Integrand size = 21, antiderivative size = 170 \[ \int \frac {x^3 (a+b \log (c x))}{d+\frac {e}{x}} \, dx=-\frac {a e^3 x}{d^4}+\frac {b e^3 x}{d^4}-\frac {b e^2 x^2}{4 d^3}+\frac {b e x^3}{9 d^2}-\frac {b x^4}{16 d}-\frac {b e^3 x \log (c x)}{d^4}+\frac {e^2 x^2 (a+b \log (c x))}{2 d^3}-\frac {e x^3 (a+b \log (c x))}{3 d^2}+\frac {x^4 (a+b \log (c x))}{4 d}+\frac {e^4 (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^5}+\frac {b e^4 \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^5} \]
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Time = 0.13 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {269, 45, 2393, 2332, 2341, 2354, 2438} \[ \int \frac {x^3 (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\frac {e^4 \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d^5}+\frac {e^2 x^2 (a+b \log (c x))}{2 d^3}-\frac {e x^3 (a+b \log (c x))}{3 d^2}+\frac {x^4 (a+b \log (c x))}{4 d}-\frac {a e^3 x}{d^4}-\frac {b e^3 x \log (c x)}{d^4}+\frac {b e^4 \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^5}+\frac {b e^3 x}{d^4}-\frac {b e^2 x^2}{4 d^3}+\frac {b e x^3}{9 d^2}-\frac {b x^4}{16 d} \]
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Rule 45
Rule 269
Rule 2332
Rule 2341
Rule 2354
Rule 2393
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {e^3 (a+b \log (c x))}{d^4}+\frac {e^2 x (a+b \log (c x))}{d^3}-\frac {e x^2 (a+b \log (c x))}{d^2}+\frac {x^3 (a+b \log (c x))}{d}+\frac {e^4 (a+b \log (c x))}{d^4 (e+d x)}\right ) \, dx \\ & = \frac {\int x^3 (a+b \log (c x)) \, dx}{d}-\frac {e \int x^2 (a+b \log (c x)) \, dx}{d^2}+\frac {e^2 \int x (a+b \log (c x)) \, dx}{d^3}-\frac {e^3 \int (a+b \log (c x)) \, dx}{d^4}+\frac {e^4 \int \frac {a+b \log (c x)}{e+d x} \, dx}{d^4} \\ & = -\frac {a e^3 x}{d^4}-\frac {b e^2 x^2}{4 d^3}+\frac {b e x^3}{9 d^2}-\frac {b x^4}{16 d}+\frac {e^2 x^2 (a+b \log (c x))}{2 d^3}-\frac {e x^3 (a+b \log (c x))}{3 d^2}+\frac {x^4 (a+b \log (c x))}{4 d}+\frac {e^4 (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^5}-\frac {\left (b e^3\right ) \int \log (c x) \, dx}{d^4}-\frac {\left (b e^4\right ) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{d^5} \\ & = -\frac {a e^3 x}{d^4}+\frac {b e^3 x}{d^4}-\frac {b e^2 x^2}{4 d^3}+\frac {b e x^3}{9 d^2}-\frac {b x^4}{16 d}-\frac {b e^3 x \log (c x)}{d^4}+\frac {e^2 x^2 (a+b \log (c x))}{2 d^3}-\frac {e x^3 (a+b \log (c x))}{3 d^2}+\frac {x^4 (a+b \log (c x))}{4 d}+\frac {e^4 (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^5}+\frac {b e^4 \text {Li}_2\left (-\frac {d x}{e}\right )}{d^5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\frac {-144 a d e^3 x+144 b d e^3 x-36 b d^2 e^2 x^2+16 b d^3 e x^3-9 b d^4 x^4-144 b d e^3 x \log (c x)+72 d^2 e^2 x^2 (a+b \log (c x))-48 d^3 e x^3 (a+b \log (c x))+36 d^4 x^4 (a+b \log (c x))+144 e^4 (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )+144 b e^4 \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{144 d^5} \]
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Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.20
method | result | size |
parts | \(a \left (\frac {\frac {1}{4} d^{3} x^{4}-\frac {1}{3} e \,d^{2} x^{3}+\frac {1}{2} d \,e^{2} x^{2}-x \,e^{3}}{d^{4}}+\frac {e^{4} \ln \left (d x +e \right )}{d^{5}}\right )+\frac {b \,x^{4} \ln \left (x c \right )}{4 d}-\frac {b \,x^{4}}{16 d}-\frac {b e \,x^{3} \ln \left (x c \right )}{3 d^{2}}+\frac {b e \,x^{3}}{9 d^{2}}+\frac {b \,e^{2} x^{2} \ln \left (x c \right )}{2 d^{3}}-\frac {b \,e^{2} x^{2}}{4 d^{3}}-\frac {b \,e^{3} x \ln \left (x c \right )}{d^{4}}+\frac {b \,e^{3} x}{d^{4}}+\frac {b \,e^{4} \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d^{5}}+\frac {b \,e^{4} \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d^{5}}\) | \(204\) |
risch | \(\frac {a \,x^{4}}{4 d}-\frac {a e \,x^{3}}{3 d^{2}}+\frac {a \,e^{2} x^{2}}{2 d^{3}}-\frac {a \,e^{3} x}{d^{4}}+\frac {a \,e^{4} \ln \left (d x +e \right )}{d^{5}}+\frac {b \,x^{4} \ln \left (x c \right )}{4 d}-\frac {b \,x^{4}}{16 d}-\frac {b e \,x^{3} \ln \left (x c \right )}{3 d^{2}}+\frac {b e \,x^{3}}{9 d^{2}}+\frac {b \,e^{2} x^{2} \ln \left (x c \right )}{2 d^{3}}-\frac {b \,e^{2} x^{2}}{4 d^{3}}-\frac {b \,e^{3} x \ln \left (x c \right )}{d^{4}}+\frac {b \,e^{3} x}{d^{4}}+\frac {b \,e^{4} \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d^{5}}+\frac {b \,e^{4} \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d^{5}}\) | \(206\) |
