3.100 \(\int \frac{1}{\left (c+d x^2\right )^{5/2}} \, dx\)

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

[Out]

x/(3*c*(c + d*x^2)^(3/2)) + (2*x)/(3*c^2*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.0191651, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 x}{3 c^2 \sqrt{c+d x^2}}+\frac{x}{3 c \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^(-5/2),x]

[Out]

x/(3*c*(c + d*x^2)^(3/2)) + (2*x)/(3*c^2*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 2.02585, size = 32, normalized size = 0.82 \[ \frac{x}{3 c \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{2 x}{3 c^{2} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x**2+c)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0216859, size = 29, normalized size = 0.74 \[ \frac{x \left (3 c+2 d x^2\right )}{3 c^2 \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^(-5/2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 26, normalized size = 0.7 \[{\frac{x \left ( 2\,d{x}^{2}+3\,c \right ) }{3\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x^2+c)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.35008, size = 42, normalized size = 1.08 \[ \frac{2 \, x}{3 \, \sqrt{d x^{2} + c} c^{2}} + \frac{x}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.210507, size = 63, normalized size = 1.62 \[ \frac{{\left (2 \, d x^{3} + 3 \, c x\right )} \sqrt{d x^{2} + c}}{3 \,{\left (c^{2} d^{2} x^{4} + 2 \, c^{3} d x^{2} + c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.73728, size = 95, normalized size = 2.44 \[ \frac{3 c x}{3 c^{\frac{7}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 3 c^{\frac{5}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{2 d x^{3}}{3 c^{\frac{7}{2}} \sqrt{1 + \frac{d x^{2}}{c}} + 3 c^{\frac{5}{2}} d x^{2} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x**2+c)**(5/2),x)

[Out]

3*c*x/(3*c**(7/2)*sqrt(1 + d*x**2/c) + 3*c**(5/2)*d*x**2*sqrt(1 + d*x**2/c)) + 2
*d*x**3/(3*c**(7/2)*sqrt(1 + d*x**2/c) + 3*c**(5/2)*d*x**2*sqrt(1 + d*x**2/c))

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GIAC/XCAS [A]  time = 0.22776, size = 36, normalized size = 0.92 \[ \frac{x{\left (\frac{2 \, d x^{2}}{c^{2}} + \frac{3}{c}\right )}}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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