \right )^{2}}-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{c^{3} \left (e i -f h \right )^{3}}-\frac {i \left (-\frac {-\frac {c \left (e i -f h \right )}{i \left (c e i -h c f -i \left (c f x +c e \right )\right )}+\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{i}}{2 c^{2} \left (e i -f h \right )^{2}}-\frac {\ln \left (c f x +c e \right ) \left (2 c e i -2 h c f -i \left (c f x +c e \right )\right ) \left (c f x +c e \right )}{2 c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{c \left (e i -f h \right )}\right )}{d}}{c f}\) | \(580\) |
default | \(\frac {-\frac {c^{4} f^{3} a \left (\frac {\ln \left (c f x +c e \right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )}-\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c^{3} \left (e i -f h \right )^{3}}+\frac {1}{2 c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{d}-\frac {c^{4} f^{3} b \left (\frac {\ln \left (c f x +c e \right )^{2}}{2 c^{3} \left (e i -f h \right )^{3}}+\frac {i \left (\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{c \left (e i -f h \right ) i}+\frac {\ln \left (c f x +c e \right ) \left (c f x +c e \right )}{c \left (e i -f h \right ) \left (c e i -h c f -i \left (c f x +c e \right )\right )}\right )}{c^{2} \left (e i -f h \right )^{2}}-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{c^{3} \left (e i -f h \right )^{3}}-\frac {i \left (-\frac {-\frac {c \left (e i -f h \right )}{i \left (c e i -h c f -i \left (c f x +c e \right )\right )}+\frac {\ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{i}}{2 c^{2} \left (e i -f h \right )^{2}}-\frac {\ln \left (c f x +c e \right ) \left (2 c e i -2 h c f -i \left (c f x +c e \right )\right ) \left (c f x +c e \right )}{2 c^{2} \left (e i -f h \right )^{2} \left (c e i -h c f -i \left (c f x +c e \right )\right )^{2}}\right )}{c \left (e i -f h \right )}\right )}{d}}{c f}\) | \(580\) |
risch | \(-\frac {a}{2 d \left (e i -f h \right ) \left (i x +h \right )^{2}}+\frac {a \,f^{2} \ln \left (i x +h \right )}{d \left (e i -f h \right )^{3}}+\frac {a f}{d \left (e i -f h \right )^{2} \left (i x +h \right )}-\frac {a \,f^{2} \ln \left (f x +e \right )}{d \left (e i -f h \right )^{3}}-\frac {b \,f^{2} \ln \left (c f x +c e \right )^{2}}{2 d \left (e i -f h \right )^{3}}-\frac {3 b \,f^{2} \ln \left (-c e i +h c f +i \left (c f x +c e \right )\right )}{2 d \left (e i -f h \right )^{3}}-\frac {b c \,f^{2} i e}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}+\frac {b c \,f^{3} h}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}+\frac {b \,c^{2} f^{4} i^{2} \ln \left (c f x +c e \right ) x^{2}}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}+\frac {b \,c^{2} f^{4} i \ln \left (c f x +c e \right ) h x}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}-\frac {b \,c^{2} f^{2} i^{2} \ln \left (c f x +c e \right ) e^{2}}{2 d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}+\frac {b \,c^{2} f^{3} i \ln \left (c f x +c e \right ) e h}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )^{2}}+\frac {b \,f^{2} \operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{3}}+\frac {b \,f^{2} \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )^{3}}+\frac {b c \,f^{3} i \ln \left (c f x +c e \right ) x}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}+\frac {b c \,f^{2} i \ln \left (c f x +c e \right ) e}{d \left (e i -f h \right )^{3} \left (c f i x +h c f \right )}\) | \(619\) |
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\[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\int { \frac {b \log \left ({\left (f x + e\right )} c\right ) + a}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{3}} \,d x } \]
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\[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\frac {\int \frac {a}{e h^{3} + 3 e h^{2} i x + 3 e h i^{2} x^{2} + e i^{3} x^{3} + f h^{3} x + 3 f h^{2} i x^{2} + 3 f h i^{2} x^{3} + f i^{3} x^{4}}\, dx + \int \frac {b \log {\left (c e + c f x \right )}}{e h^{3} + 3 e h^{2} i x + 3 e h i^{2} x^{2} + e i^{3} x^{3} + f h^{3} x + 3 f h^{2} i x^{2} + 3 f h i^{2} x^{3} + f i^{3} x^{4}}\, dx}{d} \]
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\[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\int { \frac {b \log \left ({\left (f x + e\right )} c\right ) + a}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{3}} \,d x } \]
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\[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\int { \frac {b \log \left ({\left (f x + e\right )} c\right ) + a}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log (c (e+f x))}{(d e+d f x) (h+i x)^3} \, dx=\int \frac {a+b\,\ln \left (c\,\left (e+f\,x\right )\right )}{{\left (h+i\,x\right )}^3\,\left (d\,e+d\,f\,x\right )} \,d x \]
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