Optimal. Leaf size=109 \[ -\frac{b d n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{2 b e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{b^2 d n^2}{32 x^4}-\frac{2 b^2 e n^2}{27 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.135152, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2353, 2305, 2304} \[ -\frac{b d n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{2 b e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{b^2 d n^2}{32 x^4}-\frac{2 b^2 e n^2}{27 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2353
Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx &=\int \left (\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{x^5}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{x^4}\right ) \, dx\\ &=d \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx+e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx\\ &=-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}+\frac{1}{2} (b d n) \int \frac{a+b \log \left (c x^n\right )}{x^5} \, dx+\frac{1}{3} (2 b e n) \int \frac{a+b \log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac{b^2 d n^2}{32 x^4}-\frac{2 b^2 e n^2}{27 x^3}-\frac{b d n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac{2 b e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0532247, size = 82, normalized size = 0.75 \[ -\frac{216 d \left (a+b \log \left (c x^n\right )\right )^2+27 b d n \left (4 a+4 b \log \left (c x^n\right )+b n\right )+288 e x \left (a+b \log \left (c x^n\right )\right )^2+64 b e n x \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{864 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.204, size = 1486, normalized size = 13.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.15612, size = 204, normalized size = 1.87 \begin{align*} -\frac{2}{27} \, b^{2} e{\left (\frac{n^{2}}{x^{3}} + \frac{3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac{1}{32} \, b^{2} d{\left (\frac{n^{2}}{x^{4}} + \frac{4 \, n \log \left (c x^{n}\right )}{x^{4}}\right )} - \frac{b^{2} e \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac{2 \, a b e n}{9 \, x^{3}} - \frac{2 \, a b e \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{b^{2} d \log \left (c x^{n}\right )^{2}}{4 \, x^{4}} - \frac{a b d n}{8 \, x^{4}} - \frac{a^{2} e}{3 \, x^{3}} - \frac{a b d \log \left (c x^{n}\right )}{2 \, x^{4}} - \frac{a^{2} d}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.02065, size = 467, normalized size = 4.28 \begin{align*} -\frac{27 \, b^{2} d n^{2} + 108 \, a b d n + 216 \, a^{2} d + 72 \,{\left (4 \, b^{2} e x + 3 \, b^{2} d\right )} \log \left (c\right )^{2} + 72 \,{\left (4 \, b^{2} e n^{2} x + 3 \, b^{2} d n^{2}\right )} \log \left (x\right )^{2} + 32 \,{\left (2 \, b^{2} e n^{2} + 6 \, a b e n + 9 \, a^{2} e\right )} x + 12 \,{\left (9 \, b^{2} d n + 36 \, a b d + 16 \,{\left (b^{2} e n + 3 \, a b e\right )} x\right )} \log \left (c\right ) + 12 \,{\left (9 \, b^{2} d n^{2} + 36 \, a b d n + 16 \,{\left (b^{2} e n^{2} + 3 \, a b e n\right )} x + 12 \,{\left (4 \, b^{2} e n x + 3 \, b^{2} d n\right )} \log \left (c\right )\right )} \log \left (x\right )}{864 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 4.39674, size = 311, normalized size = 2.85 \begin{align*} - \frac{a^{2} d}{4 x^{4}} - \frac{a^{2} e}{3 x^{3}} - \frac{a b d n \log{\left (x \right )}}{2 x^{4}} - \frac{a b d n}{8 x^{4}} - \frac{a b d \log{\left (c \right )}}{2 x^{4}} - \frac{2 a b e n \log{\left (x \right )}}{3 x^{3}} - \frac{2 a b e n}{9 x^{3}} - \frac{2 a b e \log{\left (c \right )}}{3 x^{3}} - \frac{b^{2} d n^{2} \log{\left (x \right )}^{2}}{4 x^{4}} - \frac{b^{2} d n^{2} \log{\left (x \right )}}{8 x^{4}} - \frac{b^{2} d n^{2}}{32 x^{4}} - \frac{b^{2} d n \log{\left (c \right )} \log{\left (x \right )}}{2 x^{4}} - \frac{b^{2} d n \log{\left (c \right )}}{8 x^{4}} - \frac{b^{2} d \log{\left (c \right )}^{2}}{4 x^{4}} - \frac{b^{2} e n^{2} \log{\left (x \right )}^{2}}{3 x^{3}} - \frac{2 b^{2} e n^{2} \log{\left (x \right )}}{9 x^{3}} - \frac{2 b^{2} e n^{2}}{27 x^{3}} - \frac{2 b^{2} e n \log{\left (c \right )} \log{\left (x \right )}}{3 x^{3}} - \frac{2 b^{2} e n \log{\left (c \right )}}{9 x^{3}} - \frac{b^{2} e \log{\left (c \right )}^{2}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.38664, size = 278, normalized size = 2.55 \begin{align*} -\frac{288 \, b^{2} n^{2} x e \log \left (x\right )^{2} + 192 \, b^{2} n^{2} x e \log \left (x\right ) + 576 \, b^{2} n x e \log \left (c\right ) \log \left (x\right ) + 216 \, b^{2} d n^{2} \log \left (x\right )^{2} + 64 \, b^{2} n^{2} x e + 192 \, b^{2} n x e \log \left (c\right ) + 288 \, b^{2} x e \log \left (c\right )^{2} + 108 \, b^{2} d n^{2} \log \left (x\right ) + 576 \, a b n x e \log \left (x\right ) + 432 \, b^{2} d n \log \left (c\right ) \log \left (x\right ) + 27 \, b^{2} d n^{2} + 192 \, a b n x e + 108 \, b^{2} d n \log \left (c\right ) + 576 \, a b x e \log \left (c\right ) + 216 \, b^{2} d \log \left (c\right )^{2} + 432 \, a b d n \log \left (x\right ) + 108 \, a b d n + 288 \, a^{2} x e + 432 \, a b d \log \left (c\right ) + 216 \, a^{2} d}{864 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]