3.439 \(\int \frac{(a+b x^2)^{9/2}}{x^{22}} \, dx\)

Optimal. Leaf size=140 \[ \frac{256 b^5 \left (a+b x^2\right )^{11/2}}{969969 a^6 x^{11}}-\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{88179 a^5 x^{13}}+\frac{32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}-\frac{80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac{10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac{\left (a+b x^2\right )^{11/2}}{21 a x^{21}} \]

[Out]

-(a + b*x^2)^(11/2)/(21*a*x^21) + (10*b*(a + b*x^2)^(11/2))/(399*a^2*x^19) - (80*b^2*(a + b*x^2)^(11/2))/(6783
*a^3*x^17) + (32*b^3*(a + b*x^2)^(11/2))/(6783*a^4*x^15) - (128*b^4*(a + b*x^2)^(11/2))/(88179*a^5*x^13) + (25
6*b^5*(a + b*x^2)^(11/2))/(969969*a^6*x^11)

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Rubi [A]  time = 0.0562635, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{256 b^5 \left (a+b x^2\right )^{11/2}}{969969 a^6 x^{11}}-\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{88179 a^5 x^{13}}+\frac{32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}-\frac{80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac{10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac{\left (a+b x^2\right )^{11/2}}{21 a x^{21}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^22,x]

[Out]

-(a + b*x^2)^(11/2)/(21*a*x^21) + (10*b*(a + b*x^2)^(11/2))/(399*a^2*x^19) - (80*b^2*(a + b*x^2)^(11/2))/(6783
*a^3*x^17) + (32*b^3*(a + b*x^2)^(11/2))/(6783*a^4*x^15) - (128*b^4*(a + b*x^2)^(11/2))/(88179*a^5*x^13) + (25
6*b^5*(a + b*x^2)^(11/2))/(969969*a^6*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^{22}} \, dx &=-\frac{\left (a+b x^2\right )^{11/2}}{21 a x^{21}}-\frac{(10 b) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx}{21 a}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac{10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}+\frac{\left (80 b^2\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx}{399 a^2}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac{10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac{80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}-\frac{\left (160 b^3\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{16}} \, dx}{2261 a^3}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac{10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac{80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac{32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}+\frac{\left (128 b^4\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx}{6783 a^4}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac{10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac{80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac{32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}-\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{88179 a^5 x^{13}}-\frac{\left (256 b^5\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{88179 a^5}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{21 a x^{21}}+\frac{10 b \left (a+b x^2\right )^{11/2}}{399 a^2 x^{19}}-\frac{80 b^2 \left (a+b x^2\right )^{11/2}}{6783 a^3 x^{17}}+\frac{32 b^3 \left (a+b x^2\right )^{11/2}}{6783 a^4 x^{15}}-\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{88179 a^5 x^{13}}+\frac{256 b^5 \left (a+b x^2\right )^{11/2}}{969969 a^6 x^{11}}\\ \end{align*}

Mathematica [A]  time = 0.0204524, size = 75, normalized size = 0.54 \[ \frac{\left (a+b x^2\right )^{11/2} \left (4576 a^2 b^3 x^6-11440 a^3 b^2 x^4+24310 a^4 b x^2-46189 a^5-1408 a b^4 x^8+256 b^5 x^{10}\right )}{969969 a^6 x^{21}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^22,x]

[Out]

((a + b*x^2)^(11/2)*(-46189*a^5 + 24310*a^4*b*x^2 - 11440*a^3*b^2*x^4 + 4576*a^2*b^3*x^6 - 1408*a*b^4*x^8 + 25
6*b^5*x^10))/(969969*a^6*x^21)

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Maple [A]  time = 0.005, size = 72, normalized size = 0.5 \begin{align*} -{\frac{-256\,{b}^{5}{x}^{10}+1408\,a{b}^{4}{x}^{8}-4576\,{a}^{2}{b}^{3}{x}^{6}+11440\,{a}^{3}{b}^{2}{x}^{4}-24310\,{a}^{4}b{x}^{2}+46189\,{a}^{5}}{969969\,{x}^{21}{a}^{6}} \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^22,x)

[Out]

-1/969969*(b*x^2+a)^(11/2)*(-256*b^5*x^10+1408*a*b^4*x^8-4576*a^2*b^3*x^6+11440*a^3*b^2*x^4-24310*a^4*b*x^2+46
189*a^5)/x^21/a^6

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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^22,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.3392, size = 319, normalized size = 2.28 \begin{align*} \frac{{\left (256 \, b^{10} x^{20} - 128 \, a b^{9} x^{18} + 96 \, a^{2} b^{8} x^{16} - 80 \, a^{3} b^{7} x^{14} + 70 \, a^{4} b^{6} x^{12} - 63 \, a^{5} b^{5} x^{10} - 80773 \, a^{6} b^{4} x^{8} - 271414 \, a^{7} b^{3} x^{6} - 351780 \, a^{8} b^{2} x^{4} - 206635 \, a^{9} b x^{2} - 46189 \, a^{10}\right )} \sqrt{b x^{2} + a}}{969969 \, a^{6} x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^22,x, algorithm="fricas")

