3.440 \(\int \frac{(a+b x^2)^{9/2}}{x^{24}} \, dx\)
Optimal. Leaf size=164 \[ -\frac{1024 b^6 \left (a+b x^2\right )^{11/2}}{7436429 a^7 x^{11}}+\frac{512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}-\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac{320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac{40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac{4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac{\left (a+b x^2\right )^{11/2}}{23 a x^{23}} \]
[Out]
-(a + b*x^2)^(11/2)/(23*a*x^23) + (4*b*(a + b*x^2)^(11/2))/(161*a^2*x^21) - (40*b^2*(a + b*x^2)^(11/2))/(3059*
a^3*x^19) + (320*b^3*(a + b*x^2)^(11/2))/(52003*a^4*x^17) - (128*b^4*(a + b*x^2)^(11/2))/(52003*a^5*x^15) + (5
12*b^5*(a + b*x^2)^(11/2))/(676039*a^6*x^13) - (1024*b^6*(a + b*x^2)^(11/2))/(7436429*a^7*x^11)
________________________________________________________________________________________
Rubi [A] time = 0.0719655, antiderivative size = 164, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used =
{271, 264} \[ -\frac{1024 b^6 \left (a+b x^2\right )^{11/2}}{7436429 a^7 x^{11}}+\frac{512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}-\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac{320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac{40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac{4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac{\left (a+b x^2\right )^{11/2}}{23 a x^{23}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x^2)^(9/2)/x^24,x]
[Out]
-(a + b*x^2)^(11/2)/(23*a*x^23) + (4*b*(a + b*x^2)^(11/2))/(161*a^2*x^21) - (40*b^2*(a + b*x^2)^(11/2))/(3059*
a^3*x^19) + (320*b^3*(a + b*x^2)^(11/2))/(52003*a^4*x^17) - (128*b^4*(a + b*x^2)^(11/2))/(52003*a^5*x^15) + (5
12*b^5*(a + b*x^2)^(11/2))/(676039*a^6*x^13) - (1024*b^6*(a + b*x^2)^(11/2))/(7436429*a^7*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^{24}} \, dx &=-\frac{\left (a+b x^2\right )^{11/2}}{23 a x^{23}}-\frac{(12 b) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{22}} \, dx}{23 a}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac{4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}+\frac{\left (40 b^2\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{20}} \, dx}{161 a^2}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac{4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac{40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}-\frac{\left (320 b^3\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx}{3059 a^3}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac{4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac{40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac{320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}+\frac{\left (1920 b^4\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{16}} \, dx}{52003 a^4}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac{4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac{40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac{320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}-\frac{\left (512 b^5\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{14}} \, dx}{52003 a^5}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac{4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac{40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac{320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac{512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}+\frac{\left (1024 b^6\right ) \int \frac{\left (a+b x^2\right )^{9/2}}{x^{12}} \, dx}{676039 a^6}\\ &=-\frac{\left (a+b x^2\right )^{11/2}}{23 a x^{23}}+\frac{4 b \left (a+b x^2\right )^{11/2}}{161 a^2 x^{21}}-\frac{40 b^2 \left (a+b x^2\right )^{11/2}}{3059 a^3 x^{19}}+\frac{320 b^3 \left (a+b x^2\right )^{11/2}}{52003 a^4 x^{17}}-\frac{128 b^4 \left (a+b x^2\right )^{11/2}}{52003 a^5 x^{15}}+\frac{512 b^5 \left (a+b x^2\right )^{11/2}}{676039 a^6 x^{13}}-\frac{1024 b^6 \left (a+b x^2\right )^{11/2}}{7436429 a^7 x^{11}}\\ \end{align*}
Mathematica [A] time = 0.0211539, size = 86, normalized size = 0.52 \[ -\frac{\left (a+b x^2\right )^{11/2} \left (18304 a^2 b^4 x^8-45760 a^3 b^3 x^6+97240 a^4 b^2 x^4-184756 a^5 b x^2+323323 a^6-5632 a b^5 x^{10}+1024 b^6 x^{12}\right )}{7436429 a^7 x^{23}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x^2)^(9/2)/x^24,x]
