3.516 \(\int \frac{1}{x^4 (a+b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{16 b^2 x}{3 a^4 \sqrt{a+b x^2}}+\frac{8 b^2 x}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac{2 b}{a^2 x \left (a+b x^2\right )^{3/2}}-\frac{1}{3 a x^3 \left (a+b x^2\right )^{3/2}} \]

[Out]

-1/(3*a*x^3*(a + b*x^2)^(3/2)) + (2*b)/(a^2*x*(a + b*x^2)^(3/2)) + (8*b^2*x)/(3*a^3*(a + b*x^2)^(3/2)) + (16*b
^2*x)/(3*a^4*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.0224556, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {271, 192, 191} \[ \frac{16 b^2 x}{3 a^4 \sqrt{a+b x^2}}+\frac{8 b^2 x}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac{2 b}{a^2 x \left (a+b x^2\right )^{3/2}}-\frac{1}{3 a x^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^2)^(5/2)),x]

[Out]

-1/(3*a*x^3*(a + b*x^2)^(3/2)) + (2*b)/(a^2*x*(a + b*x^2)^(3/2)) + (8*b^2*x)/(3*a^3*(a + b*x^2)^(3/2)) + (16*b
^2*x)/(3*a^4*Sqrt[a + b*x^2])

Rule 271

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

Rule 192

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]

Rule 191

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]

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (a+b x^2\right )^{5/2}} \, dx &=-\frac{1}{3 a x^3 \left (a+b x^2\right )^{3/2}}-\frac{(2 b) \int \frac{1}{x^2 \left (a+b x^2\right )^{5/2}} \, dx}{a}\\ &=-\frac{1}{3 a x^3 \left (a+b x^2\right )^{3/2}}+\frac{2 b}{a^2 x \left (a+b x^2\right )^{3/2}}+\frac{\left (8 b^2\right ) \int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx}{a^2}\\ &=-\frac{1}{3 a x^3 \left (a+b x^2\right )^{3/2}}+\frac{2 b}{a^2 x \left (a+b x^2\right )^{3/2}}+\frac{8 b^2 x}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac{\left (16 b^2\right ) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a^3}\\ &=-\frac{1}{3 a x^3 \left (a+b x^2\right )^{3/2}}+\frac{2 b}{a^2 x \left (a+b x^2\right )^{3/2}}+\frac{8 b^2 x}{3 a^3 \left (a+b x^2\right )^{3/2}}+\frac{16 b^2 x}{3 a^4 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0098215, size = 53, normalized size = 0.62 \[ \frac{6 a^2 b x^2-a^3+24 a b^2 x^4+16 b^3 x^6}{3 a^4 x^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^2)^(5/2)),x]

[Out]

(-a^3 + 6*a^2*b*x^2 + 24*a*b^2*x^4 + 16*b^3*x^6)/(3*a^4*x^3*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.005, size = 48, normalized size = 0.6 \begin{align*} -{\frac{-16\,{b}^{3}{x}^{6}-24\,a{b}^{2}{x}^{4}-6\,{a}^{2}b{x}^{2}+{a}^{3}}{3\,{x}^{3}{a}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^2+a)^(5/2),x)

[Out]

-1/3*(-16*b^3*x^6-24*a*b^2*x^4-6*a^2*b*x^2+a^3)/x^3/(b*x^2+a)^(3/2)/a^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.31881, size = 144, normalized size = 1.67 \begin{align*} \frac{{\left (16 \, b^{3} x^{6} + 24 \, a b^{2} x^{4} + 6 \, a^{2} b x^{2} - a^{3}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/3*(16*b^3*x^6 + 24*a*b^2*x^4 + 6*a^2*b*x^2 - a^3)*sqrt(b*x^2 + a)/(a^4*b^2*x^7 + 2*a^5*b*x^5 + a^6*x^3)

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Sympy [B]  time = 1.88013, size = 354, normalized size = 4.12 \begin{align*} - \frac{a^{4} b^{\frac{19}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac{5 a^{3} b^{\frac{21}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac{30 a^{2} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac{40 a b^{\frac{25}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} + \frac{16 b^{\frac{27}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{7} b^{9} x^{2} + 9 a^{6} b^{10} x^{4} + 9 a^{5} b^{11} x^{6} + 3 a^{4} b^{12} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**2+a)**(5/2),x)

[Out]

-a**4*b**(19/2)*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*
x**8) + 5*a**3*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 +
3*a**4*b**12*x**8) + 30*a**2*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a*
*5*b**11*x**6 + 3*a**4*b**12*x**8) + 40*a*b**(25/2)*x**6*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10
*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8) + 16*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*
a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8)

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Giac [A]  time = 2.75468, size = 163, normalized size = 1.9 \begin{align*} \frac{x{\left (\frac{8 \, b^{3} x^{2}}{a^{4}} + \frac{9 \, b^{2}}{a^{3}}\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{4 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} b^{\frac{3}{2}} - 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a b^{\frac{3}{2}} + 4 \, a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

1/3*x*(8*b^3*x^2/a^4 + 9*b^2/a^3)/(b*x^2 + a)^(3/2) - 4/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(3/2) - 9*(sqrt
(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2) + 4*a^2*b^(3/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3*a^3)