3.1019 \(\int (d+e x)^2 \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx\)

Optimal. Leaf size=39 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c e} \]

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))/(4*c*e)

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Rubi [A]  time = 0.0736582, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 c e} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))/(4*c*e)

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Rubi in Sympy [A]  time = 18.9858, size = 34, normalized size = 0.87 \[ \frac{\left (d + e x\right )^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{4 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**3*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(4*e)

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Mathematica [A]  time = 0.0243158, size = 28, normalized size = 0.72 \[ \frac{(d+e x) \left (c (d+e x)^2\right )^{3/2}}{4 c e} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)*(c*(d + e*x)^2)^(3/2))/(4*c*e)

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Maple [A]  time = 0.005, size = 62, normalized size = 1.6 \[{\frac{x \left ({e}^{3}{x}^{3}+4\,d{e}^{2}{x}^{2}+6\,{d}^{2}ex+4\,{d}^{3} \right ) }{4\,ex+4\,d}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x*(e^3*x^3+4*d*e^2*x^2+6*d^2*e*x+4*d^3)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)/(e
*x+d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.209587, size = 85, normalized size = 2.18 \[ \frac{{\left (e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \,{\left (e x + d\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)*sqrt(c*e^2*x^2 + 2*c*d*e*x +
c*d^2)/(e*x + d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c \left (d + e x\right )^{2}} \left (d + e x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218858, size = 69, normalized size = 1.77 \[ \frac{1}{4} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (d^{3} e^{\left (-1\right )} +{\left (3 \, d^{2} +{\left (x e^{2} + 3 \, d e\right )} x\right )} x\right )} \]

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)^2,x, algorithm="giac")

[Out]

1/4*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)*(d^3*e^(-1) + (3*d^2 + (x*e^2 + 3*d*e)*x
)*x)