3.437 \(\int \frac{(a+b x^2)^{9/2}}{x^{18}} \, dx\)

Optimal. Leaf size=92 \[ \frac{16 b^3 \left (a+b x^2\right )^{11/2}}{12155 a^4 x^{11}}-\frac{8 b^2 \left (a+b x^2\right )^{11/2}}{1105 a^3 x^{13}}+\frac{2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}-\frac{\left (a+b x^2\right )^{11/2}}{17 a x^{17}} \]

[Out]

-(a + b*x^2)^(11/2)/(17*a*x^17) + (2*b*(a + b*x^2)^(11/2))/(85*a^2*x^15) - (8*b^2*(a + b*x^2)^(11/2))/(1105*a^
3*x^13) + (16*b^3*(a + b*x^2)^(11/2))/(12155*a^4*x^11)

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Rubi [A]  time = 0.0296989, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{16 b^3 \left (a+b x^2\right )^{11/2}}{12155 a^4 x^{11}}-\frac{8 b^2 \left (a+b x^2\right )^{11/2}}{1105 a^3 x^{13}}+\frac{2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}-\frac{\left (a+b x^2\right )^{11/2}}{17 a x^{17}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^18,x]

[Out]

-(a + b*x^2)^(11/2)/(17*a*x^17) + (2*b*(a + b*x^2)^(11/2))/(85*a^2*x^15) - (8*b^2*(a + b*x^2)^(11/2))/(1105*a^
3*x^13) + (16*b^3*(a + b*x^2)^(11/2))/(12155*a^4*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^{18}} \, dx &=-\frac{\left (a+b x^2\right )^{11/2}}{17 a x^{17}}-\frac{(6 b) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{16}} \, dx}{17 a}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{17 a x^{17}}+\frac{2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}+\frac{\left (8 b^2\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx}{85 a^2}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{17 a x^{17}}+\frac{2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}-\frac{8 b^2 \left (a+b x^2\right )^{11/2}}{1105 a^3 x^{13}}-\frac{\left (16 b^3\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{1105 a^3}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{17 a x^{17}}+\frac{2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}-\frac{8 b^2 \left (a+b x^2\right )^{11/2}}{1105 a^3 x^{13}}+\frac{16 b^3 \left (a+b x^2\right )^{11/2}}{12155 a^4 x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.017672, size = 53, normalized size = 0.58 \[ \frac{\left (a+b x^2\right )^{11/2} \left (286 a^2 b x^2-715 a^3-88 a b^2 x^4+16 b^3 x^6\right )}{12155 a^4 x^{17}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^18,x]

[Out]

((a + b*x^2)^(11/2)*(-715*a^3 + 286*a^2*b*x^2 - 88*a*b^2*x^4 + 16*b^3*x^6))/(12155*a^4*x^17)

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Maple [A]  time = 0.003, size = 50, normalized size = 0.5 \begin{align*} -{\frac{-16\,{b}^{3}{x}^{6}+88\,a{b}^{2}{x}^{4}-286\,{a}^{2}b{x}^{2}+715\,{a}^{3}}{12155\,{x}^{17}{a}^{4}} \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^18,x)

[Out]

-1/12155*(b*x^2+a)^(11/2)*(-16*b^3*x^6+88*a*b^2*x^4-286*a^2*b*x^2+715*a^3)/x^17/a^4

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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^18,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.76667, size = 247, normalized size = 2.68 \begin{align*} \frac{{\left (16 \, b^{8} x^{16} - 8 \, a b^{7} x^{14} + 6 \, a^{2} b^{6} x^{12} - 5 \, a^{3} b^{5} x^{10} - 1515 \, a^{4} b^{4} x^{8} - 4714 \, a^{5} b^{3} x^{6} - 5808 \, a^{6} b^{2} x^{4} - 3289 \, a^{7} b x^{2} - 715 \, a^{8}\right )} \sqrt{b x^{2} + a}}{12155 \, a^{4} x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^18,x, algorithm="fricas")

