3.436 \(\int \frac {(a+b x^2)^{9/2}}{x^{16}} \, dx\)

Optimal. Leaf size=68 \[ -\frac {8 b^2 \left (a+b x^2\right )^{11/2}}{2145 a^3 x^{11}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{195 a^2 x^{13}}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}} \]

[Out]

-1/15*(b*x^2+a)^(11/2)/a/x^15+4/195*b*(b*x^2+a)^(11/2)/a^2/x^13-8/2145*b^2*(b*x^2+a)^(11/2)/a^3/x^11

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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} \[ -\frac {8 b^2 \left (a+b x^2\right )^{11/2}}{2145 a^3 x^{11}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{195 a^2 x^{13}}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^16,x]

[Out]

-(a + b*x^2)^(11/2)/(15*a*x^15) + (4*b*(a + b*x^2)^(11/2))/(195*a^2*x^13) - (8*b^2*(a + b*x^2)^(11/2))/(2145*a
^3*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]

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 )^{9/2}}{x^{16}} \, dx &=-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}-\frac {(4 b) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx}{15 a}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{195 a^2 x^{13}}+\frac {\left (8 b^2\right ) \int \frac {\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{195 a^2}\\ &=-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}+\frac {4 b \left (a+b x^2\right )^{11/2}}{195 a^2 x^{13}}-\frac {8 b^2 \left (a+b x^2\right )^{11/2}}{2145 a^3 x^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 42, normalized size = 0.62 \[ -\frac {\left (a+b x^2\right )^{11/2} \left (143 a^2-44 a b x^2+8 b^2 x^4\right )}{2145 a^3 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^16,x]

[Out]

-1/2145*((a + b*x^2)^(11/2)*(143*a^2 - 44*a*b*x^2 + 8*b^2*x^4))/(a^3*x^15)

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fricas [A]  time = 1.15, size = 93, normalized size = 1.37 \[ -\frac {{\left (8 \, b^{7} x^{14} - 4 \, a b^{6} x^{12} + 3 \, a^{2} b^{5} x^{10} + 355 \, a^{3} b^{4} x^{8} + 1030 \, a^{4} b^{3} x^{6} + 1218 \, a^{5} b^{2} x^{4} + 671 \, a^{6} b x^{2} + 143 \, a^{7}\right )} \sqrt {b x^{2} + a}}{2145 \, a^{3} x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^16,x, algorithm="fricas")

[Out]

-1/2145*(8*b^7*x^14 - 4*a*b^6*x^12 + 3*a^2*b^5*x^10 + 355*a^3*b^4*x^8 + 1030*a^4*b^3*x^6 + 1218*a^5*b^2*x^4 +
671*a^6*b*x^2 + 143*a^7)*sqrt(b*x^2 + a)/(a^3*x^15)

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giac [B]  time = 1.22, size = 354, normalized size = 5.21 \[ \frac {16 \, {\left (1430 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} b^{\frac {15}{2}} + 6435 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a b^{\frac {15}{2}} + 24453 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{2} b^{\frac {15}{2}} + 45045 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{3} b^{\frac {15}{2}} + 70785 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{4} b^{\frac {15}{2}} + 64350 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{5} b^{\frac {15}{2}} + 50050 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{6} b^{\frac {15}{2}} + 21450 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{7} b^{\frac {15}{2}} + 7800 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{8} b^{\frac {15}{2}} + 975 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{9} b^{\frac {15}{2}} + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{10} b^{\frac {15}{2}} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{11} b^{\frac {15}{2}} + a^{12} b^{\frac {15}{2}}\right )}}{2145 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^16,x, algorithm="giac")

[Out]

16/2145*(1430*(sqrt(b)*x - sqrt(b*x^2 + a))^24*b^(15/2) + 6435*(sqrt(b)*x - sqrt(b*x^2 + a))^22*a*b^(15/2) + 2
4453*(sqrt(b)*x - sqrt(b*x^2 + a))^20*a^2*b^(15/2) + 45045*(sqrt(b)*x - sqrt(b*x^2 + a))^18*a^3*b^(15/2) + 707
85*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^4*b^(15/2) + 64350*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^5*b^(15/2) + 50050
*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^6*b^(15/2) + 21450*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^7*b^(15/2) + 7800*(s
qrt(b)*x - sqrt(b*x^2 + a))^8*a^8*b^(15/2) + 975*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^9*b^(15/2) + 105*(sqrt(b)*x
- sqrt(b*x^2 + a))^4*a^10*b^(15/2) - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^11*b^(15/2) + a^12*b^(15/2))/((sqrt
(b)*x - sqrt(b*x^2 + a))^2 - a)^15

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maple [A]  time = 0.00, size = 39, normalized size = 0.57 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (8 b^{2} x^{4}-44 a b \,x^{2}+143 a^{2}\right )}{2145 a^{3} x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^16,x)

[Out]

-1/2145*(b*x^2+a)^(11/2)*(8*b^2*x^4-44*a*b*x^2+143*a^2)/x^15/a^3

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maxima [A]  time = 1.51, size = 56, normalized size = 0.82 \[ -\frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{2145 \, a^{3} x^{11}} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{195 \, a^{2} x^{13}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{15 \, a x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^16,x, algorithm="maxima")

[Out]

-8/2145*(b*x^2 + a)^(11/2)*b^2/(a^3*x^11) + 4/195*(b*x^2 + a)^(11/2)*b/(a^2*x^13) - 1/15*(b*x^2 + a)^(11/2)/(a
*x^15)