[Out]
-((a + b*x^2)^(11/2)*(323323*a^6 - 184756*a^5*b*x^2 + 97240*a^4*b^2*x^4 - 45760*a^3*b^3*x^6 + 18304*a^2*b^4*x^
8 - 5632*a*b^5*x^10 + 1024*b^6*x^12))/(7436429*a^7*x^23)
________________________________________________________________________________________
Maple [A] time = 0.005, size = 83, normalized size = 0.5 \begin{align*} -{\frac{1024\,{b}^{6}{x}^{12}-5632\,{b}^{5}{x}^{10}a+18304\,{b}^{4}{x}^{8}{a}^{2}-45760\,{b}^{3}{x}^{6}{a}^{3}+97240\,{b}^{2}{x}^{4}{a}^{4}-184756\,b{x}^{2}{a}^{5}+323323\,{a}^{6}}{7436429\,{x}^{23}{a}^{7}} \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^24,x)
[Out]
-1/7436429*(b*x^2+a)^(11/2)*(1024*b^6*x^12-5632*a*b^5*x^10+18304*a^2*b^4*x^8-45760*a^3*b^3*x^6+97240*a^4*b^2*x
^4-184756*a^5*b*x^2+323323*a^6)/x^23/a^7
________________________________________________________________________________________
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^24,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [A] time = 5.68819, size = 363, normalized size = 2.21 \begin{align*} -\frac{{\left (1024 \, b^{11} x^{22} - 512 \, a b^{10} x^{20} + 384 \, a^{2} b^{9} x^{18} - 320 \, a^{3} b^{8} x^{16} + 280 \, a^{4} b^{7} x^{14} - 252 \, a^{5} b^{6} x^{12} + 231 \, a^{6} b^{5} x^{10} + 530959 \, a^{7} b^{4} x^{8} + 1826110 \, a^{8} b^{3} x^{6} + 2406690 \, a^{9} b^{2} x^{4} + 1431859 \, a^{10} b x^{2} + 323323 \, a^{11}\right )} \sqrt{b x^{2} + a}}{7436429 \, a^{7} x^{23}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x^2+a)^(9/2)/x^24,x, algorithm="fricas")
[Out]
-1/7436429*(1024*b^11*x^22 - 512*a*b^10*x^20 + 384*a^2*b^9*x^18 - 320*a^3*b^8*x^16 + 280*a^4*b^7*x^14 - 252*a^
5*b^6*x^12 + 231*a^6*b^5*x^10 + 530959*a^7*b^4*x^8 + 1826110*a^8*b^3*x^6 + 2406690*a^9*b^2*x^4 + 1431859*a^10*
b*x^2 + 323323*a^11)*sqrt(b*x^2 + a)/(a^7*x^23)
________________________________________________________________________________________
Sympy [B] time = 23.7845, size = 1950, normalized size = 11.89 \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**24,x)
[Out]
-323323*a**17*b**(73/2)*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 1115464
35*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 +
7436429*a**7*b**42*x**34) - 3371797*a**16*b**(75/2)*x**2*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 446
18574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**
30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 15847689*a**15*b**(77/2)*x**4*sqrt(a/(b*x**2) + 1
)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**3
9*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 44210595*a**14*
b**(79/2)*x**6*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*
b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a
**7*b**42*x**34) - 81074994*a**13*b**(81/2)*x**8*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a*
*12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 446
18574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 102129258*a**12*b**(83/2)*x**10*sqrt(a/(b*x**2) + 1)/(743
6429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**2
8 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 89502546*a**11*b**(85
/2)*x**12*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38
*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b
**42*x**34) - 53885062*a**10*b**(87/2)*x**14*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*
b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 4461857
4*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 21329935*a**9*b**(89/2)*x**16*sqrt(a/(b*x**2) + 1)/(7436429*a
**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 11
1546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 5012953*a**8*b**(91/2)*x**1
8*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 +
148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**
34) - 531157*a**7*b**(93/2)*x**20*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24
+ 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**4
1*x**32 + 7436429*a**7*b**42*x**34) - 231*a**6*b**(95/2)*x**22*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22