[Out]

1/969969*(256*b^10*x^20 - 128*a*b^9*x^18 + 96*a^2*b^8*x^16 - 80*a^3*b^7*x^14 + 70*a^4*b^6*x^12 - 63*a^5*b^5*x^
10 - 80773*a^6*b^4*x^8 - 271414*a^7*b^3*x^6 - 351780*a^8*b^2*x^4 - 206635*a^9*b*x^2 - 46189*a^10)*sqrt(b*x^2 +
a)/(a^6*x^21)

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Sympy [B]  time = 18.8148, size = 1540, normalized size = 11. \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**22,x)

[Out]

-46189*a**15*b**(51/2)*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a*
*9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 437580*a**14
*b**(53/2)*x**2*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**2
7*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 1846845*a**13*b**(5
5/2)*x**4*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**2
4 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 4558554*a**12*b**(57/2)*x
**6*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 96
99690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 7252938*a**11*b**(59/2)*x**8*sq
rt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*
a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 7715232*a**10*b**(61/2)*x**10*sqrt(a/
(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*
b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 5487650*a**9*b**(63/2)*x**12*sqrt(a/(b*x**
2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*
x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 2516940*a**8*b**(65/2)*x**14*sqrt(a/(b*x**2) + 1
)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x**26
+ 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 675513*a**7*b**(67/2)*x**16*sqrt(a/(b*x**2) + 1)/(9699
69*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 48498
45*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) - 80836*a**6*b**(69/2)*x**18*sqrt(a/(b*x**2) + 1)/(969969*a**11
*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*
b**29*x**28 + 969969*a**6*b**30*x**30) + 63*a**5*b**(71/2)*x**20*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**2
0 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28
+ 969969*a**6*b**30*x**30) + 630*a**4*b**(73/2)*x**22*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 484984
5*a**10*b**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*
a**6*b**30*x**30) + 1680*a**3*b**(75/2)*x**24*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b
**26*x**22 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**3
0*x**30) + 2016*a**2*b**(77/2)*x**26*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**2
2 + 9699690*a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30)
+ 1152*a*b**(79/2)*x**28*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*
a**9*b**27*x**24 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30) + 256*b**(81
/2)*x**30*sqrt(a/(b*x**2) + 1)/(969969*a**11*b**25*x**20 + 4849845*a**10*b**26*x**22 + 9699690*a**9*b**27*x**2
4 + 9699690*a**8*b**28*x**26 + 4849845*a**7*b**29*x**28 + 969969*a**6*b**30*x**30)

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Giac [B]  time = 2.50813, size = 589, normalized size = 4.21 \begin{align*} \frac{512 \,{\left (646646 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{30} b^{\frac{21}{2}} + 4157010 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{28} a b^{\frac{21}{2}} + 14549535 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{26} a^{2} b^{\frac{21}{2}} + 30715685 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{24} a^{3} b^{\frac{21}{2}} + 44618574 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{22} a^{4} b^{\frac{21}{2}} + 44265858 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{20} a^{5} b^{\frac{21}{2}} + 31009615 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{18} a^{6} b^{\frac{21}{2}} + 14346045 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a^{7} b^{\frac{21}{2}} + 4273290 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a^{8} b^{\frac{21}{2}} + 592382 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{9} b^{\frac{21}{2}} + 20349 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{10} b^{\frac{21}{2}} - 5985 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{11} b^{\frac{21}{2}} + 1330 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{12} b^{\frac{21}{2}} - 210 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{13} b^{\frac{21}{2}} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{14} b^{\frac{21}{2}} - a^{15} b^{\frac{21}{2}}\right )}}{969969 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^22,x, algorithm="giac")

[Out]

512/969969*(646646*(sqrt(b)*x - sqrt(b*x^2 + a))^30*b^(21/2) + 4157010*(sqrt(b)*x - sqrt(b*x^2 + a))^28*a*b^(2
1/2) + 14549535*(sqrt(b)*x - sqrt(b*x^2 + a))^26*a^2*b^(21/2) + 30715685*(sqrt(b)*x - sqrt(b*x^2 + a))^24*a^3*
b^(21/2) + 44618574*(sqrt(b)*x - sqrt(b*x^2 + a))^22*a^4*b^(21/2) + 44265858*(sqrt(b)*x - sqrt(b*x^2 + a))^20*
a^5*b^(21/2) + 31009615*(sqrt(b)*x - sqrt(b*x^2 + a))^18*a^6*b^(21/2) + 14346045*(sqrt(b)*x - sqrt(b*x^2 + a))
^16*a^7*b^(21/2) + 4273290*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^8*b^(21/2) + 592382*(sqrt(b)*x - sqrt(b*x^2 + a)
)^12*a^9*b^(21/2) + 20349*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^10*b^(21/2) - 5985*(sqrt(b)*x - sqrt(b*x^2 + a))^
8*a^11*b^(21/2) + 1330*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^12*b^(21/2) - 210*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^1
3*b^(21/2) + 21*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^14*b^(21/2) - a^15*b^(21/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^
2 - a)^21