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mupad [B]  time = 7.42, size = 151, normalized size = 2.22 \[ \frac {4\,b^6\,\sqrt {b\,x^2+a}}{2145\,a^2\,x^3}-\frac {71\,b^4\,\sqrt {b\,x^2+a}}{429\,x^7}-\frac {206\,a\,b^3\,\sqrt {b\,x^2+a}}{429\,x^9}-\frac {61\,a^3\,b\,\sqrt {b\,x^2+a}}{195\,x^{13}}-\frac {b^5\,\sqrt {b\,x^2+a}}{715\,a\,x^5}-\frac {a^4\,\sqrt {b\,x^2+a}}{15\,x^{15}}-\frac {8\,b^7\,\sqrt {b\,x^2+a}}{2145\,a^3\,x}-\frac {406\,a^2\,b^2\,\sqrt {b\,x^2+a}}{715\,x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x^16,x)

[Out]

(4*b^6*(a + b*x^2)^(1/2))/(2145*a^2*x^3) - (71*b^4*(a + b*x^2)^(1/2))/(429*x^7) - (206*a*b^3*(a + b*x^2)^(1/2)
)/(429*x^9) - (61*a^3*b*(a + b*x^2)^(1/2))/(195*x^13) - (b^5*(a + b*x^2)^(1/2))/(715*a*x^5) - (a^4*(a + b*x^2)
^(1/2))/(15*x^15) - (8*b^7*(a + b*x^2)^(1/2))/(2145*a^3*x) - (406*a^2*b^2*(a + b*x^2)^(1/2))/(715*x^11)

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sympy [B]  time = 3.93, size = 604, normalized size = 8.88 \[ - \frac {143 a^{9} b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{x^{6} \left (2145 a^{5} b^{4} x^{8} + 4290 a^{4} b^{5} x^{10} + 2145 a^{3} b^{6} x^{12}\right )} - \frac {957 a^{8} b^{\frac {11}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{x^{4} \left (2145 a^{5} b^{4} x^{8} + 4290 a^{4} b^{5} x^{10} + 2145 a^{3} b^{6} x^{12}\right )} - \frac {2703 a^{7} b^{\frac {13}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{x^{2} \left (2145 a^{5} b^{4} x^{8} + 4290 a^{4} b^{5} x^{10} + 2145 a^{3} b^{6} x^{12}\right )} - \frac {4137 a^{6} b^{\frac {15}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2145 a^{5} b^{4} x^{8} + 4290 a^{4} b^{5} x^{10} + 2145 a^{3} b^{6} x^{12}} - \frac {3633 a^{5} b^{\frac {17}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{2145 a^{5} b^{4} x^{8} + 4290 a^{4} b^{5} x^{10} + 2145 a^{3} b^{6} x^{12}} - \frac {1743 a^{4} b^{\frac {19}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{2145 a^{5} b^{4} x^{8} + 4290 a^{4} b^{5} x^{10} + 2145 a^{3} b^{6} x^{12}} - \frac {357 a^{3} b^{\frac {21}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{2145 a^{5} b^{4} x^{8} + 4290 a^{4} b^{5} x^{10} + 2145 a^{3} b^{6} x^{12}} - \frac {3 a^{2} b^{\frac {23}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{2145 a^{5} b^{4} x^{8} + 4290 a^{4} b^{5} x^{10} + 2145 a^{3} b^{6} x^{12}} - \frac {12 a b^{\frac {25}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{2145 a^{5} b^{4} x^{8} + 4290 a^{4} b^{5} x^{10} + 2145 a^{3} b^{6} x^{12}} - \frac {8 b^{\frac {27}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{2145 a^{5} b^{4} x^{8} + 4290 a^{4} b^{5} x^{10} + 2145 a^{3} b^{6} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**16,x)

[Out]

-143*a**9*b**(9/2)*sqrt(a/(b*x**2) + 1)/(x**6*(2145*a**5*b**4*x**8 + 4290*a**4*b**5*x**10 + 2145*a**3*b**6*x**
12)) - 957*a**8*b**(11/2)*sqrt(a/(b*x**2) + 1)/(x**4*(2145*a**5*b**4*x**8 + 4290*a**4*b**5*x**10 + 2145*a**3*b
**6*x**12)) - 2703*a**7*b**(13/2)*sqrt(a/(b*x**2) + 1)/(x**2*(2145*a**5*b**4*x**8 + 4290*a**4*b**5*x**10 + 214
5*a**3*b**6*x**12)) - 4137*a**6*b**(15/2)*sqrt(a/(b*x**2) + 1)/(2145*a**5*b**4*x**8 + 4290*a**4*b**5*x**10 + 2
145*a**3*b**6*x**12) - 3633*a**5*b**(17/2)*x**2*sqrt(a/(b*x**2) + 1)/(2145*a**5*b**4*x**8 + 4290*a**4*b**5*x**
10 + 2145*a**3*b**6*x**12) - 1743*a**4*b**(19/2)*x**4*sqrt(a/(b*x**2) + 1)/(2145*a**5*b**4*x**8 + 4290*a**4*b*
*5*x**10 + 2145*a**3*b**6*x**12) - 357*a**3*b**(21/2)*x**6*sqrt(a/(b*x**2) + 1)/(2145*a**5*b**4*x**8 + 4290*a*
*4*b**5*x**10 + 2145*a**3*b**6*x**12) - 3*a**2*b**(23/2)*x**8*sqrt(a/(b*x**2) + 1)/(2145*a**5*b**4*x**8 + 4290
*a**4*b**5*x**10 + 2145*a**3*b**6*x**12) - 12*a*b**(25/2)*x**10*sqrt(a/(b*x**2) + 1)/(2145*a**5*b**4*x**8 + 42
90*a**4*b**5*x**10 + 2145*a**3*b**6*x**12) - 8*b**(27/2)*x**12*sqrt(a/(b*x**2) + 1)/(2145*a**5*b**4*x**8 + 429
0*a**4*b**5*x**10 + 2145*a**3*b**6*x**12)

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