Optimal. Leaf size=50 \[ \frac{\left (\frac{b^2}{4 c}+b x+c x^2\right )^n \left (\frac{b e}{2 c}+e x\right )^{m+1}}{e (m+2 n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0205833, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {644, 32} \[ \frac{\left (\frac{b^2}{4 c}+b x+c x^2\right )^n \left (\frac{b e}{2 c}+e x\right )^{m+1}}{e (m+2 n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 644
Rule 32
Rubi steps
\begin{align*} \int \left (\frac{b e}{2 c}+e x\right )^m \left (\frac{b^2}{4 c}+b x+c x^2\right )^n \, dx &=\left (\left (\frac{b e}{2 c}+e x\right )^{-2 n} \left (\frac{b^2}{4 c}+b x+c x^2\right )^n\right ) \int \left (\frac{b e}{2 c}+e x\right )^{m+2 n} \, dx\\ &=\frac{\left (\frac{b e}{2 c}+e x\right )^{1+m} \left (\frac{b^2}{4 c}+b x+c x^2\right )^n}{e (1+m+2 n)}\\ \end{align*}
Mathematica [A] time = 0.0271691, size = 54, normalized size = 1.08 \[ \frac{2^{-2 n-1} (b+2 c x) \left (\frac{(b+2 c x)^2}{c}\right )^n \left (\frac{b e}{2 c}+e x\right )^m}{c (m+2 n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 58, normalized size = 1.2 \begin{align*}{\frac{2\,cx+b}{2\,c \left ( 1+m+2\,n \right ) } \left ({\frac{e \left ( 2\,cx+b \right ) }{2\,c}} \right ) ^{m} \left ({\frac{4\,{c}^{2}{x}^{2}+4\,bcx+{b}^{2}}{4\,c}} \right ) ^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.66517, size = 107, normalized size = 2.14 \begin{align*} \frac{{\left (2 \, c e^{m} x + b e^{m}\right )} c^{-m - n - 1} e^{\left (m \log \left (2 \, c x + b\right ) + 2 \, n \log \left (2 \, c x + b\right )\right )}}{{\left (2^{2 \, n + 2} n + 2^{2 \, n + 1}\right )} 2^{m} + 2^{m + 2 \, n + 1} m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.19233, size = 150, normalized size = 3. \begin{align*} \frac{{\left (2 \, c x + b\right )} \left (\frac{2 \, c e x + b e}{2 \, c}\right )^{m} e^{\left (2 \, n \log \left (\frac{2 \, c e x + b e}{2 \, c}\right ) + n \log \left (\frac{c}{e^{2}}\right )\right )}}{2 \,{\left (c m + 2 \, c n + c\right )}} \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 = 1.18686, size = 140, normalized size = 2.8 \begin{align*} \frac{2 \, c x e^{\left (-m \log \left (2\right ) - 2 \, n \log \left (2\right ) + m \log \left (2 \, c x + b\right ) + 2 \, n \log \left (2 \, c x + b\right ) - m \log \left (c\right ) - n \log \left (c\right ) + m\right )} + b e^{\left (-m \log \left (2\right ) - 2 \, n \log \left (2\right ) + m \log \left (2 \, c x + b\right ) + 2 \, n \log \left (2 \, c x + b\right ) - m \log \left (c\right ) - n \log \left (c\right ) + m\right )}}{2 \,{\left (c m + 2 \, c n + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]