Integrand size = 28, antiderivative size = 79 \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {a i x}{d f}-\frac {b i x}{d f}+\frac {b i (e+f x) \log (c (e+f x))}{d f^2}+\frac {(f h-e i) (a+b \log (c (e+f x)))^2}{2 b d f^2} \]
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Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2458, 12, 2388, 2338, 2332} \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {(f h-e i) (a+b \log (c (e+f x)))^2}{2 b d f^2}+\frac {a i x}{d f}+\frac {b i (e+f x) \log (c (e+f x))}{d f^2}-\frac {b i x}{d f} \]
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Rule 12
Rule 2332
Rule 2338
Rule 2388
Rule 2458
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right ) (a+b \log (c x))}{d x} \, dx,x,e+f x\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {\left (\frac {f h-e i}{f}+\frac {i x}{f}\right ) (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{d f} \\ & = \frac {i \text {Subst}(\int (a+b \log (c x)) \, dx,x,e+f x)}{d f^2}+\frac {(f h-e i) \text {Subst}\left (\int \frac {a+b \log (c x)}{x} \, dx,x,e+f x\right )}{d f^2} \\ & = \frac {a i x}{d f}+\frac {(f h-e i) (a+b \log (c (e+f x)))^2}{2 b d f^2}+\frac {(b i) \text {Subst}(\int \log (c x) \, dx,x,e+f x)}{d f^2} \\ & = \frac {a i x}{d f}-\frac {b i x}{d f}+\frac {b i (e+f x) \log (c (e+f x))}{d f^2}+\frac {(f h-e i) (a+b \log (c (e+f x)))^2}{2 b d f^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84 \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {2 a f i x-2 b f i x+2 b i (e+f x) \log (c (e+f x))+\frac {(f h-e i) (a+b \log (c (e+f x)))^2}{b}}{2 d f^2} \]
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Time = 0.57 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.16
method | result | size |
norman | \(\frac {i \left (a -b \right ) x}{d f}+\frac {b i x \ln \left (c \left (f x +e \right )\right )}{d f}-\frac {\left (a e i -a f h -b e i \right ) \ln \left (c \left (f x +e \right )\right )}{d \,f^{2}}-\frac {b \left (e i -f h \right ) \ln \left (c \left (f x +e \right )\right )^{2}}{2 d \,f^{2}}\) | \(92\) |
parts | \(\frac {a \left (\frac {x i}{f}+\frac {\left (-e i +f h \right ) \ln \left (f x +e \right )}{f^{2}}\right )}{d}+\frac {b \left (-\frac {c e i \ln \left (c f x +c e \right )^{2}}{2 f}+\frac {c h \ln \left (c f x +c e \right )^{2}}{2}+\frac {i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f}\right )}{d c f}\) | \(111\) |
parallelrisch | \(\frac {2 x \ln \left (c \left (f x +e \right )\right ) b \,e^{2} f i -\ln \left (c \left (f x +e \right )\right )^{2} b \,e^{3} i +\ln \left (c \left (f x +e \right )\right )^{2} b \,e^{2} f h +2 x a \,e^{2} f i -2 x b \,e^{2} f i -2 \ln \left (c \left (f x +e \right )\right ) a \,e^{3} i +2 \ln \left (c \left (f x +e \right )\right ) a \,e^{2} f h +2 \ln \left (c \left (f x +e \right )\right ) b \,e^{3} i}{2 d \,e^{2} f^{2}}\) | \(128\) |
risch | \(-\frac {b \ln \left (c \left (f x +e \right )\right )^{2} e i}{2 d \,f^{2}}+\frac {b \ln \left (c \left (f x +e \right )\right )^{2} h}{2 d f}+\frac {b i x \ln \left (c \left (f x +e \right )\right )}{d f}-\frac {\ln \left (f x +e \right ) a e i}{d \,f^{2}}+\frac {\ln \left (f x +e \right ) a h}{d f}+\frac {\ln \left (f x +e \right ) b e i}{d \,f^{2}}+\frac {a i x}{d f}-\frac {b i x}{d f}\) | \(130\) |