[Out]

1/12155*(16*b^8*x^16 - 8*a*b^7*x^14 + 6*a^2*b^6*x^12 - 5*a^3*b^5*x^10 - 1515*a^4*b^4*x^8 - 4714*a^5*b^3*x^6 -
5808*a^6*b^2*x^4 - 3289*a^7*b*x^2 - 715*a^8)*sqrt(b*x^2 + a)/(a^4*x^17)

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Sympy [B]  time = 13.0886, size = 867, normalized size = 9.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**18,x)

[Out]

-715*a**11*b**(19/2)*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x
**20 + 12155*a**4*b**12*x**22) - 5434*a**10*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465
*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22) - 17820*a**9*b**(23/2)*x**4*sqrt(a/(b*x**
2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22) - 3
2720*a**8*b**(25/2)*x**6*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**
11*x**20 + 12155*a**4*b**12*x**22) - 36370*a**7*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 3
6465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22) - 24500*a**6*b**(29/2)*x**10*sqrt(a/(
b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22
) - 9268*a**5*b**(31/2)*x**12*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**
5*b**11*x**20 + 12155*a**4*b**12*x**22) - 1520*a**4*b**(33/2)*x**14*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**1
6 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22) + 5*a**3*b**(35/2)*x**16*sqrt(a/
(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**2
2) + 30*a**2*b**(37/2)*x**18*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5
*b**11*x**20 + 12155*a**4*b**12*x**22) + 40*a*b**(39/2)*x**20*sqrt(a/(b*x**2) + 1)/(12155*a**7*b**9*x**16 + 36
465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22) + 16*b**(41/2)*x**22*sqrt(a/(b*x**2) +
1)/(12155*a**7*b**9*x**16 + 36465*a**6*b**10*x**18 + 36465*a**5*b**11*x**20 + 12155*a**4*b**12*x**22)

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Giac [B]  time = 2.85873, size = 516, normalized size = 5.61 \begin{align*} \frac{32 \,{\left (12155 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{26} b^{\frac{17}{2}} + 65637 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{24} a b^{\frac{17}{2}} + 233376 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{22} a^{2} b^{\frac{17}{2}} + 466752 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{20} a^{3} b^{\frac{17}{2}} + 692835 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{18} a^{4} b^{\frac{17}{2}} + 668525 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a^{5} b^{\frac{17}{2}} + 486200 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a^{6} b^{\frac{17}{2}} + 221000 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{7} b^{\frac{17}{2}} + 71825 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{8} b^{\frac{17}{2}} + 9775 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{9} b^{\frac{17}{2}} + 680 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{10} b^{\frac{17}{2}} - 136 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{11} b^{\frac{17}{2}} + 17 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{12} b^{\frac{17}{2}} - a^{13} b^{\frac{17}{2}}\right )}}{12155 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^18,x, algorithm="giac")

[Out]

32/12155*(12155*(sqrt(b)*x - sqrt(b*x^2 + a))^26*b^(17/2) + 65637*(sqrt(b)*x - sqrt(b*x^2 + a))^24*a*b^(17/2)
+ 233376*(sqrt(b)*x - sqrt(b*x^2 + a))^22*a^2*b^(17/2) + 466752*(sqrt(b)*x - sqrt(b*x^2 + a))^20*a^3*b^(17/2)
+ 692835*(sqrt(b)*x - sqrt(b*x^2 + a))^18*a^4*b^(17/2) + 668525*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^5*b^(17/2)
+ 486200*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^6*b^(17/2) + 221000*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^7*b^(17/2)
+ 71825*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^8*b^(17/2) + 9775*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^9*b^(17/2) + 68
0*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^10*b^(17/2) - 136*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^11*b^(17/2) + 17*(sqrt
(b)*x - sqrt(b*x^2 + a))^2*a^12*b^(17/2) - a^13*b^(17/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^17