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]

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

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \[ -\frac {\left (a+b x^2\right )^{3/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x^2]/x^4,x]

[Out]

-1/3*(a + b*x^2)^(3/2)/(a*x^3)

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fricas [A]  time = 0.81, size = 17, normalized size = 0.81 \[ -\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, a x^{3}} \]

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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giac [B]  time = 0.68, size = 59, normalized size = 2.81 \[ \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {3}{2}} + a^{2} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

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

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maple [A]  time = 0.00, size = 18, normalized size = 0.86 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3 a \,x^{3}} \]

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 [A]  time = 1.32, size = 17, normalized size = 0.81 \[ -\frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b*x^2 + a)^(3/2)/(a*x^3)

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mupad [B]  time = 4.57, size = 17, normalized size = 0.81 \[ -\frac {{\left (b\,x^2+a\right )}^{3/2}}{3\,a\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(1/2)/x^4,x)

[Out]

-(a + b*x^2)^(3/2)/(3*a*x^3)

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sympy [B]  time = 0.71, size = 42, normalized size = 2.00 \[ - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3 a} \]

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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