3.383 \(\int \frac{(a+b x^2)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^2\right )^{5/2}}{5 a x^5} \]

[Out]

-(a + b*x^2)^(5/2)/(5*a*x^5)

________________________________________________________________________________________

Rubi [A]  time = 0.0049624, 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} \[ -\frac{\left (a+b x^2\right )^{5/2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(3/2)/x^6,x]

[Out]

-(a + b*x^2)^(5/2)/(5*a*x^5)

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{\left (a+b x^2\right )^{3/2}}{x^6} \, dx &=-\frac{\left (a+b x^2\right )^{5/2}}{5 a x^5}\\ \end{align*}

Mathematica [A]  time = 0.00613, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^2\right )^{5/2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(3/2)/x^6,x]

[Out]

-(a + b*x^2)^(5/2)/(5*a*x^5)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,a{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(3/2)/x^6,x)

[Out]

-1/5*(b*x^2+a)^(5/2)/a/x^5

________________________________________________________________________________________

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)^(3/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.4898, size = 78, normalized size = 3.71 \begin{align*} -\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{b x^{2} + a}}{5 \, a x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/x^6,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.967514, size = 68, normalized size = 3.24 \begin{align*} - \frac{a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{2}} - \frac{b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(3/2)/x**6,x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 2.67829, size = 116, normalized size = 5.52 \begin{align*} \frac{2 \,{\left (5 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} b^{\frac{5}{2}} + 10 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{2} b^{\frac{5}{2}} + a^{4} b^{\frac{5}{2}}\right )}}{5 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)/x^6,x, algorithm="giac")

[Out]

2/5*(5*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(5/2) + 10*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(5/2) + a^4*b^(5/2))
/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5