3.1018 \(\int (d+e x)^3 \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx\)

Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c^2 e} \]

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(5*c^2*e)

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Rubi [A]  time = 0.0712381, antiderivative size = 34, 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 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c^2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*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)^(5/2)/(5*c^2*e)

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Rubi in Sympy [A]  time = 19.7707, size = 31, normalized size = 0.91 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{5}{2}}}{5 c^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(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)**(5/2)/(5*c**2*e)

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Mathematica [A]  time = 0.0200009, size = 27, normalized size = 0.79 \[ \frac{(d+e x)^4 \sqrt{c (d+e x)^2}}{5 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2],x]

[Out]

((d + e*x)^4*Sqrt[c*(d + e*x)^2])/(5*e)

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Maple [B]  time = 0.004, size = 73, normalized size = 2.2 \[{\frac{x \left ({e}^{4}{x}^{4}+5\,d{e}^{3}{x}^{3}+10\,{d}^{2}{e}^{2}{x}^{2}+10\,{d}^{3}ex+5\,{d}^{4} \right ) }{5\,ex+5\,d}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2),x)

[Out]

1/5*x*(e^4*x^4+5*d*e^3*x^3+10*d^2*e^2*x^2+10*d^3*e*x+5*d^4)*(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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215341, size = 100, normalized size = 2.94 \[ \frac{{\left (e^{4} x^{5} + 5 \, d e^{3} x^{4} + 10 \, d^{2} e^{2} x^{3} + 10 \, d^{3} e x^{2} + 5 \, d^{4} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{5 \,{\left (e x + d\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/5*(e^4*x^5 + 5*d*e^3*x^4 + 10*d^2*e^2*x^3 + 10*d^3*e*x^2 + 5*d^4*x)*sqrt(c*e^2
*x^2 + 2*c*d*e*x + c*d^2)/(e*x + d)

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Sympy [A]  time = 1.39223, size = 187, normalized size = 5.5 \[ \begin{cases} \frac{d^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5 e} + \frac{4 d^{3} x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{6 d^{2} e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{4 d e^{2} x^{3} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac{e^{3} x^{4} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} & \text{for}\: e \neq 0 \\d^{3} x \sqrt{c d^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(c*e**2*x**2+2*c*d*e*x+c*d**2)**(1/2),x)

[Out]

Piecewise((d**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(5*e) + 4*d**3*x*sqrt(c*d
**2 + 2*c*d*e*x + c*e**2*x**2)/5 + 6*d**2*e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**
2*x**2)/5 + 4*d*e**2*x**3*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5 + e**3*x**4*s
qrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5, Ne(e, 0)), (d**3*x*sqrt(c*d**2), True))

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GIAC/XCAS [A]  time = 0.221044, size = 82, normalized size = 2.41 \[ \frac{1}{5} \,{\left (d^{4} e^{\left (-1\right )} +{\left (4 \, d^{3} +{\left (6 \, d^{2} e +{\left (x e^{3} + 4 \, d e^{2}\right )} x\right )} x\right )} x\right )} \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} \]

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)^3,x, algorithm="giac")

[Out]

1/5*(d^4*e^(-1) + (4*d^3 + (6*d^2*e + (x*e^3 + 4*d*e^2)*x)*x)*x)*sqrt(c*x^2*e^2
+ 2*c*d*x*e + c*d^2)