3.505 \(\int \frac{1}{x^4 (a+b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{8 b^2 x}{3 a^3 \sqrt{a+b x^2}}+\frac{4 b}{3 a^2 x \sqrt{a+b x^2}}-\frac{1}{3 a x^3 \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0157467, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 191} \[ \frac{8 b^2 x}{3 a^3 \sqrt{a+b x^2}}+\frac{4 b}{3 a^2 x \sqrt{a+b x^2}}-\frac{1}{3 a x^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^2)^(3/2)),x]

[Out]

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

Mathematica [A]  time = 0.0080129, size = 40, normalized size = 0.61 \[ -\frac{a^2-4 a b x^2-8 b^2 x^4}{3 a^3 x^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^2)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 37, normalized size = 0.6 \begin{align*} -{\frac{-8\,{b}^{2}{x}^{4}-4\,ab{x}^{2}+{a}^{2}}{3\,{a}^{3}{x}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^2+a)^(3/2),x)

[Out]

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

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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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.21503, size = 99, normalized size = 1.5 \begin{align*} \frac{{\left (8 \, b^{2} x^{4} + 4 \, a b x^{2} - a^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

1/3*(8*b^2*x^4 + 4*a*b*x^2 - a^2)*sqrt(b*x^2 + a)/(a^3*b*x^5 + a^4*x^3)

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Sympy [B]  time = 1.19645, size = 233, normalized size = 3.53 \begin{align*} - \frac{a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{3 a^{2} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{12 a b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac{8 b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**2+a)**(3/2),x)

[Out]

-a**3*b**(9/2)*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 3*a**2*b**(11/2
)*x**2*sqrt(a/(b*x**2) + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 12*a*b**(13/2)*x**4*sqr
t(a/(b*x**2) + 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6) + 8*b**(15/2)*x**6*sqrt(a/(b*x**2)
+ 1)/(3*a**5*b**4*x**2 + 6*a**4*b**5*x**4 + 3*a**3*b**6*x**6)

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Giac [A]  time = 2.16911, size = 143, normalized size = 2.17 \begin{align*} \frac{b^{2} x}{\sqrt{b x^{2} + a} a^{3}} - \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} b^{\frac{3}{2}} - 12 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a b^{\frac{3}{2}} + 5 \, a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

b^2*x/(sqrt(b*x^2 + a)*a^3) - 2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(3/2) - 12*(sqrt(b)*x - sqrt(b*x^2 + a)
)^2*a*b^(3/2) + 5*a^2*b^(3/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3*a^2)