Integrand size = 15, antiderivative size = 92 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx=-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}+\frac {2 b \left (a+b x^2\right )^{11/2}}{85 a^2 x^{15}}-\frac {8 b^2 \left (a+b x^2\right )^{11/2}}{1105 a^3 x^{13}}+\frac {16 b^3 \left (a+b x^2\right )^{11/2}}{12155 a^4 x^{11}} \] Output:
-1/17*(b*x^2+a)^(11/2)/a/x^17+2/85*b*(b*x^2+a)^(11/2)/a^2/x^15-8/1105*b^2* (b*x^2+a)^(11/2)/a^3/x^13+16/12155*b^3*(b*x^2+a)^(11/2)/a^4/x^11
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx=\frac {\left (a+b x^2\right )^{11/2} \left (-715 a^3+286 a^2 b x^2-88 a b^2 x^4+16 b^3 x^6\right )}{12155 a^4 x^{17}} \] Input:
Integrate[(a + b*x^2)^(9/2)/x^18,x]
Output:
((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)
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {245, 245, 245, 242}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {6 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{16}}dx}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {6 b \left (-\frac {4 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{14}}dx}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\) |
\(\Big \downarrow \) 245 |
\(\displaystyle -\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a\right )^{9/2}}{x^{12}}dx}{13 a}-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {6 b \left (-\frac {4 b \left (\frac {2 b \left (a+b x^2\right )^{11/2}}{143 a^2 x^{11}}-\frac {\left (a+b x^2\right )^{11/2}}{13 a x^{13}}\right )}{15 a}-\frac {\left (a+b x^2\right )^{11/2}}{15 a x^{15}}\right )}{17 a}-\frac {\left (a+b x^2\right )^{11/2}}{17 a x^{17}}\) |
Input:
Int[(a + b*x^2)^(9/2)/x^18,x]
Output:
-1/17*(a + b*x^2)^(11/2)/(a*x^17) - (6*b*(-1/15*(a + b*x^2)^(11/2)/(a*x^15 ) - (4*b*(-1/13*(a + b*x^2)^(11/2)/(a*x^13) + (2*b*(a + b*x^2)^(11/2))/(14 3*a^2*x^11)))/(15*a)))/(17*a)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Time = 2.36 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-16 b^{3} x^{6}+88 a \,b^{2} x^{4}-286 a^{2} b \,x^{2}+715 a^{3}\right )}{12155 x^{17} a^{4}}\) | \(50\) |
pseudoelliptic | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-16 b^{3} x^{6}+88 a \,b^{2} x^{4}-286 a^{2} b \,x^{2}+715 a^{3}\right )}{12155 x^{17} a^{4}}\) | \(50\) |
orering | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-16 b^{3} x^{6}+88 a \,b^{2} x^{4}-286 a^{2} b \,x^{2}+715 a^{3}\right )}{12155 x^{17} a^{4}}\) | \(50\) |
default | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{17 a \,x^{17}}-\frac {6 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{15 a \,x^{15}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{13 a \,x^{13}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{143 a^{2} x^{11}}\right )}{15 a}\right )}{17 a}\) | \(85\) |
trager | \(-\frac {\left (-16 b^{8} x^{16}+8 a \,b^{7} x^{14}-6 a^{2} b^{6} x^{12}+5 a^{3} b^{5} x^{10}+1515 a^{4} b^{4} x^{8}+4714 a^{5} b^{3} x^{6}+5808 a^{6} b^{2} x^{4}+3289 a^{7} b \,x^{2}+715 a^{8}\right ) \sqrt {b \,x^{2}+a}}{12155 x^{17} a^{4}}\) | \(105\) |
risch | \(-\frac {\left (-16 b^{8} x^{16}+8 a \,b^{7} x^{14}-6 a^{2} b^{6} x^{12}+5 a^{3} b^{5} x^{10}+1515 a^{4} b^{4} x^{8}+4714 a^{5} b^{3} x^{6}+5808 a^{6} b^{2} x^{4}+3289 a^{7} b \,x^{2}+715 a^{8}\right ) \sqrt {b \,x^{2}+a}}{12155 x^{17} a^{4}}\) | \(105\) |
Input:
int((b*x^2+a)^(9/2)/x^18,x,method=_RETURNVERBOSE)
Output:
-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
Time = 0.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx=\frac {{\left (16 \, b^{8} x^{16} - 8 \, a b^{7} x^{14} + 6 \, a^{2} b^{6} x^{12} - 5 \, a^{3} b^{5} x^{10} - 1515 \, a^{4} b^{4} x^{8} - 4714 \, a^{5} b^{3} x^{6} - 5808 \, a^{6} b^{2} x^{4} - 3289 \, a^{7} b x^{2} - 715 \, a^{8}\right )} \sqrt {b x^{2} + a}}{12155 \, a^{4} x^{17}} \] Input:
integrate((b*x^2+a)^(9/2)/x^18,x, algorithm="fricas")
Output:
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 - 15 15*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 - 71 5*a^8)*sqrt(b*x^2 + a)/(a^4*x^17)
Leaf count of result is larger than twice the leaf count of optimal. 867 vs. \(2 (85) = 170\).
Time = 2.51 (sec) , antiderivative size = 867, normalized size of antiderivative = 9.42 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx =\text {Too large to display} \] Input:
integrate((b*x**2+a)**(9/2)/x**18,x)
Output:
-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) - 32720 *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 + 36465*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**16 + 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**22) + 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...
