Integrand size = 15, antiderivative size = 161 \[ \int x^{15} \left (a+b x^2\right )^{9/2} \, dx=-\frac {a^7 \left (a+b x^2\right )^{11/2}}{11 b^8}+\frac {7 a^6 \left (a+b x^2\right )^{13/2}}{13 b^8}-\frac {7 a^5 \left (a+b x^2\right )^{15/2}}{5 b^8}+\frac {35 a^4 \left (a+b x^2\right )^{17/2}}{17 b^8}-\frac {35 a^3 \left (a+b x^2\right )^{19/2}}{19 b^8}+\frac {a^2 \left (a+b x^2\right )^{21/2}}{b^8}-\frac {7 a \left (a+b x^2\right )^{23/2}}{23 b^8}+\frac {\left (a+b x^2\right )^{25/2}}{25 b^8} \]
-1/11*a^7*(b*x^2+a)^(11/2)/b^8+7/13*a^6*(b*x^2+a)^(13/2)/b^8-7/5*a^5*(b*x^ 2+a)^(15/2)/b^8+35/17*a^4*(b*x^2+a)^(17/2)/b^8-35/19*a^3*(b*x^2+a)^(19/2)/ b^8+a^2*(b*x^2+a)^(21/2)/b^8-7/23*a*(b*x^2+a)^(23/2)/b^8+1/25*(b*x^2+a)^(2 5/2)/b^8
Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.58 \[ \int x^{15} \left (a+b x^2\right )^{9/2} \, dx=\frac {\left (a+b x^2\right )^{11/2} \left (-2048 a^7+11264 a^6 b x^2-36608 a^5 b^2 x^4+91520 a^4 b^3 x^6-194480 a^3 b^4 x^8+369512 a^2 b^5 x^{10}-646646 a b^6 x^{12}+1062347 b^7 x^{14}\right )}{26558675 b^8} \]
((a + b*x^2)^(11/2)*(-2048*a^7 + 11264*a^6*b*x^2 - 36608*a^5*b^2*x^4 + 915 20*a^4*b^3*x^6 - 194480*a^3*b^4*x^8 + 369512*a^2*b^5*x^10 - 646646*a*b^6*x ^12 + 1062347*b^7*x^14))/(26558675*b^8)
Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {243, 53, 2009}
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.
| \(\displaystyle \int x^{15} \left (a+b x^2\right )^{9/2} \, dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \int x^{14} \left (b x^2+a\right )^{9/2}dx^2\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\left (b x^2+a\right )^{23/2}}{b^7}-\frac {7 a \left (b x^2+a\right )^{21/2}}{b^7}+\frac {21 a^2 \left (b x^2+a\right )^{19/2}}{b^7}-\frac {35 a^3 \left (b x^2+a\right )^{17/2}}{b^7}+\frac {35 a^4 \left (b x^2+a\right )^{15/2}}{b^7}-\frac {21 a^5 \left (b x^2+a\right )^{13/2}}{b^7}+\frac {7 a^6 \left (b x^2+a\right )^{11/2}}{b^7}-\frac {a^7 \left (b x^2+a\right )^{9/2}}{b^7}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 a^7 \left (a+b x^2\right )^{11/2}}{11 b^8}+\frac {14 a^6 \left (a+b x^2\right )^{13/2}}{13 b^8}-\frac {14 a^5 \left (a+b x^2\right )^{15/2}}{5 b^8}+\frac {70 a^4 \left (a+b x^2\right )^{17/2}}{17 b^8}-\frac {70 a^3 \left (a+b x^2\right )^{19/2}}{19 b^8}+\frac {2 a^2 \left (a+b x^2\right )^{21/2}}{b^8}+\frac {2 \left (a+b x^2\right )^{25/2}}{25 b^8}-\frac {14 a \left (a+b x^2\right )^{23/2}}{23 b^8}\right )\) |
((-2*a^7*(a + b*x^2)^(11/2))/(11*b^8) + (14*a^6*(a + b*x^2)^(13/2))/(13*b^ 8) - (14*a^5*(a + b*x^2)^(15/2))/(5*b^8) + (70*a^4*(a + b*x^2)^(17/2))/(17 *b^8) - (70*a^3*(a + b*x^2)^(19/2))/(19*b^8) + (2*a^2*(a + b*x^2)^(21/2))/ b^8 - (14*a*(a + b*x^2)^(23/2))/(23*b^8) + (2*(a + b*x^2)^(25/2))/(25*b^8) )/2
3.5.9.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 2.13 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.57
| method | result | size |
| gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-1062347 b^{7} x^{14}+646646 a \,b^{6} x^{12}-369512 a^{2} b^{5} x^{10}+194480 a^{3} b^{4} x^{8}-91520 a^{4} b^{3} x^{6}+36608 a^{5} b^{2} x^{4}-11264 a^{6} b \,x^{2}+2048 a^{7}\right )}{26558675 b^{8}}\) | \(91\) |
| pseudoelliptic | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-1062347 b^{7} x^{14}+646646 a \,b^{6} x^{12}-369512 a^{2} b^{5} x^{10}+194480 a^{3} b^{4} x^{8}-91520 a^{4} b^{3} x^{6}+36608 a^{5} b^{2} x^{4}-11264 a^{6} b \,x^{2}+2048 a^{7}\right )}{26558675 b^{8}}\) | \(91\) |
| trager | \(-\frac {\left (-1062347 b^{12} x^{24}-4665089 a \,b^{11} x^{22}-7759752 a^{2} b^{10} x^{20}-5810090 a^{3} b^{9} x^{18}-1659515 a^{4} b^{8} x^{16}-429 a^{5} b^{7} x^{14}+462 a^{6} b^{6} x^{12}-504 a^{7} b^{5} x^{10}+560 a^{8} b^{4} x^{8}-640 a^{9} b^{3} x^{6}+768 a^{10} b^{2} x^{4}-1024 a^{11} b \,x^{2}+2048 a^{12}\right ) \sqrt {b \,x^{2}+a}}{26558675 b^{8}}\) | \(146\) |
| risch | \(-\frac {\left (-1062347 b^{12} x^{24}-4665089 a \,b^{11} x^{22}-7759752 a^{2} b^{10} x^{20}-5810090 a^{3} b^{9} x^{18}-1659515 a^{4} b^{8} x^{16}-429 a^{5} b^{7} x^{14}+462 a^{6} b^{6} x^{12}-504 a^{7} b^{5} x^{10}+560 a^{8} b^{4} x^{8}-640 a^{9} b^{3} x^{6}+768 a^{10} b^{2} x^{4}-1024 a^{11} b \,x^{2}+2048 a^{12}\right ) \sqrt {b \,x^{2}+a}}{26558675 b^{8}}\) | \(146\) |
| default | \(\frac {x^{14} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{25 b}-\frac {14 a \left (\frac {x^{12} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{23 b}-\frac {12 a \left (\frac {x^{10} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{21 b}-\frac {10 a \left (\frac {x^{8} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{19 b}-\frac {8 a \left (\frac {x^{6} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{17 b}-\frac {6 a \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{15 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{13 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{143 b^{2}}\right )}{15 b}\right )}{17 b}\right )}{19 b}\right )}{21 b}\right )}{23 b}\right )}{25 b}\) | \(178\) |
-1/26558675*(b*x^2+a)^(11/2)*(-1062347*b^7*x^14+646646*a*b^6*x^12-369512*a ^2*b^5*x^10+194480*a^3*b^4*x^8-91520*a^4*b^3*x^6+36608*a^5*b^2*x^4-11264*a ^6*b*x^2+2048*a^7)/b^8
Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90 \[ \int x^{15} \left (a+b x^2\right )^{9/2} \, dx=\frac {{\left (1062347 \, b^{12} x^{24} + 4665089 \, a b^{11} x^{22} + 7759752 \, a^{2} b^{10} x^{20} + 5810090 \, a^{3} b^{9} x^{18} + 1659515 \, a^{4} b^{8} x^{16} + 429 \, a^{5} b^{7} x^{14} - 462 \, a^{6} b^{6} x^{12} + 504 \, a^{7} b^{5} x^{10} - 560 \, a^{8} b^{4} x^{8} + 640 \, a^{9} b^{3} x^{6} - 768 \, a^{10} b^{2} x^{4} + 1024 \, a^{11} b x^{2} - 2048 \, a^{12}\right )} \sqrt {b x^{2} + a}}{26558675 \, b^{8}} \]
1/26558675*(1062347*b^12*x^24 + 4665089*a*b^11*x^22 + 7759752*a^2*b^10*x^2 0 + 5810090*a^3*b^9*x^18 + 1659515*a^4*b^8*x^16 + 429*a^5*b^7*x^14 - 462*a ^6*b^6*x^12 + 504*a^7*b^5*x^10 - 560*a^8*b^4*x^8 + 640*a^9*b^3*x^6 - 768*a ^10*b^2*x^4 + 1024*a^11*b*x^2 - 2048*a^12)*sqrt(b*x^2 + a)/b^8
Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (150) = 300\).
