3.438 \(\int \frac{(a+b x^2)^{9/2}}{x^{20}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{230945 a^5 x^{11}}+\frac{64 b^3 \left (a+b x^2\right )^{11/2}}{20995 a^4 x^{13}}-\frac{16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}+\frac{8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac{\left (a+b x^2\right )^{11/2}}{19 a x^{19}} \]

[Out]

-(a + b*x^2)^(11/2)/(19*a*x^19) + (8*b*(a + b*x^2)^(11/2))/(323*a^2*x^17) - (16*b^2*(a + b*x^2)^(11/2))/(1615*
a^3*x^15) + (64*b^3*(a + b*x^2)^(11/2))/(20995*a^4*x^13) - (128*b^4*(a + b*x^2)^(11/2))/(230945*a^5*x^11)

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Rubi [A]  time = 0.0445839, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{230945 a^5 x^{11}}+\frac{64 b^3 \left (a+b x^2\right )^{11/2}}{20995 a^4 x^{13}}-\frac{16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}+\frac{8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac{\left (a+b x^2\right )^{11/2}}{19 a x^{19}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^20,x]

[Out]

-(a + b*x^2)^(11/2)/(19*a*x^19) + (8*b*(a + b*x^2)^(11/2))/(323*a^2*x^17) - (16*b^2*(a + b*x^2)^(11/2))/(1615*
a^3*x^15) + (64*b^3*(a + b*x^2)^(11/2))/(20995*a^4*x^13) - (128*b^4*(a + b*x^2)^(11/2))/(230945*a^5*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^{20}} \, dx &=-\frac{\left (a+b x^2\right )^{11/2}}{19 a x^{19}}-\frac{(8 b) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx}{19 a}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{19 a x^{19}}+\frac{8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}+\frac{\left (48 b^2\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{16}} \, dx}{323 a^2}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{19 a x^{19}}+\frac{8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac{16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}-\frac{\left (64 b^3\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx}{1615 a^3}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{19 a x^{19}}+\frac{8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac{16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}+\frac{64 b^3 \left (a+b x^2\right )^{11/2}}{20995 a^4 x^{13}}+\frac{\left (128 b^4\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{20995 a^4}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{19 a x^{19}}+\frac{8 b \left (a+b x^2\right )^{11/2}}{323 a^2 x^{17}}-\frac{16 b^2 \left (a+b x^2\right )^{11/2}}{1615 a^3 x^{15}}+\frac{64 b^3 \left (a+b x^2\right )^{11/2}}{20995 a^4 x^{13}}-\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{230945 a^5 x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.0168273, size = 64, normalized size = 0.55 \[ -\frac{\left (a+b x^2\right )^{11/2} \left (2288 a^2 b^2 x^4-5720 a^3 b x^2+12155 a^4-704 a b^3 x^6+128 b^4 x^8\right )}{230945 a^5 x^{19}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^20,x]

[Out]

-((a + b*x^2)^(11/2)*(12155*a^4 - 5720*a^3*b*x^2 + 2288*a^2*b^2*x^4 - 704*a*b^3*x^6 + 128*b^4*x^8))/(230945*a^
5*x^19)

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Maple [A]  time = 0.003, size = 61, normalized size = 0.5 \begin{align*} -{\frac{128\,{b}^{4}{x}^{8}-704\,{b}^{3}{x}^{6}a+2288\,{b}^{2}{x}^{4}{a}^{2}-5720\,b{x}^{2}{a}^{3}+12155\,{a}^{4}}{230945\,{x}^{19}{a}^{5}} \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^20,x)

[Out]

-1/230945*(b*x^2+a)^(11/2)*(128*b^4*x^8-704*a*b^3*x^6+2288*a^2*b^2*x^4-5720*a^3*b*x^2+12155*a^4)/x^19/a^5

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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^20,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.32294, size = 288, normalized size = 2.48 \begin{align*} -\frac{{\left (128 \, b^{9} x^{18} - 64 \, a b^{8} x^{16} + 48 \, a^{2} b^{7} x^{14} - 40 \, a^{3} b^{6} x^{12} + 35 \, a^{4} b^{5} x^{10} + 23063 \, a^{5} b^{4} x^{8} + 75086 \, a^{6} b^{3} x^{6} + 95238 \, a^{7} b^{2} x^{4} + 55055 \, a^{8} b x^{2} + 12155 \, a^{9}\right )} \sqrt{b x^{2} + a}}{230945 \, a^{5} x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^20,x, algorithm="fricas")

