Optimal. Leaf size=60 \[ \frac{(2 x+3) (2 d-3 e) \left (4 x^2+12 x+9\right )^p}{4 (2 p+1)}+\frac{e \left (4 x^2+12 x+9\right )^{p+1}}{8 (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0149501, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {640, 609} \[ \frac{(2 x+3) (2 d-3 e) \left (4 x^2+12 x+9\right )^p}{4 (2 p+1)}+\frac{e \left (4 x^2+12 x+9\right )^{p+1}}{8 (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 640
Rule 609
Rubi steps
\begin{align*} \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx &=\frac{e \left (9+12 x+4 x^2\right )^{1+p}}{8 (1+p)}+\frac{1}{2} (2 d-3 e) \int \left (9+12 x+4 x^2\right )^p \, dx\\ &=\frac{(2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^p}{4 (1+2 p)}+\frac{e \left (9+12 x+4 x^2\right )^{1+p}}{8 (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0246458, size = 48, normalized size = 0.8 \[ \frac{(2 x+3) \left ((2 x+3)^2\right )^p (4 d (p+1)+e ((4 p+2) x-3))}{8 (p+1) (2 p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.042, size = 52, normalized size = 0.9 \begin{align*}{\frac{ \left ( 4\,{x}^{2}+12\,x+9 \right ) ^{p} \left ( 4\,epx+4\,dp+2\,ex+4\,d-3\,e \right ) \left ( 3+2\,x \right ) }{16\,{p}^{2}+24\,p+8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.07015, size = 88, normalized size = 1.47 \begin{align*} \frac{{\left (4 \,{\left (2 \, p + 1\right )} x^{2} + 12 \, p x - 9\right )} e{\left (2 \, x + 3\right )}^{2 \, p}}{8 \,{\left (2 \, p^{2} + 3 \, p + 1\right )}} + \frac{d{\left (2 \, x + 3\right )}^{2 \, p}{\left (2 \, x + 3\right )}}{2 \,{\left (2 \, p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.8246, size = 154, normalized size = 2.57 \begin{align*} \frac{{\left (4 \,{\left (2 \, e p + e\right )} x^{2} + 12 \, d p + 4 \,{\left ({\left (2 \, d + 3 \, e\right )} p + 2 \, d\right )} x + 12 \, d - 9 \, e\right )}{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p}}{8 \,{\left (2 \, p^{2} + 3 \, p + 1\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 = 1.16227, size = 205, normalized size = 3.42 \begin{align*} \frac{8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} p x^{2} e + 8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d p x + 12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} p x e + 4 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} x^{2} e + 12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d p + 8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d x + 12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d - 9 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} e}{8 \,{\left (2 \, p^{2} + 3 \, p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]