derivativedivides | \(\frac {-\frac {a c e i \ln \left (c f x +c e \right )}{f d}+\frac {a h c \ln \left (c f x +c e \right )}{d}+\frac {a i \left (c f x +c e \right )}{f d}-\frac {b c e i \ln \left (c f x +c e \right )^{2}}{2 f d}+\frac {b h c \ln \left (c f x +c e \right )^{2}}{2 d}+\frac {b i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f d}}{c f}\) | \(142\) |
default | \(\frac {-\frac {a c e i \ln \left (c f x +c e \right )}{f d}+\frac {a h c \ln \left (c f x +c e \right )}{d}+\frac {a i \left (c f x +c e \right )}{f d}-\frac {b c e i \ln \left (c f x +c e \right )^{2}}{2 f d}+\frac {b h c \ln \left (c f x +c e \right )^{2}}{2 d}+\frac {b i \left (\left (c f x +c e \right ) \ln \left (c f x +c e \right )-c f x -c e \right )}{f d}}{c f}\) | \(142\) |
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Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {2 \, {\left (a - b\right )} f i x + {\left (b f h - b e i\right )} \log \left (c f x + c e\right )^{2} + 2 \, {\left (b f i x + a f h - {\left (a - b\right )} e i\right )} \log \left (c f x + c e\right )}{2 \, d f^{2}} \]
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Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.08 \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {b i x \log {\left (c \left (e + f x\right ) \right )}}{d f} + x \left (\frac {a i}{d f} - \frac {b i}{d f}\right ) + \frac {\left (- b e i + b f h\right ) \log {\left (c \left (e + f x\right ) \right )}^{2}}{2 d f^{2}} - \frac {\left (a e i - a f h - b e i\right ) \log {\left (e + f x \right )}}{d f^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (77) = 154\).
Time = 0.23 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.54 \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))}{d e+d f x} \, dx=b i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} \log \left (c f x + c e\right ) - \frac {1}{2} \, b h {\left (\frac {2 \, \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} - \frac {\log \left (f x + e\right )^{2} + 2 \, \log \left (f x + e\right ) \log \left (c\right )}{d f}\right )} + a i {\left (\frac {x}{d f} - \frac {e \log \left (f x + e\right )}{d f^{2}}\right )} + \frac {b h \log \left (c f x + c e\right ) \log \left (d f x + d e\right )}{d f} + \frac {a h \log \left (d f x + d e\right )}{d f} + \frac {{\left (e \log \left (f x + e\right )^{2} - 2 \, f x + 2 \, e \log \left (f x + e\right )\right )} b i}{2 \, d f^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.16 \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {b i x \log \left (c f x + c e\right )}{d f} + \frac {{\left (a i - b i\right )} x}{d f} + \frac {{\left (b f h - b e i\right )} \log \left (c f x + c e\right )^{2}}{2 \, d f^{2}} + \frac {{\left (a f h - a e i + b e i\right )} \log \left (f x + e\right )}{d f^{2}} \]
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Time = 1.75 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.27 \[ \int \frac {(h+i x) (a+b \log (c (e+f x)))}{d e+d f x} \, dx=\frac {2\,a\,f\,i\,x-2\,b\,f\,i\,x-b\,e\,i\,{\ln \left (c\,e+c\,f\,x\right )}^2+b\,f\,h\,{\ln \left (c\,e+c\,f\,x\right )}^2-2\,a\,e\,i\,\ln \left (e+f\,x\right )+2\,a\,f\,h\,\ln \left (e+f\,x\right )+2\,b\,e\,i\,\ln \left (e+f\,x\right )+2\,b\,f\,i\,x\,\ln \left (c\,e+c\,f\,x\right )}{2\,d\,f^2} \]
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