3.1.61 \(\int \frac {1}{(a+c x^2)^{5/2}} \, dx\) [61]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {198, 197} \begin {gather*} \frac {2 x}{3 a^2 \sqrt {a+c x^2}}+\frac {x}{3 a \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^(-5/2),x]

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+c x^2\right )^{5/2}} \, dx &=\frac {x}{3 a \left (a+c x^2\right )^{3/2}}+\frac {2 \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a}\\ &=\frac {x}{3 a \left (a+c x^2\right )^{3/2}}+\frac {2 x}{3 a^2 \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 29, normalized size = 0.74 \begin {gather*} \frac {3 a x+2 c x^3}{3 a^2 \left (a+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^(-5/2),x]

[Out]

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

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Maple [A]
time = 0.37, size = 32, normalized size = 0.82

method	result	size
gosper	\(\frac {x \left (2 c \,x^{2}+3 a \right )}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\)	\(26\)
trager	\(\frac {x \left (2 c \,x^{2}+3 a \right )}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\)	\(26\)
default	\(\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 31, normalized size = 0.79 \begin {gather*} \frac {2 \, x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.65, size = 47, normalized size = 1.21 \begin {gather*} \frac {{\left (2 \, c x^{3} + 3 \, a x\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{2} x^{4} + 2 \, a^{3} c x^{2} + a^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (32) = 64\).
time = 0.43, size = 95, normalized size = 2.44 \begin {gather*} \frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {5}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {2 c x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {5}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x**2+a)**(5/2),x)

[Out]

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

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Giac [A]
time = 1.99, size = 27, normalized size = 0.69 \begin {gather*} \frac {x {\left (\frac {2 \, c x^{2}}{a^{2}} + \frac {3}{a}\right )}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.19, size = 28, normalized size = 0.72 \begin {gather*} \frac {2\,x\,\left (c\,x^2+a\right )+a\,x}{3\,a^2\,{\left (c\,x^2+a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + c*x^2)^(5/2),x)

[Out]

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

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