### 3.1032 $$\int \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx$$

Optimal. Leaf size=36 $\frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e}$

[Out]

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

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Rubi [A]  time = 0.0061876, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {609} $\frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0133653, size = 31, normalized size = 0.86 $\frac{c x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 40, normalized size = 1.1 \begin{align*}{\frac{x \left ( ex+2\,d \right ) }{2\,ex+2\,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((c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)

[Out]

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

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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((c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.02898, size = 90, normalized size = 2.5 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x^{2} + 2 \, d x\right )}}{2 \,{\left (e x + d\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)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}\, dx \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)**(1/2),x)

[Out]

Integral(sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2), x)

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Giac [A]  time = 1.26372, size = 41, normalized size = 1.14 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (d e^{\left (-1\right )} + x\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)^(1/2),x, algorithm="giac")

[Out]

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