3.403 \(\int \frac{(a+b x^2)^{5/2}}{x^8} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7} \]

[Out]

-(a + b*x^2)^(7/2)/(7*a*x^7)

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Rubi [A]  time = 0.0046399, 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} \[ -\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/x^8,x]

[Out]

-(a + b*x^2)^(7/2)/(7*a*x^7)

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 )^{5/2}}{x^8} \, dx &=-\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7}\\ \end{align*}

Mathematica [A]  time = 0.0080252, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^2\right )^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/x^8,x]

[Out]

-(a + b*x^2)^(7/2)/(7*a*x^7)

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Maple [A]  time = 0.002, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,a{x}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/x^8,x)

[Out]

-1/7*(b*x^2+a)^(7/2)/a/x^7

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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)^(5/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.58271, size = 100, normalized size = 4.76 \begin{align*} -\frac{{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{b x^{2} + a}}{7 \, a x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^8,x, algorithm="fricas")

[Out]

-1/7*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*sqrt(b*x^2 + a)/(a*x^7)

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Sympy [B]  time = 1.695, size = 95, normalized size = 4.52 \begin{align*} - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{6}} - \frac{3 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{4}} - \frac{3 b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{7 x^{2}} - \frac{b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{7 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**8,x)

[Out]

-a**2*sqrt(b)*sqrt(a/(b*x**2) + 1)/(7*x**6) - 3*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/(7*x**4) - 3*b**(5/2)*sqrt(a/(
b*x**2) + 1)/(7*x**2) - b**(7/2)*sqrt(a/(b*x**2) + 1)/(7*a)

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Giac [B]  time = 2.08011, size = 153, normalized size = 7.29 \begin{align*} \frac{2 \,{\left (7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} b^{\frac{7}{2}} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{2} b^{\frac{7}{2}} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{4} b^{\frac{7}{2}} + a^{6} b^{\frac{7}{2}}\right )}}{7 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^8,x, algorithm="giac")

[Out]

2/7*(7*(sqrt(b)*x - sqrt(b*x^2 + a))^12*b^(7/2) + 35*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(7/2) + 21*(sqrt(b)
*x - sqrt(b*x^2 + a))^4*a^4*b^(7/2) + a^6*b^(7/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7