### 3.1102 $$\int (d+e x) (c d^2+2 c d e x+c e^2 x^2)^p \, dx$$

Optimal. Leaf size=39 $\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{p+1}}{2 c e (p+1)}$

[Out]

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

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Rubi [A]  time = 0.0103326, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.036, Rules used = {629} $\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{p+1}}{2 c e (p+1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 629

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

Rubi steps

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

Mathematica [A]  time = 0.0143138, size = 28, normalized size = 0.72 $\frac{\left (c (d+e x)^2\right )^{p+1}}{2 c e (p+1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 40, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) ^{2} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{2\,e \left ( 1+p \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.622, size = 101, normalized size = 2.59 \begin{align*} \frac{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \,{\left (e p + e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.472735, size = 139, normalized size = 3.56 \begin{align*} \begin{cases} \frac{x}{c d} & \text{for}\: e = 0 \wedge p = -1 \\d x \left (c d^{2}\right )^{p} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{c e} & \text{for}\: p = -1 \\\frac{d^{2} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} + \frac{2 d e x \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} + \frac{e^{2} x^{2} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/(c*d), Eq(e, 0) & Eq(p, -1)), (d*x*(c*d**2)**p, Eq(e, 0)), (log(d/e + x)/(c*e), Eq(p, -1)), (d**2
*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**p/(2*e*p + 2*e) + 2*d*e*x*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**p/(2*e*p +
2*e) + e**2*x**2*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**p/(2*e*p + 2*e), True))

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Giac [B]  time = 1.19729, size = 127, normalized size = 3.26 \begin{align*} \frac{{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} x^{2} e^{2} + 2 \,{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d x e +{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d^{2}}{2 \,{\left (p e + e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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