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

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

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.68 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{3/2}}{3 c e} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 23, normalized size = 0.68 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{3/2}}{3 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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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fricas [A]  time = 0.40, size = 52, normalized size = 1.53 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.23, size = 41, normalized size = 1.21 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 51, normalized size = 1.50 \begin {gather*} \frac {\left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) \sqrt {c \,e^{2} x^{2}+2 c d e x +c \,d^{2}}\, x}{3 e x +3 d} \end {gather*}

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 = 1.39, size = 30, normalized size = 0.88 \begin {gather*} \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}}}{3 \, c e} \end {gather*}

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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mupad [B]  time = 0.53, size = 34, normalized size = 1.00 \begin {gather*} \frac {{\left (d+e\,x\right )}^2\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{3\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.34, size = 107, normalized size = 3.15 \begin {gather*} \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 {gather*}

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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