### 3.1031 $$\int (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx$$

Optimal. Leaf size=34 $\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e}$

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(3*c*e)

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Rubi [A]  time = 0.0087348, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.033, Rules used = {629} $\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(3*c*e)

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) \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e}\\ \end{align*}

Mathematica [A]  time = 0.0086244, size = 23, normalized size = 0.68 $\frac{\left (c (d+e x)^2\right )^{3/2}}{3 c e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c*(d + e*x)^2)^(3/2)/(3*c*e)

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Maple [A]  time = 0.04, size = 51, normalized size = 1.5 \begin{align*}{\frac{x \left ({e}^{2}{x}^{2}+3\,dex+3\,{d}^{2} \right ) }{3\,ex+3\,d}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \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)^(1/2),x)

[Out]

1/3*x*(e^2*x^2+3*d*e*x+3*d^2)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)/(e*x+d)

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Maxima [A]  time = 0.985503, size = 41, normalized size = 1.21 \begin{align*} \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}}}{3 \, c e} \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)^(1/2),x, algorithm="maxima")

[Out]

1/3*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)/(c*e)

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Fricas [A]  time = 2.04189, size = 112, normalized size = 3.29 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )}}{3 \,{\left (e x + d\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)^(1/2),x, algorithm="fricas")

[Out]

1/3*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e^2*x^3 + 3*d*e*x^2 + 3*d^2*x)/(e*x + d)

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Sympy [A]  time = 0.292331, size = 107, normalized size = 3.15 \begin{align*} \begin{cases} \frac{d^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 e} + \frac{2 d x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3} + \frac{e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3} & \text{for}\: e \neq 0 \\d x \sqrt{c d^{2}} & \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)**(1/2),x)

[Out]

Piecewise((d**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(3*e) + 2*d*x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/3
+ e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/3, Ne(e, 0)), (d*x*sqrt(c*d**2), True))

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Giac [A]  time = 1.16425, size = 55, normalized size = 1.62 \begin{align*} \frac{1}{3} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (d^{2} e^{\left (-1\right )} +{\left (x e + 2 \, d\right )} x\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)^(1/2),x, algorithm="giac")

[Out]

1/3*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)*(d^2*e^(-1) + (x*e + 2*d)*x)