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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.01, 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]

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

________________________________________________________________________________________

fricas [A]  time = 0.93, size = 50, normalized size = 0.76 \[ \frac {{\left (8 \, b^{2} x^{4} + 4 \, a b x^{2} - a^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 1.16, size = 106, normalized size = 1.61 \[ \frac {b^{2} x}{\sqrt {b x^{2} + a} a^{3}} - \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {3}{2}} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} + 5 \, a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a^{2}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 37, normalized size = 0.56 \[ -\frac {-8 b^{2} x^{4}-4 a b \,x^{2}+a^{2}}{3 \sqrt {b \,x^{2}+a}\, a^{3} x^{3}} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.31, size = 54, normalized size = 0.82 \[ \frac {8 \, b^{2} x}{3 \, \sqrt {b x^{2} + a} a^{3}} + \frac {4 \, b}{3 \, \sqrt {b x^{2} + a} a^{2} x} - \frac {1}{3 \, \sqrt {b x^{2} + a} a x^{3}} \]

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]

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

________________________________________________________________________________________

mupad [B]  time = 5.12, size = 38, normalized size = 0.58 \[ \frac {-a^2+4\,a\,b\,x^2+8\,b^2\,x^4}{3\,a^3\,x^3\,\sqrt {b\,x^2+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^2)^(3/2)),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 1.23, size = 233, normalized size = 3.53 \[ - \frac {a^{3} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {3 a^{2} b^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {12 a b^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} + \frac {8 b^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{5} b^{4} x^{2} + 6 a^{4} b^{5} x^{4} + 3 a^{3} b^{6} x^{6}} \]

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)

________________________________________________________________________________________