3.1969 \(\int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0653799, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ \frac{2 c d \sqrt{d+e x}}{e^2}-\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{\sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.0848, size = 41, normalized size = 1.05 \[ \frac{2 c d \sqrt{d + e x}}{e^{2}} - \frac{2 \left (a e^{2} - c d^{2}\right )}{e^{2} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*d*sqrt(d + e*x)/e**2 - 2*(a*e**2 - c*d**2)/(e**2*sqrt(d + e*x))

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Mathematica [A]  time = 0.0370588, size = 31, normalized size = 0.79 \[ \frac{2 c d (2 d+e x)-2 a e^2}{e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 31, normalized size = 0.8 \[ -2\,{\frac{-cdex+a{e}^{2}-2\,c{d}^{2}}{\sqrt{ex+d}{e}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.777577, size = 57, normalized size = 1.46 \[ \frac{2 \,{\left (\frac{\sqrt{e x + d} c d}{e} + \frac{c d^{2} - a e^{2}}{\sqrt{e x + d} e}\right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.218931, size = 41, normalized size = 1.05 \[ \frac{2 \,{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )}}{\sqrt{e x + d} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.12797, size = 58, normalized size = 1.49 \[ \begin{cases} - \frac{2 a}{\sqrt{d + e x}} + \frac{4 c d^{2}}{e^{2} \sqrt{d + e x}} + \frac{2 c d x}{e \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c x^{2}}{2 \sqrt{d}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d)**(5/2),x)

[Out]

Piecewise((-2*a/sqrt(d + e*x) + 4*c*d**2/(e**2*sqrt(d + e*x)) + 2*c*d*x/(e*sqrt(
d + e*x)), Ne(e, 0)), (c*x**2/(2*sqrt(d)), True))

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GIAC/XCAS [A]  time = 0.202185, size = 68, normalized size = 1.74 \[ 2 \, \sqrt{x e + d} c d e^{\left (-2\right )} + \frac{2 \,{\left ({\left (x e + d\right )} c d^{2} -{\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

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