3.1020 \(\int (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx\)

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 = 0.0217368, 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 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e} \]

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)

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Rubi in Sympy [A]  time = 10.1614, size = 29, normalized size = 0.85 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}{3 c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0155253, 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 = 0.004, size = 51, normalized size = 1.5 \[{\frac{x \left ({e}^{2}{x}^{2}+3\,dex+3\,{d}^{2} \right ) }{3\,ex+3\,d}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]

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 = 0.695062, size = 41, normalized size = 1.21 \[ \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}}}{3 \, c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d),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 = 0.21527, size = 70, normalized size = 2.06 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )}}{3 \,{\left (e x + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d),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 = 0.564285, size = 107, normalized size = 3.15 \[ \begin{cases} \frac{d^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 e} + \frac{2 d x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3} + \frac{e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3} & \text{for}\: e \neq 0 \\d x \sqrt{c d^{2}} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.218985, size = 55, normalized size = 1.62 \[ \frac{1}{3} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (d^{2} e^{\left (-1\right )} +{\left (x e + 2 \, d\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d),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)