[Out]

-1/230945*(128*b^9*x^18 - 64*a*b^8*x^16 + 48*a^2*b^7*x^14 - 40*a^3*b^6*x^12 + 35*a^4*b^5*x^10 + 23063*a^5*b^4*
x^8 + 75086*a^6*b^3*x^6 + 95238*a^7*b^2*x^4 + 55055*a^8*b*x^2 + 12155*a^9)*sqrt(b*x^2 + a)/(a^5*x^19)

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Sympy [B]  time = 17.0971, size = 1182, normalized size = 10.19 \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**20,x)

[Out]

-12155*a**13*b**(33/2)*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*
b**18*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 103675*a**12*b**(35/2)*x**2*sqrt(a/(b*x**2)
+ 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24
+ 230945*a**5*b**20*x**26) - 388388*a**11*b**(37/2)*x**4*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 92378
0*a**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 834988*a*
*10*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**1
8*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 1127210*a**9*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1
)/(230945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 23
0945*a**5*b**20*x**26) - 978810*a**8*b**(43/2)*x**10*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a*
*8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 534060*a**7*b
**(45/2)*x**12*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**18*x*
*22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 167436*a**6*b**(47/2)*x**14*sqrt(a/(b*x**2) + 1)/(2
30945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 230945
*a**5*b**20*x**26) - 23091*a**5*b**(49/2)*x**16*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b*
*17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 35*a**4*b**(51/2)*
x**18*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923
780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 280*a**3*b**(53/2)*x**20*sqrt(a/(b*x**2) + 1)/(230945*a**9*b
**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*
x**26) - 560*a**2*b**(55/2)*x**22*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 13
85670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26) - 448*a*b**(57/2)*x**24*sqrt(a/(b*
x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x
**24 + 230945*a**5*b**20*x**26) - 128*b**(59/2)*x**26*sqrt(a/(b*x**2) + 1)/(230945*a**9*b**16*x**18 + 923780*a
**8*b**17*x**20 + 1385670*a**7*b**18*x**22 + 923780*a**6*b**19*x**24 + 230945*a**5*b**20*x**26)

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Giac [B]  time = 3.05115, size = 551, normalized size = 4.75 \begin{align*} \frac{256 \,{\left (92378 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{28} b^{\frac{19}{2}} + 554268 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{26} a b^{\frac{19}{2}} + 1939938 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{24} a^{2} b^{\frac{19}{2}} + 4018443 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{22} a^{3} b^{\frac{19}{2}} + 5866003 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{20} a^{4} b^{\frac{19}{2}} + 5773625 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{18} a^{5} b^{\frac{19}{2}} + 4094025 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a^{6} b^{\frac{19}{2}} + 1889550 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a^{7} b^{\frac{19}{2}} + 581400 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{8} b^{\frac{19}{2}} + 80750 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{9} b^{\frac{19}{2}} + 3876 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{10} b^{\frac{19}{2}} - 969 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{11} b^{\frac{19}{2}} + 171 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{12} b^{\frac{19}{2}} - 19 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{13} b^{\frac{19}{2}} + a^{14} b^{\frac{19}{2}}\right )}}{230945 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^20,x, algorithm="giac")

[Out]

256/230945*(92378*(sqrt(b)*x - sqrt(b*x^2 + a))^28*b^(19/2) + 554268*(sqrt(b)*x - sqrt(b*x^2 + a))^26*a*b^(19/
2) + 1939938*(sqrt(b)*x - sqrt(b*x^2 + a))^24*a^2*b^(19/2) + 4018443*(sqrt(b)*x - sqrt(b*x^2 + a))^22*a^3*b^(1
9/2) + 5866003*(sqrt(b)*x - sqrt(b*x^2 + a))^20*a^4*b^(19/2) + 5773625*(sqrt(b)*x - sqrt(b*x^2 + a))^18*a^5*b^
(19/2) + 4094025*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^6*b^(19/2) + 1889550*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^7*
b^(19/2) + 581400*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^8*b^(19/2) + 80750*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^9*b
^(19/2) + 3876*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^10*b^(19/2) - 969*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^11*b^(19/
2) + 171*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^12*b^(19/2) - 19*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^13*b^(19/2) + a^
14*b^(19/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^19