3.434 \(\int \frac{(a+b x^2)^{9/2}}{x^{12}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^2\right )^{11/2}}{11 a x^{11}} \]

[Out]

-(a + b*x^2)^(11/2)/(11*a*x^11)

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Rubi [A]  time = 0.0049336, 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} \[ -\frac{\left (a+b x^2\right )^{11/2}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^12,x]

[Out]

-(a + b*x^2)^(11/2)/(11*a*x^11)

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 )^{9/2}}{x^{12}} \, dx &=-\frac{\left (a+b x^2\right )^{11/2}}{11 a x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.0089415, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^2\right )^{11/2}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^12,x]

[Out]

-(a + b*x^2)^(11/2)/(11*a*x^11)

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Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{11\,a{x}^{11}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^12,x)

[Out]

-1/11*(b*x^2+a)^(11/2)/a/x^11

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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)^(9/2)/x^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.83294, size = 150, normalized size = 7.14 \begin{align*} -\frac{{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{b x^{2} + a}}{11 \, a x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^12,x, algorithm="fricas")

[Out]

-1/11*(b^5*x^10 + 5*a*b^4*x^8 + 10*a^2*b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x^2 + a^5)*sqrt(b*x^2 + a)/(a*x^11)

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Sympy [B]  time = 4.84791, size = 150, normalized size = 7.14 \begin{align*} - \frac{a^{4} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{11 x^{10}} - \frac{5 a^{3} b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{11 x^{8}} - \frac{10 a^{2} b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{11 x^{6}} - \frac{10 a b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{11 x^{4}} - \frac{5 b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{11 x^{2}} - \frac{b^{\frac{11}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{11 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**12,x)

[Out]

-a**4*sqrt(b)*sqrt(a/(b*x**2) + 1)/(11*x**10) - 5*a**3*b**(3/2)*sqrt(a/(b*x**2) + 1)/(11*x**8) - 10*a**2*b**(5
/2)*sqrt(a/(b*x**2) + 1)/(11*x**6) - 10*a*b**(7/2)*sqrt(a/(b*x**2) + 1)/(11*x**4) - 5*b**(9/2)*sqrt(a/(b*x**2)
+ 1)/(11*x**2) - b**(11/2)*sqrt(a/(b*x**2) + 1)/(11*a)

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Giac [B]  time = 1.59613, size = 225, normalized size = 10.71 \begin{align*} \frac{2 \,{\left (11 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{20} b^{\frac{11}{2}} + 165 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a^{2} b^{\frac{11}{2}} + 462 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{4} b^{\frac{11}{2}} + 330 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{6} b^{\frac{11}{2}} + 55 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{8} b^{\frac{11}{2}} + a^{10} b^{\frac{11}{2}}\right )}}{11 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^12,x, algorithm="giac")

[Out]

2/11*(11*(sqrt(b)*x - sqrt(b*x^2 + a))^20*b^(11/2) + 165*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^2*b^(11/2) + 462*(
sqrt(b)*x - sqrt(b*x^2 + a))^12*a^4*b^(11/2) + 330*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^6*b^(11/2) + 55*(sqrt(b)*
x - sqrt(b*x^2 + a))^4*a^8*b^(11/2) + a^10*b^(11/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^11