### 3.1103 $$\int (c d^2+2 c d e x+c e^2 x^2)^p \, dx$$

Optimal. Leaf size=38 $\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)}$

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p)/(e*(1 + 2*p))

________________________________________________________________________________________

Rubi [A]  time = 0.0097498, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {609} $\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p)/(e*(1 + 2*p))

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rubi steps

\begin{align*} \int \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx &=\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (1+2 p)}\\ \end{align*}

Mathematica [A]  time = 0.0101832, size = 25, normalized size = 0.66 $\frac{(d+e x) \left (c (d+e x)^2\right )^p}{2 e p+e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]

[Out]

((d + e*x)*(c*(d + e*x)^2)^p)/(e + 2*e*p)

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 39, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{e \left ( 1+2\,p \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^p,x)

[Out]

(e*x+d)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^p/e/(1+2*p)

________________________________________________________________________________________

Maxima [A]  time = 1.24028, size = 43, normalized size = 1.13 \begin{align*} \frac{{\left (c^{p} e x + c^{p} d\right )}{\left (e x + d\right )}^{2 \, p}}{e{\left (2 \, p + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^p,x, algorithm="maxima")

[Out]

(c^p*e*x + c^p*d)*(e*x + d)^(2*p)/(e*(2*p + 1))

________________________________________________________________________________________

Fricas [A]  time = 2.52531, size = 77, normalized size = 2.03 \begin{align*} \frac{{\left (e x + d\right )}{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, e p + e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^p,x, algorithm="fricas")

[Out]

(e*x + d)*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^p/(2*e*p + e)

________________________________________________________________________________________

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]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**p,x)

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [A]  time = 1.32154, size = 84, normalized size = 2.21 \begin{align*} \frac{{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} x e +{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d}{2 \, p e + e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^p,x, algorithm="giac")

[Out]

((c*x^2*e^2 + 2*c*d*x*e + c*d^2)^p*x*e + (c*x^2*e^2 + 2*c*d*x*e + c*d^2)^p*d)/(2*p*e + e)