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]

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

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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} \[ -\frac {\left (a+b x^2\right )^{11/2}}{11 a x^{11}} \]

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*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \[ -\frac {\left (a+b x^2\right )^{11/2}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^12,x]

[Out]

-1/11*(a + b*x^2)^(11/2)/(a*x^11)

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fricas [B]  time = 0.98, size = 68, normalized size = 3.24 \[ -\frac {{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {b x^{2} + a}}{11 \, a x^{11}} \]

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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giac [B]  time = 1.06, size = 167, normalized size = 7.95 \[ \frac {2 \, {\left (11 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} b^{\frac {11}{2}} + 165 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{2} b^{\frac {11}{2}} + 462 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{4} b^{\frac {11}{2}} + 330 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{6} b^{\frac {11}{2}} + 55 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{8} b^{\frac {11}{2}} + a^{10} b^{\frac {11}{2}}\right )}}{11 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{11}} \]

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

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maple [A]  time = 0.00, size = 18, normalized size = 0.86 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{11 a \,x^{11}} \]

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 [A]  time = 1.50, size = 17, normalized size = 0.81 \[ -\frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{11 \, a x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11*(b*x^2 + a)^(11/2)/(a*x^11)

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mupad [B]  time = 6.28, size = 111, normalized size = 5.29 \[ -\frac {a^4\,\sqrt {b\,x^2+a}}{11\,x^{11}}-\frac {5\,b^4\,\sqrt {b\,x^2+a}}{11\,x^3}-\frac {10\,a\,b^3\,\sqrt {b\,x^2+a}}{11\,x^5}-\frac {5\,a^3\,b\,\sqrt {b\,x^2+a}}{11\,x^9}-\frac {b^5\,\sqrt {b\,x^2+a}}{11\,a\,x}-\frac {10\,a^2\,b^2\,\sqrt {b\,x^2+a}}{11\,x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x^12,x)

[Out]

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

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sympy [B]  time = 2.45, size = 150, normalized size = 7.14 \[ - \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{11 x^{10}} - \frac {5 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{11 x^{8}} - \frac {10 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{11 x^{6}} - \frac {10 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{11 x^{4}} - \frac {5 b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{11 x^{2}} - \frac {b^{\frac {11}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{11 a} \]

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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