Optimal. Leaf size=71 \[ -\frac{d \left (a+b \log \left (c x^n\right )\right )}{5 x^5}-\frac{e x^{r-5} \left (a+b \log \left (c x^n\right )\right )}{5-r}-\frac{b d n}{25 x^5}-\frac{b e n x^{r-5}}{(5-r)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0721168, antiderivative size = 63, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {14, 2334} \[ -\frac{1}{5} \left (\frac{d}{x^5}+\frac{5 e x^{r-5}}{5-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b d n}{25 x^5}-\frac{b e n x^{r-5}}{(5-r)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2334
Rubi steps
\begin{align*} \int \frac{\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx &=-\frac{1}{5} \left (\frac{d}{x^5}+\frac{5 e x^{-5+r}}{5-r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{d}{5 x^6}+\frac{e x^{-6+r}}{-5+r}\right ) \, dx\\ &=-\frac{b d n}{25 x^5}-\frac{b e n x^{-5+r}}{(5-r)^2}-\frac{1}{5} \left (\frac{d}{x^5}+\frac{5 e x^{-5+r}}{5-r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.093609, size = 72, normalized size = 1.01 \[ -\frac{5 a (r-5) \left (d (r-5)-5 e x^r\right )+5 b (r-5) \log \left (c x^n\right ) \left (d (r-5)-5 e x^r\right )+b n \left (d (r-5)^2+25 e x^r\right )}{25 (r-5)^2 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.167, size = 614, normalized size = 8.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.33956, size = 367, normalized size = 5.17 \begin{align*} -\frac{25 \, b d n +{\left (b d n + 5 \, a d\right )} r^{2} + 125 \, a d - 10 \,{\left (b d n + 5 \, a d\right )} r + 25 \,{\left (b e n - a e r + 5 \, a e -{\left (b e r - 5 \, b e\right )} \log \left (c\right ) -{\left (b e n r - 5 \, b e n\right )} \log \left (x\right )\right )} x^{r} + 5 \,{\left (b d r^{2} - 10 \, b d r + 25 \, b d\right )} \log \left (c\right ) + 5 \,{\left (b d n r^{2} - 10 \, b d n r + 25 \, b d n\right )} \log \left (x\right )}{25 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.2723, size = 536, normalized size = 7.55 \begin{align*} -\frac{b d n r^{2} \log \left (x\right )}{5 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{b n r x^{r} e \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{b d n r^{2}}{25 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{b d r^{2} \log \left (c\right )}{5 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{b r x^{r} e \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{2 \, b d n r \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, b n x^{r} e \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{2 \, b d n r}{5 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{a d r^{2}}{5 \,{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{b n x^{r} e}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{a r x^{r} e}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{2 \, b d r \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, b x^{r} e \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, b d n \log \left (x\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{b d n}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} + \frac{2 \, a d r}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, a x^{r} e}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, b d \log \left (c\right )}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} - \frac{5 \, a d}{{\left (r^{2} - 10 \, r + 25\right )} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]