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

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

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {(d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 0.86 \begin {gather*} \frac {c x (d+e x) (2 d+e x)}{2 \sqrt {c (d+e x)^2}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 41, normalized size = 1.14 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.17, size = 30, normalized size = 0.83 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.04, size = 40, normalized size = 1.11 \begin {gather*} \frac {\left (e x +2 d \right ) \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, x}{2 e x +2 d} \end {gather*}

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 [A]  time = 1.32, size = 54, normalized size = 1.50 \begin {gather*} \frac {1}{2} \, \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x + \frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d}{2 \, e} \end {gather*}

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]

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

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mupad [B]  time = 0.43, size = 33, normalized size = 0.92 \begin {gather*} \left (\frac {x}{2}+\frac {d}{2\,e}\right )\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}\, dx \end {gather*}

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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