3.1060 \(\int \frac{(d+e x)^4}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e} \]

[Out]

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

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Rubi [A]  time = 0.0737094, 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 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c^2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.6231, size = 34, normalized size = 0.87 \[ \frac{\left (d + e x\right )^{5}}{2 e \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)

[Out]

(d + e*x)**5/(2*e*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(3/2))

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Mathematica [A]  time = 0.0119574, size = 33, normalized size = 0.85 \[ \frac{x (d+e x) (2 d+e x)}{2 c \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 40, normalized size = 1. \[{\frac{x \left ( ex+2\,d \right ) \left ( ex+d \right ) ^{3}}{2} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.684827, size = 134, normalized size = 3.44 \[ \frac{e^{2} x^{3}}{2 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} + \frac{3 \, d e x^{2}}{2 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} - \frac{d^{3}}{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.213076, size = 65, normalized size = 1.67 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x^{2} + 2 \, d x\right )}}{2 \,{\left (c^{2} e x + c^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),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)/(c^2*e*x + c^2*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{4}}{\left (c \left (d + e x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.270071, size = 88, normalized size = 2.26 \[ \frac{4 \, C_{0} d e^{\left (-1\right )} - \frac{2 \, d^{3} e^{\left (-1\right )}}{c} +{\left (x{\left (\frac{x e^{2}}{c} + \frac{3 \, d e}{c}\right )} + 4 \, C_{0}\right )} x}{2 \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2),x, algorithm="giac")

[Out]

1/2*(4*C_0*d*e^(-1) - 2*d^3*e^(-1)/c + (x*(x*e^2/c + 3*d*e/c) + 4*C_0)*x)/sqrt(c
*x^2*e^2 + 2*c*d*x*e + c*d^2)