+ 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**
40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 2772*a**5*b**(97/2)*x**24*sqrt(a/(b*x**2) +
1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b*
*39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 9240*a**4*b**
(99/2)*x**26*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b*
*38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**
7*b**42*x**34) - 14784*a**3*b**(101/2)*x**28*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*
b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 4461857
4*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 12672*a**2*b**(103/2)*x**30*sqrt(a/(b*x**2) + 1)/(7436429*a**
13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 1115
46435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 5632*a*b**(105/2)*x**32*sqrt(
a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*a**11*b**38*x**26 + 148728
580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 7436429*a**7*b**42*x**34) - 1
024*b**(107/2)*x**34*sqrt(a/(b*x**2) + 1)/(7436429*a**13*b**36*x**22 + 44618574*a**12*b**37*x**24 + 111546435*
a**11*b**38*x**26 + 148728580*a**10*b**39*x**28 + 111546435*a**9*b**40*x**30 + 44618574*a**8*b**41*x**32 + 743
6429*a**7*b**42*x**34)
________________________________________________________________________________________
Giac [B] time = 3.05804, size = 624, normalized size = 3.8 \begin{align*} \frac{2048 \,{\left (4249388 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{32} b^{\frac{23}{2}} + 28683369 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{30} a b^{\frac{23}{2}} + 100922965 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{28} a^{2} b^{\frac{23}{2}} + 215656441 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{26} a^{3} b^{\frac{23}{2}} + 313006057 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{24} a^{4} b^{\frac{23}{2}} + 311653979 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{22} a^{5} b^{\frac{23}{2}} + 216800507 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{20} a^{6} b^{\frac{23}{2}} + 100105775 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{18} a^{7} b^{\frac{23}{2}} + 29173683 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{16} a^{8} b^{\frac{23}{2}} + 4004231 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} a^{9} b^{\frac{23}{2}} + 100947 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a^{10} b^{\frac{23}{2}} - 33649 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{11} b^{\frac{23}{2}} + 8855 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{12} b^{\frac{23}{2}} - 1771 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{13} b^{\frac{23}{2}} + 253 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{14} b^{\frac{23}{2}} - 23 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{15} b^{\frac{23}{2}} + a^{16} b^{\frac{23}{2}}\right )}}{7436429 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{23}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x^2+a)^(9/2)/x^24,x, algorithm="giac")
[Out]
2048/7436429*(4249388*(sqrt(b)*x - sqrt(b*x^2 + a))^32*b^(23/2) + 28683369*(sqrt(b)*x - sqrt(b*x^2 + a))^30*a*
b^(23/2) + 100922965*(sqrt(b)*x - sqrt(b*x^2 + a))^28*a^2*b^(23/2) + 215656441*(sqrt(b)*x - sqrt(b*x^2 + a))^2
6*a^3*b^(23/2) + 313006057*(sqrt(b)*x - sqrt(b*x^2 + a))^24*a^4*b^(23/2) + 311653979*(sqrt(b)*x - sqrt(b*x^2 +
a))^22*a^5*b^(23/2) + 216800507*(sqrt(b)*x - sqrt(b*x^2 + a))^20*a^6*b^(23/2) + 100105775*(sqrt(b)*x - sqrt(b
*x^2 + a))^18*a^7*b^(23/2) + 29173683*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^8*b^(23/2) + 4004231*(sqrt(b)*x - sqr
t(b*x^2 + a))^14*a^9*b^(23/2) + 100947*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^10*b^(23/2) - 33649*(sqrt(b)*x - sqr
t(b*x^2 + a))^10*a^11*b^(23/2) + 8855*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^12*b^(23/2) - 1771*(sqrt(b)*x - sqrt(b
*x^2 + a))^6*a^13*b^(23/2) + 253*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^14*b^(23/2) - 23*(sqrt(b)*x - sqrt(b*x^2 +
a))^2*a^15*b^(23/2) + a^16*b^(23/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^23