3.405 \(\int \frac {(a+b x^2)^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=68 \[ -\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{693 a^3 x^7}+\frac {4 b \left (a+b x^2\right )^{7/2}}{99 a^2 x^9}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}} \]

[Out]

-1/11*(b*x^2+a)^(7/2)/a/x^11+4/99*b*(b*x^2+a)^(7/2)/a^2/x^9-8/693*b^2*(b*x^2+a)^(7/2)/a^3/x^7

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Rubi [A]  time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{693 a^3 x^7}+\frac {4 b \left (a+b x^2\right )^{7/2}}{99 a^2 x^9}-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/x^12,x]

[Out]

-(a + b*x^2)^(7/2)/(11*a*x^11) + (4*b*(a + b*x^2)^(7/2))/(99*a^2*x^9) - (8*b^2*(a + b*x^2)^(7/2))/(693*a^3*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]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2}}{x^{12}} \, dx &=-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}-\frac {(4 b) \int \frac {\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx}{11 a}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}+\frac {4 b \left (a+b x^2\right )^{7/2}}{99 a^2 x^9}+\frac {\left (8 b^2\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{99 a^2}\\ &=-\frac {\left (a+b x^2\right )^{7/2}}{11 a x^{11}}+\frac {4 b \left (a+b x^2\right )^{7/2}}{99 a^2 x^9}-\frac {8 b^2 \left (a+b x^2\right )^{7/2}}{693 a^3 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 42, normalized size = 0.62 \[ -\frac {\left (a+b x^2\right )^{7/2} \left (63 a^2-28 a b x^2+8 b^2 x^4\right )}{693 a^3 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/x^12,x]

[Out]

-1/693*((a + b*x^2)^(7/2)*(63*a^2 - 28*a*b*x^2 + 8*b^2*x^4))/(a^3*x^11)

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fricas [A]  time = 0.94, size = 71, normalized size = 1.04 \[ -\frac {{\left (8 \, b^{5} x^{10} - 4 \, a b^{4} x^{8} + 3 \, a^{2} b^{3} x^{6} + 113 \, a^{3} b^{2} x^{4} + 161 \, a^{4} b x^{2} + 63 \, a^{5}\right )} \sqrt {b x^{2} + a}}{693 \, a^{3} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^12,x, algorithm="fricas")

[Out]

-1/693*(8*b^5*x^10 - 4*a*b^4*x^8 + 3*a^2*b^3*x^6 + 113*a^3*b^2*x^4 + 161*a^4*b*x^2 + 63*a^5)*sqrt(b*x^2 + a)/(
a^3*x^11)

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giac [B]  time = 1.16, size = 246, normalized size = 3.62 \[ \frac {16 \, {\left (462 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} b^{\frac {11}{2}} + 1155 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a b^{\frac {11}{2}} + 2541 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{2} b^{\frac {11}{2}} + 2079 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{3} b^{\frac {11}{2}} + 1485 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} b^{\frac {11}{2}} + 297 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} b^{\frac {11}{2}} + 55 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} b^{\frac {11}{2}} - 11 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} b^{\frac {11}{2}} + a^{8} b^{\frac {11}{2}}\right )}}{693 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^12,x, algorithm="giac")

[Out]

16/693*(462*(sqrt(b)*x - sqrt(b*x^2 + a))^16*b^(11/2) + 1155*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a*b^(11/2) + 254
1*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^2*b^(11/2) + 2079*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*b^(11/2) + 1485*(s
qrt(b)*x - sqrt(b*x^2 + a))^8*a^4*b^(11/2) + 297*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^5*b^(11/2) + 55*(sqrt(b)*x
- sqrt(b*x^2 + a))^4*a^6*b^(11/2) - 11*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^7*b^(11/2) + a^8*b^(11/2))/((sqrt(b)*
x - sqrt(b*x^2 + a))^2 - a)^11

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maple [A]  time = 0.01, size = 39, normalized size = 0.57 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (8 b^{2} x^{4}-28 a b \,x^{2}+63 a^{2}\right )}{693 a^{3} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/x^12,x)

[Out]

-1/693*(b*x^2+a)^(7/2)*(8*b^2*x^4-28*a*b*x^2+63*a^2)/x^11/a^3

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maxima [A]  time = 1.41, size = 56, normalized size = 0.82 \[ -\frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}}{693 \, a^{3} x^{7}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b}{99 \, a^{2} x^{9}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}}}{11 \, a x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^12,x, algorithm="maxima")

[Out]

-8/693*(b*x^2 + a)^(7/2)*b^2/(a^3*x^7) + 4/99*(b*x^2 + a)^(7/2)*b/(a^2*x^9) - 1/11*(b*x^2 + a)^(7/2)/(a*x^11)

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mupad [B]  time = 5.62, size = 111, normalized size = 1.63 \[ \frac {4\,b^4\,\sqrt {b\,x^2+a}}{693\,a^2\,x^3}-\frac {113\,b^2\,\sqrt {b\,x^2+a}}{693\,x^7}-\frac {b^3\,\sqrt {b\,x^2+a}}{231\,a\,x^5}-\frac {a^2\,\sqrt {b\,x^2+a}}{11\,x^{11}}-\frac {8\,b^5\,\sqrt {b\,x^2+a}}{693\,a^3\,x}-\frac {23\,a\,b\,\sqrt {b\,x^2+a}}{99\,x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(5/2)/x^12,x)

[Out]

(4*b^4*(a + b*x^2)^(1/2))/(693*a^2*x^3) - (113*b^2*(a + b*x^2)^(1/2))/(693*x^7) - (b^3*(a + b*x^2)^(1/2))/(231
*a*x^5) - (a^2*(a + b*x^2)^(1/2))/(11*x^11) - (8*b^5*(a + b*x^2)^(1/2))/(693*a^3*x) - (23*a*b*(a + b*x^2)^(1/2
))/(99*x^9)

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sympy [B]  time = 2.16, size = 481, normalized size = 7.07 \[ - \frac {63 a^{7} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{x^{2} \left (693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}\right )} - \frac {287 a^{6} b^{\frac {11}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {498 a^{5} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {390 a^{4} b^{\frac {15}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {115 a^{3} b^{\frac {17}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {3 a^{2} b^{\frac {19}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {12 a b^{\frac {21}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} - \frac {8 b^{\frac {23}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{693 a^{5} b^{4} x^{8} + 1386 a^{4} b^{5} x^{10} + 693 a^{3} b^{6} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**12,x)

[Out]

-63*a**7*b**(9/2)*sqrt(a/(b*x**2) + 1)/(x**2*(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x**12)
) - 287*a**6*b**(11/2)*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x**12)
- 498*a**5*b**(13/2)*x**2*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x**1
2) - 390*a**4*b**(15/2)*x**4*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*x
**12) - 115*a**3*b**(17/2)*x**6*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**
6*x**12) - 3*a**2*b**(19/2)*x**8*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b*
*6*x**12) - 12*a*b**(21/2)*x**10*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b*
*6*x**12) - 8*b**(23/2)*x**12*sqrt(a/(b*x**2) + 1)/(693*a**5*b**4*x**8 + 1386*a**4*b**5*x**10 + 693*a**3*b**6*
x**12)

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