Optimal. Leaf size=117 \[ -\frac{a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}+\frac{b n \log (x)}{6 d^2 e^2}-\frac{b n \log (d+e x)}{6 d^2 e^2}+\frac{b n}{6 d e^2 (d+e x)}-\frac{b n}{6 e^2 (d+e x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0871504, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {43, 2350, 12, 77} \[ -\frac{a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}+\frac{b n \log (x)}{6 d^2 e^2}-\frac{b n \log (d+e x)}{6 d^2 e^2}+\frac{b n}{6 d e^2 (d+e x)}-\frac{b n}{6 e^2 (d+e x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2350
Rule 12
Rule 77
Rubi steps
\begin{align*} \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac{a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}-(b n) \int \frac{-d-3 e x}{6 e^2 x (d+e x)^3} \, dx\\ &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac{a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}-\frac{(b n) \int \frac{-d-3 e x}{x (d+e x)^3} \, dx}{6 e^2}\\ &=\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac{a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}-\frac{(b n) \int \left (-\frac{1}{d^2 x}-\frac{2 e}{(d+e x)^3}+\frac{e}{d (d+e x)^2}+\frac{e}{d^2 (d+e x)}\right ) \, dx}{6 e^2}\\ &=-\frac{b n}{6 e^2 (d+e x)^2}+\frac{b n}{6 d e^2 (d+e x)}+\frac{b n \log (x)}{6 d^2 e^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac{a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}-\frac{b n \log (d+e x)}{6 d^2 e^2}\\ \end{align*}
Mathematica [A] time = 0.0897124, size = 135, normalized size = 1.15 \[ -\frac{a+b \log \left (c x^n\right )}{2 e^2 (d+e x)^2}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^3}-\frac{b n \left (-\frac{2 \log (d+e x)}{d^2}+\frac{2 \log (x)}{d^2}+\frac{2}{d (d+e x)}+\frac{1}{(d+e x)^2}\right )}{6 e^2}+\frac{b n \left (-\frac{\log (d+e x)}{d^2}+\frac{\log (x)}{d^2}+\frac{1}{d (d+e x)}\right )}{2 e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.115, size = 403, normalized size = 3.4 \begin{align*} -{\frac{b \left ( 3\,ex+d \right ) \ln \left ({x}^{n} \right ) }{6\, \left ( ex+d \right ) ^{3}{e}^{2}}}-{\frac{3\,i\pi \,b{d}^{2}ex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \,b{d}^{3}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,b{d}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +3\,i\pi \,b{d}^{2}ex{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-3\,i\pi \,b{d}^{2}ex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \,b{d}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-i\pi \,b{d}^{3}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -3\,i\pi \,b{d}^{2}ex{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +2\,\ln \left ( ex+d \right ) b{e}^{3}n{x}^{3}-2\,\ln \left ( -x \right ) b{e}^{3}n{x}^{3}+6\,\ln \left ( ex+d \right ) bd{e}^{2}n{x}^{2}-6\,\ln \left ( -x \right ) bd{e}^{2}n{x}^{2}+6\,\ln \left ( ex+d \right ) b{d}^{2}enx-6\,\ln \left ( -x \right ) b{d}^{2}enx-2\,bd{e}^{2}n{x}^{2}+6\,\ln \left ( c \right ) b{d}^{2}ex+2\,\ln \left ( ex+d \right ) b{d}^{3}n-2\,\ln \left ( -x \right ) b{d}^{3}n-2\,b{d}^{2}enx+2\,\ln \left ( c \right ) b{d}^{3}+6\,a{d}^{2}ex+2\,a{d}^{3}}{12\,{e}^{2}{d}^{2} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.15431, size = 203, normalized size = 1.74 \begin{align*} \frac{1}{6} \, b n{\left (\frac{x}{d e^{3} x^{2} + 2 \, d^{2} e^{2} x + d^{3} e} - \frac{\log \left (e x + d\right )}{d^{2} e^{2}} + \frac{\log \left (x\right )}{d^{2} e^{2}}\right )} - \frac{{\left (3 \, e x + d\right )} b \log \left (c x^{n}\right )}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac{{\left (3 \, e x + d\right )} a}{6 \,{\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.07109, size = 346, normalized size = 2.96 \begin{align*} \frac{b d e^{2} n x^{2} - a d^{3} +{\left (b d^{2} e n - 3 \, a d^{2} e\right )} x -{\left (b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2} + 3 \, b d^{2} e n x + b d^{3} n\right )} \log \left (e x + d\right ) -{\left (3 \, b d^{2} e x + b d^{3}\right )} \log \left (c\right ) +{\left (b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2}\right )} \log \left (x\right )}{6 \,{\left (d^{2} e^{5} x^{3} + 3 \, d^{3} e^{4} x^{2} + 3 \, d^{4} e^{3} x + d^{5} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 11.6869, size = 799, normalized size = 6.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27931, size = 238, normalized size = 2.03 \begin{align*} -\frac{b n x^{3} e^{3} \log \left (x e + d\right ) + 3 \, b d n x^{2} e^{2} \log \left (x e + d\right ) + 3 \, b d^{2} n x e \log \left (x e + d\right ) - b n x^{3} e^{3} \log \left (x\right ) - 3 \, b d n x^{2} e^{2} \log \left (x\right ) - b d n x^{2} e^{2} - b d^{2} n x e + b d^{3} n \log \left (x e + d\right ) + 3 \, b d^{2} x e \log \left (c\right ) + 3 \, a d^{2} x e + b d^{3} \log \left (c\right ) + a d^{3}}{6 \,{\left (d^{2} x^{3} e^{5} + 3 \, d^{3} x^{2} e^{4} + 3 \, d^{4} x e^{3} + d^{5} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]