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx=\frac {16 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{12155 \, a^{4} x^{11}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{1105 \, a^{3} x^{13}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{85 \, a^{2} x^{15}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{17 \, a x^{17}} \] Input:
integrate((b*x^2+a)^(9/2)/x^18,x, algorithm="maxima")
Output:
16/12155*(b*x^2 + a)^(11/2)*b^3/(a^4*x^11) - 8/1105*(b*x^2 + a)^(11/2)*b^2 /(a^3*x^13) + 2/85*(b*x^2 + a)^(11/2)*b/(a^2*x^15) - 1/17*(b*x^2 + a)^(11/ 2)/(a*x^17)
Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (76) = 152\).
Time = 0.14 (sec) , antiderivative size = 382, normalized size of antiderivative = 4.15 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx=\frac {32 \, {\left (12155 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{26} b^{\frac {17}{2}} + 65637 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{24} a b^{\frac {17}{2}} + 233376 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{22} a^{2} b^{\frac {17}{2}} + 466752 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{20} a^{3} b^{\frac {17}{2}} + 692835 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{18} a^{4} b^{\frac {17}{2}} + 668525 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a^{5} b^{\frac {17}{2}} + 486200 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{6} b^{\frac {17}{2}} + 221000 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{7} b^{\frac {17}{2}} + 71825 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{8} b^{\frac {17}{2}} + 9775 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{9} b^{\frac {17}{2}} + 680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{10} b^{\frac {17}{2}} - 136 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{11} b^{\frac {17}{2}} + 17 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{12} b^{\frac {17}{2}} - a^{13} b^{\frac {17}{2}}\right )}}{12155 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{17}} \] Input:
integrate((b*x^2+a)^(9/2)/x^18,x, algorithm="giac")
Output:
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))^1 4*a^6*b^(17/2) + 221000*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^7*b^(17/2) + 71 825*(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) + 680*(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
Time = 4.09 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx=\frac {6\,b^6\,\sqrt {b\,x^2+a}}{12155\,a^2\,x^5}-\frac {303\,b^4\,\sqrt {b\,x^2+a}}{2431\,x^9}-\frac {4714\,a\,b^3\,\sqrt {b\,x^2+a}}{12155\,x^{11}}-\frac {23\,a^3\,b\,\sqrt {b\,x^2+a}}{85\,x^{15}}-\frac {b^5\,\sqrt {b\,x^2+a}}{2431\,a\,x^7}-\frac {a^4\,\sqrt {b\,x^2+a}}{17\,x^{17}}-\frac {8\,b^7\,\sqrt {b\,x^2+a}}{12155\,a^3\,x^3}+\frac {16\,b^8\,\sqrt {b\,x^2+a}}{12155\,a^4\,x}-\frac {528\,a^2\,b^2\,\sqrt {b\,x^2+a}}{1105\,x^{13}} \] Input:
int((a + b*x^2)^(9/2)/x^18,x)
Output:
(6*b^6*(a + b*x^2)^(1/2))/(12155*a^2*x^5) - (303*b^4*(a + b*x^2)^(1/2))/(2 431*x^9) - (4714*a*b^3*(a + b*x^2)^(1/2))/(12155*x^11) - (23*a^3*b*(a + b* x^2)^(1/2))/(85*x^15) - (b^5*(a + b*x^2)^(1/2))/(2431*a*x^7) - (a^4*(a + b *x^2)^(1/2))/(17*x^17) - (8*b^7*(a + b*x^2)^(1/2))/(12155*a^3*x^3) + (16*b ^8*(a + b*x^2)^(1/2))/(12155*a^4*x) - (528*a^2*b^2*(a + b*x^2)^(1/2))/(110 5*x^13)
Time = 0.24 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.92 \[ \int \frac {\left (a+b x^2\right )^{9/2}}{x^{18}} \, dx=\frac {-715 \sqrt {b \,x^{2}+a}\, a^{8}-3289 \sqrt {b \,x^{2}+a}\, a^{7} b \,x^{2}-5808 \sqrt {b \,x^{2}+a}\, a^{6} b^{2} x^{4}-4714 \sqrt {b \,x^{2}+a}\, a^{5} b^{3} x^{6}-1515 \sqrt {b \,x^{2}+a}\, a^{4} b^{4} x^{8}-5 \sqrt {b \,x^{2}+a}\, a^{3} b^{5} x^{10}+6 \sqrt {b \,x^{2}+a}\, a^{2} b^{6} x^{12}-8 \sqrt {b \,x^{2}+a}\, a \,b^{7} x^{14}+16 \sqrt {b \,x^{2}+a}\, b^{8} x^{16}-16 \sqrt {b}\, b^{8} x^{17}}{12155 a^{4} x^{17}} \] Input:
int((b*x^2+a)^(9/2)/x^18,x)
Output:
( - 715*sqrt(a + b*x**2)*a**8 - 3289*sqrt(a + b*x**2)*a**7*b*x**2 - 5808*s qrt(a + b*x**2)*a**6*b**2*x**4 - 4714*sqrt(a + b*x**2)*a**5*b**3*x**6 - 15 15*sqrt(a + b*x**2)*a**4*b**4*x**8 - 5*sqrt(a + b*x**2)*a**3*b**5*x**10 + 6*sqrt(a + b*x**2)*a**2*b**6*x**12 - 8*sqrt(a + b*x**2)*a*b**7*x**14 + 16* sqrt(a + b*x**2)*b**8*x**16 - 16*sqrt(b)*b**8*x**17)/(12155*a**4*x**17)