Optimal. Leaf size=135 \[ \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2389, 2297, 2300, 2178} \[ \frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2178
Rule 2297
Rule 2300
Rule 2389
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx,x,d+e x\right )}{e}\\ &=-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{2 b e n}\\ &=-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e n^2}\\ &=-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac {\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e n^3}\\ &=\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac {d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac {d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 144, normalized size = 1.07 \[ -\frac {e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (b n e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (a+b \log \left (c (d+e x)^n\right )+b n\right )-\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Ei}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )\right )}{2 b^3 e n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.68, size = 263, normalized size = 1.95 \[ -\frac {{\left ({\left (b^{2} d n^{2} + a b d n + {\left (b^{2} e n^{2} + a b e n\right )} x + {\left (b^{2} e n^{2} x + b^{2} d n^{2}\right )} \log \left (e x + d\right ) + {\left (b^{2} e n x + b^{2} d n\right )} \log \relax (c)\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )} - {\left (b^{2} n^{2} \log \left (e x + d\right )^{2} + b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c) + a^{2} + 2 \, {\left (b^{2} n \log \relax (c) + a b n\right )} \log \left (e x + d\right )\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \relax (c) + a}{b n}\right )}\right )\right )} e^{\left (-\frac {b \log \relax (c) + a}{b n}\right )}}{2 \, {\left (b^{5} e n^{5} \log \left (e x + d\right )^{2} + b^{5} e n^{3} \log \relax (c)^{2} + 2 \, a b^{4} e n^{3} \log \relax (c) + a^{2} b^{3} e n^{3} + 2 \, {\left (b^{5} e n^{4} \log \relax (c) + a b^{4} e n^{4}\right )} \log \left (e x + d\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.27, size = 1322, normalized size = 9.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.66, size = 734, normalized size = 5.44 \[ -\frac {\left (e x +d \right ) c^{-\frac {1}{n}} \left (\left (e x +d \right )^{n}\right )^{-\frac {1}{n}} \Ei \left (1, -\ln \left (e x +d \right )-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b \ln \relax (c )+2 a +2 \left (-n \ln \left (e x +d \right )+\ln \left (\left (e x +d \right )^{n}\right )\right ) b}{2 b n}\right ) {\mathrm e}^{-\frac {-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 a}{2 b n}}}{2 b^{3} e \,n^{3}}-\frac {-i \pi b e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b e x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b e x \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b e x \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b d \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b d \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b e n x +2 b e x \ln \relax (c )+2 b e x \ln \left (\left (e x +d \right )^{n}\right )+2 a e x +2 b d n +2 b d \ln \relax (c )+2 b d \ln \left (\left (e x +d \right )^{n}\right )+2 a d}{\left (-i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )+i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+2 b \ln \relax (c )+2 b \ln \left (\left (e x +d \right )^{n}\right )+2 a \right )^{2} b^{2} e \,n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (d n + d \log \relax (c)\right )} b + a d + {\left ({\left (e n + e \log \relax (c)\right )} b + a e\right )} x + {\left (b e x + b d\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{2 \, {\left (b^{4} e n^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{4} e n^{2} \log \relax (c)^{2} + 2 \, a b^{3} e n^{2} \log \relax (c) + a^{2} b^{2} e n^{2} + 2 \, {\left (b^{4} e n^{2} \log \relax (c) + a b^{3} e n^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )\right )}} + \int \frac {1}{2 \, {\left (b^{3} n^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________