derivativedivides | \(\frac {a \left (-\frac {c^{4} e^{3} x -\frac {1}{2} d \,c^{4} e^{2} x^{2}+\frac {1}{3} e \,c^{4} x^{3} d^{2}-\frac {1}{4} x^{4} c^{4} d^{3}}{d^{4}}+\frac {c^{4} e^{4} \ln \left (c d x +c e \right )}{d^{5}}\right )+b \left (\frac {\frac {x^{4} c^{4} \ln \left (x c \right )}{4}-\frac {x^{4} c^{4}}{16}}{d}-\frac {c e \left (\frac {x^{3} c^{3} \ln \left (x c \right )}{3}-\frac {x^{3} c^{3}}{9}\right )}{d^{2}}+\frac {c^{2} e^{2} \left (\frac {x^{2} c^{2} \ln \left (x c \right )}{2}-\frac {x^{2} c^{2}}{4}\right )}{d^{3}}-\frac {c^{3} e^{3} \left (x c \ln \left (x c \right )-x c \right )}{d^{4}}+\frac {c^{4} e^{4} \left (\frac {\operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {\ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\right )}{d^{4}}\right )}{c^{4}}\) | \(242\) |
default | \(\frac {a \left (-\frac {c^{4} e^{3} x -\frac {1}{2} d \,c^{4} e^{2} x^{2}+\frac {1}{3} e \,c^{4} x^{3} d^{2}-\frac {1}{4} x^{4} c^{4} d^{3}}{d^{4}}+\frac {c^{4} e^{4} \ln \left (c d x +c e \right )}{d^{5}}\right )+b \left (\frac {\frac {x^{4} c^{4} \ln \left (x c \right )}{4}-\frac {x^{4} c^{4}}{16}}{d}-\frac {c e \left (\frac {x^{3} c^{3} \ln \left (x c \right )}{3}-\frac {x^{3} c^{3}}{9}\right )}{d^{2}}+\frac {c^{2} e^{2} \left (\frac {x^{2} c^{2} \ln \left (x c \right )}{2}-\frac {x^{2} c^{2}}{4}\right )}{d^{3}}-\frac {c^{3} e^{3} \left (x c \ln \left (x c \right )-x c \right )}{d^{4}}+\frac {c^{4} e^{4} \left (\frac {\operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {\ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\right )}{d^{4}}\right )}{c^{4}}\) | \(242\) |
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\[ \int \frac {x^3 (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\int { \frac {{\left (b \log \left (c x\right ) + a\right )} x^{3}}{d + \frac {e}{x}} \,d x } \]
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Time = 82.83 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.76 \[ \int \frac {x^3 (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\frac {a x^{4}}{4 d} - \frac {a e x^{3}}{3 d^{2}} + \frac {a e^{2} x^{2}}{2 d^{3}} + \frac {a e^{4} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{4}} - \frac {a e^{3} x}{d^{4}} + \frac {b x^{4} \log {\left (c x \right )}}{4 d} - \frac {b x^{4}}{16 d} - \frac {b e x^{3} \log {\left (c x \right )}}{3 d^{2}} + \frac {b e x^{3}}{9 d^{2}} + \frac {b e^{2} x^{2} \log {\left (c x \right )}}{2 d^{3}} - \frac {b e^{2} x^{2}}{4 d^{3}} - \frac {b e^{4} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d^{4}} + \frac {b e^{4} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x \right )}}{d^{4}} - \frac {b e^{3} x \log {\left (c x \right )}}{d^{4}} + \frac {b e^{3} x}{d^{4}} \]
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Time = 0.24 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.24 \[ \int \frac {x^3 (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\frac {{\left (\log \left (\frac {d x}{e} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {d x}{e}\right )\right )} b e^{4}}{d^{5}} + \frac {9 \, {\left (4 \, a d^{3} + {\left (4 \, d^{3} \log \left (c\right ) - d^{3}\right )} b\right )} x^{4} - 16 \, {\left (3 \, a d^{2} e + {\left (3 \, d^{2} e \log \left (c\right ) - d^{2} e\right )} b\right )} x^{3} + 36 \, {\left (2 \, a d e^{2} + {\left (2 \, d e^{2} \log \left (c\right ) - d e^{2}\right )} b\right )} x^{2} - 144 \, {\left (a e^{3} + {\left (e^{3} \log \left (c\right ) - e^{3}\right )} b\right )} x + 12 \, {\left (3 \, b d^{3} x^{4} - 4 \, b d^{2} e x^{3} + 6 \, b d e^{2} x^{2} - 12 \, b e^{3} x\right )} \log \left (x\right )}{144 \, d^{4}} + \frac {{\left (b e^{4} \log \left (c\right ) + a e^{4}\right )} \log \left (d x + e\right )}{d^{5}} \]
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\[ \int \frac {x^3 (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\int { \frac {{\left (b \log \left (c x\right ) + a\right )} x^{3}}{d + \frac {e}{x}} \,d x } \]
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Timed out. \[ \int \frac {x^3 (a+b \log (c x))}{d+\frac {e}{x}} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,x\right )\right )}{d+\frac {e}{x}} \,d x \]
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