3.365 \(\int \frac{\sqrt{a+b x^2}}{x^4} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^2\right )^{3/2}}{3 a x^3} \]

[Out]

-(a + b*x^2)^(3/2)/(3*a*x^3)

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Rubi [A]  time = 0.0044524, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\left (a+b x^2\right )^{3/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x^2]/x^4,x]

[Out]

-(a + b*x^2)^(3/2)/(3*a*x^3)

Rule 264

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

Rubi steps

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

Mathematica [A]  time = 0.0049409, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^2\right )^{3/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x^2]/x^4,x]

[Out]

-(a + b*x^2)^(3/2)/(3*a*x^3)

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Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.44487, size = 43, normalized size = 2.05 \begin{align*} -\frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{3 \, a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b*x^2 + a)^(3/2)/(a*x^3)

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Sympy [B]  time = 0.627931, size = 42, normalized size = 2. \begin{align*} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.82172, size = 80, normalized size = 3.81 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} b^{\frac{3}{2}} + a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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