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

Optimal. Leaf size=59 \[ -\frac{2 \left (a e^2+c d^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^3}+\frac{4 c d}{e^3 \sqrt{d+e x}} \]

[Out]

(-2*(c*d^2 + a*e^2))/(3*e^3*(d + e*x)^(3/2)) + (4*c*d)/(e^3*Sqrt[d + e*x]) + (2*
c*Sqrt[d + e*x])/e^3

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Rubi [A]  time = 0.0685352, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 \left (a e^2+c d^2\right )}{3 e^3 (d+e x)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^3}+\frac{4 c d}{e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(c*d^2 + a*e^2))/(3*e^3*(d + e*x)^(3/2)) + (4*c*d)/(e^3*Sqrt[d + e*x]) + (2*
c*Sqrt[d + e*x])/e^3

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Rubi in Sympy [A]  time = 11.2506, size = 58, normalized size = 0.98 \[ \frac{4 c d}{e^{3} \sqrt{d + e x}} + \frac{2 c \sqrt{d + e x}}{e^{3}} - \frac{2 \left (a e^{2} + c d^{2}\right )}{3 e^{3} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0489948, size = 44, normalized size = 0.75 \[ \frac{2 \left (c \left (8 d^2+12 d e x+3 e^2 x^2\right )-a e^2\right )}{3 e^3 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(a*e^2) + c*(8*d^2 + 12*d*e*x + 3*e^2*x^2)))/(3*e^3*(d + e*x)^(3/2))

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Maple [A]  time = 0.005, size = 40, normalized size = 0.7 \[ -{\frac{-6\,c{e}^{2}{x}^{2}-24\,cdex+2\,a{e}^{2}-16\,c{d}^{2}}{3\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)/(e*x+d)^(5/2),x)

[Out]

-2/3/(e*x+d)^(3/2)*(-3*c*e^2*x^2-12*c*d*e*x+a*e^2-8*c*d^2)/e^3

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Maxima [A]  time = 0.697296, size = 70, normalized size = 1.19 \[ \frac{2 \,{\left (\frac{3 \, \sqrt{e x + d} c}{e^{2}} + \frac{6 \,{\left (e x + d\right )} c d - c d^{2} - a e^{2}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{2}}\right )}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(3*sqrt(e*x + d)*c/e^2 + (6*(e*x + d)*c*d - c*d^2 - a*e^2)/((e*x + d)^(3/2)*
e^2))/e

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Fricas [A]  time = 0.207595, size = 68, normalized size = 1.15 \[ \frac{2 \,{\left (3 \, c e^{2} x^{2} + 12 \, c d e x + 8 \, c d^{2} - a e^{2}\right )}}{3 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(3*c*e^2*x^2 + 12*c*d*e*x + 8*c*d^2 - a*e^2)/((e^4*x + d*e^3)*sqrt(e*x + d))

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Sympy [A]  time = 3.80626, size = 168, normalized size = 2.85 \[ \begin{cases} - \frac{2 a e^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{16 c d^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{24 c d e x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{6 c e^{2} x^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*a*e**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 16*c*d*
*2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 24*c*d*e*x/(3*d*e**3*sqrt
(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 6*c*e**2*x**2/(3*d*e**3*sqrt(d + e*x) + 3*
e**4*x*sqrt(d + e*x)), Ne(e, 0)), ((a*x + c*x**3/3)/d**(5/2), True))

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GIAC/XCAS [A]  time = 0.212676, size = 65, normalized size = 1.1 \[ 2 \, \sqrt{x e + d} c e^{\left (-3\right )} + \frac{2 \,{\left (6 \,{\left (x e + d\right )} c d - c d^{2} - a e^{2}\right )} e^{\left (-3\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x*e + d)*c*e^(-3) + 2/3*(6*(x*e + d)*c*d - c*d^2 - a*e^2)*e^(-3)/(x*e + d
)^(3/2)