Optimal. Leaf size=66 \[ \frac{x^{m+1} \log \left (d \left (b x+c x^2\right )^n\right )}{m+1}+\frac{n x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{c x}{b}\right )}{(m+1)^2}-\frac{2 n x^{m+1}}{(m+1)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.059421, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2525, 80, 64} \[ \frac{x^{m+1} \log \left (d \left (b x+c x^2\right )^n\right )}{m+1}+\frac{n x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{c x}{b}\right )}{(m+1)^2}-\frac{2 n x^{m+1}}{(m+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2525
Rule 80
Rule 64
Rubi steps
\begin{align*} \int x^m \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=\frac{x^{1+m} \log \left (d \left (b x+c x^2\right )^n\right )}{1+m}-\frac{n \int \frac{x^m (b+2 c x)}{b+c x} \, dx}{1+m}\\ &=-\frac{2 n x^{1+m}}{(1+m)^2}+\frac{x^{1+m} \log \left (d \left (b x+c x^2\right )^n\right )}{1+m}+\frac{(b n) \int \frac{x^m}{b+c x} \, dx}{1+m}\\ &=-\frac{2 n x^{1+m}}{(1+m)^2}+\frac{n x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{c x}{b}\right )}{(1+m)^2}+\frac{x^{1+m} \log \left (d \left (b x+c x^2\right )^n\right )}{1+m}\\ \end{align*}
Mathematica [A] time = 0.0209516, size = 48, normalized size = 0.73 \[ \frac{x^{m+1} \left ((m+1) \log \left (d (x (b+c x))^n\right )+n \, _2F_1\left (1,m+1;m+2;-\frac{c x}{b}\right )-2 n\right )}{(m+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\ln \left ( d \left ( c{x}^{2}+bx \right ) ^{n} \right ) \, dx \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \log{\left (d \left (b x + c x^{2}\right )^{n} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]