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]

-(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)

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Rubi [A]  time = 0.021, antiderivative size = 68, normalized size of antiderivative = 1., 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 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]

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^{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*}

Mathematica [A]  time = 0.0144015, 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]

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

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Maple [A]  time = 0.003, size = 39, normalized size = 0.6 \begin{align*} -{\frac{8\,{b}^{2}{x}^{4}-44\,ab{x}^{2}+143\,{a}^{2}}{2145\,{x}^{15}{a}^{3}} \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^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 [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^16,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.36999, size = 220, normalized size = 3.24 \begin{align*} -\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}} \end{align*}

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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Sympy [B]  time = 10.0584, size = 604, normalized size = 8.88 \begin{align*} - \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}} \end{align*}

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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Giac [B]  time = 3.06728, size = 478, normalized size = 7.03 \begin{align*} \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}} \end{align*}

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