Optimal. Leaf size=75 \[ \frac{d (c d-b e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0356838, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {698} \[ \frac{d (c d-b e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin{align*} \int (d+e x)^m \left (b x+c x^2\right ) \, dx &=\int \left (\frac{d (c d-b e) (d+e x)^m}{e^2}+\frac{(-2 c d+b e) (d+e x)^{1+m}}{e^2}+\frac{c (d+e x)^{2+m}}{e^2}\right ) \, dx\\ &=\frac{d (c d-b e) (d+e x)^{1+m}}{e^3 (1+m)}-\frac{(2 c d-b e) (d+e x)^{2+m}}{e^3 (2+m)}+\frac{c (d+e x)^{3+m}}{e^3 (3+m)}\\ \end{align*}
Mathematica [A] time = 0.0529326, size = 75, normalized size = 1. \[ \frac{d (c d-b e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac{c (d+e x)^{m+3}}{e^3 (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.053, size = 116, normalized size = 1.6 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{1+m} \left ( -c{e}^{2}{m}^{2}{x}^{2}-b{e}^{2}{m}^{2}x-3\,c{e}^{2}m{x}^{2}-4\,b{e}^{2}mx+2\,cdemx-2\,c{e}^{2}{x}^{2}+bdem-3\,b{e}^{2}x+2\,cdex+3\,bde-2\,c{d}^{2} \right ) }{{e}^{3} \left ({m}^{3}+6\,{m}^{2}+11\,m+6 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.21144, size = 153, normalized size = 2.04 \begin{align*} \frac{{\left (e^{2}{\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )}{\left (e x + d\right )}^{m} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac{{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} +{\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )}{\left (e x + d\right )}^{m} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.02048, size = 323, normalized size = 4.31 \begin{align*} -\frac{{\left (b d^{2} e m - 2 \, c d^{3} + 3 \, b d^{2} e -{\left (c e^{3} m^{2} + 3 \, c e^{3} m + 2 \, c e^{3}\right )} x^{3} -{\left (3 \, b e^{3} +{\left (c d e^{2} + b e^{3}\right )} m^{2} +{\left (c d e^{2} + 4 \, b e^{3}\right )} m\right )} x^{2} -{\left (b d e^{2} m^{2} -{\left (2 \, c d^{2} e - 3 \, b d e^{2}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.31757, size = 1095, normalized size = 14.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.34015, size = 355, normalized size = 4.73 \begin{align*} \frac{{\left (x e + d\right )}^{m} c m^{2} x^{3} e^{3} +{\left (x e + d\right )}^{m} c d m^{2} x^{2} e^{2} +{\left (x e + d\right )}^{m} b m^{2} x^{2} e^{3} + 3 \,{\left (x e + d\right )}^{m} c m x^{3} e^{3} +{\left (x e + d\right )}^{m} b d m^{2} x e^{2} +{\left (x e + d\right )}^{m} c d m x^{2} e^{2} - 2 \,{\left (x e + d\right )}^{m} c d^{2} m x e + 4 \,{\left (x e + d\right )}^{m} b m x^{2} e^{3} + 2 \,{\left (x e + d\right )}^{m} c x^{3} e^{3} + 3 \,{\left (x e + d\right )}^{m} b d m x e^{2} -{\left (x e + d\right )}^{m} b d^{2} m e + 2 \,{\left (x e + d\right )}^{m} c d^{3} + 3 \,{\left (x e + d\right )}^{m} b x^{2} e^{3} - 3 \,{\left (x e + d\right )}^{m} b d^{2} e}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]