Optimal. Leaf size=57 \[ -\frac{d \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{b d n}{9 x^3}-\frac{b e n}{4 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0447902, antiderivative size = 48, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {43, 2334, 12} \[ -\frac{1}{6} \left (\frac{2 d}{x^3}+\frac{3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d n}{9 x^3}-\frac{b e n}{4 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2334
Rule 12
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac{1}{6} \left (\frac{2 d}{x^3}+\frac{3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{-2 d-3 e x}{6 x^4} \, dx\\ &=-\frac{1}{6} \left (\frac{2 d}{x^3}+\frac{3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \int \frac{-2 d-3 e x}{x^4} \, dx\\ &=-\frac{1}{6} \left (\frac{2 d}{x^3}+\frac{3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{6} (b n) \int \left (-\frac{2 d}{x^4}-\frac{3 e}{x^3}\right ) \, dx\\ &=-\frac{b d n}{9 x^3}-\frac{b e n}{4 x^2}-\frac{1}{6} \left (\frac{2 d}{x^3}+\frac{3 e}{x^2}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0229789, size = 47, normalized size = 0.82 \[ -\frac{6 a (2 d+3 e x)+6 b (2 d+3 e x) \log \left (c x^n\right )+b n (4 d+9 e x)}{36 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.11, size = 235, normalized size = 4.1 \begin{align*} -{\frac{b \left ( 3\,ex+2\,d \right ) \ln \left ({x}^{n} \right ) }{6\,{x}^{3}}}-{\frac{9\,i\pi \,bex{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-9\,i\pi \,bex{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -9\,i\pi \,bex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+9\,i\pi \,bex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +18\,\ln \left ( c \right ) bex+9\,benx+18\,aex+6\,i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-6\,i\pi \,bd{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -6\,i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+6\,i\pi \,bd \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +12\,\ln \left ( c \right ) bd+4\,bdn+12\,ad}{36\,{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.15926, size = 77, normalized size = 1.35 \begin{align*} -\frac{b e n}{4 \, x^{2}} - \frac{b e \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{b d n}{9 \, x^{3}} - \frac{a e}{2 \, x^{2}} - \frac{b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{a d}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.04731, size = 154, normalized size = 2.7 \begin{align*} -\frac{4 \, b d n + 12 \, a d + 9 \,{\left (b e n + 2 \, a e\right )} x + 6 \,{\left (3 \, b e x + 2 \, b d\right )} \log \left (c\right ) + 6 \,{\left (3 \, b e n x + 2 \, b d n\right )} \log \left (x\right )}{36 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.55703, size = 88, normalized size = 1.54 \begin{align*} - \frac{a d}{3 x^{3}} - \frac{a e}{2 x^{2}} - \frac{b d n \log{\left (x \right )}}{3 x^{3}} - \frac{b d n}{9 x^{3}} - \frac{b d \log{\left (c \right )}}{3 x^{3}} - \frac{b e n \log{\left (x \right )}}{2 x^{2}} - \frac{b e n}{4 x^{2}} - \frac{b e \log{\left (c \right )}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.2936, size = 78, normalized size = 1.37 \begin{align*} -\frac{18 \, b n x e \log \left (x\right ) + 9 \, b n x e + 18 \, b x e \log \left (c\right ) + 12 \, b d n \log \left (x\right ) + 4 \, b d n + 18 \, a x e + 12 \, b d \log \left (c\right ) + 12 \, a d}{36 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]