Time = 2.83 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.87 \[ \int x^{15} \left (a+b x^2\right )^{9/2} \, dx=\begin {cases} - \frac {2048 a^{12} \sqrt {a + b x^{2}}}{26558675 b^{8}} + \frac {1024 a^{11} x^{2} \sqrt {a + b x^{2}}}{26558675 b^{7}} - \frac {768 a^{10} x^{4} \sqrt {a + b x^{2}}}{26558675 b^{6}} + \frac {128 a^{9} x^{6} \sqrt {a + b x^{2}}}{5311735 b^{5}} - \frac {112 a^{8} x^{8} \sqrt {a + b x^{2}}}{5311735 b^{4}} + \frac {504 a^{7} x^{10} \sqrt {a + b x^{2}}}{26558675 b^{3}} - \frac {42 a^{6} x^{12} \sqrt {a + b x^{2}}}{2414425 b^{2}} + \frac {3 a^{5} x^{14} \sqrt {a + b x^{2}}}{185725 b} + \frac {2321 a^{4} x^{16} \sqrt {a + b x^{2}}}{37145} + \frac {478 a^{3} b x^{18} \sqrt {a + b x^{2}}}{2185} + \frac {168 a^{2} b^{2} x^{20} \sqrt {a + b x^{2}}}{575} + \frac {101 a b^{3} x^{22} \sqrt {a + b x^{2}}}{575} + \frac {b^{4} x^{24} \sqrt {a + b x^{2}}}{25} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{16}}{16} & \text {otherwise} \end {cases} \]
Piecewise((-2048*a**12*sqrt(a + b*x**2)/(26558675*b**8) + 1024*a**11*x**2* sqrt(a + b*x**2)/(26558675*b**7) - 768*a**10*x**4*sqrt(a + b*x**2)/(265586 75*b**6) + 128*a**9*x**6*sqrt(a + b*x**2)/(5311735*b**5) - 112*a**8*x**8*s qrt(a + b*x**2)/(5311735*b**4) + 504*a**7*x**10*sqrt(a + b*x**2)/(26558675 *b**3) - 42*a**6*x**12*sqrt(a + b*x**2)/(2414425*b**2) + 3*a**5*x**14*sqrt (a + b*x**2)/(185725*b) + 2321*a**4*x**16*sqrt(a + b*x**2)/37145 + 478*a** 3*b*x**18*sqrt(a + b*x**2)/2185 + 168*a**2*b**2*x**20*sqrt(a + b*x**2)/575 + 101*a*b**3*x**22*sqrt(a + b*x**2)/575 + b**4*x**24*sqrt(a + b*x**2)/25, Ne(b, 0)), (a**(9/2)*x**16/16, True))
Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.95 \[ \int x^{15} \left (a+b x^2\right )^{9/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} x^{14}}{25 \, b} - \frac {14 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a x^{12}}{575 \, b^{2}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{2} x^{10}}{575 \, b^{3}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{3} x^{8}}{2185 \, b^{4}} + \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{4} x^{6}}{37145 \, b^{5}} - \frac {256 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{5} x^{4}}{185725 \, b^{6}} + \frac {1024 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{6} x^{2}}{2414425 \, b^{7}} - \frac {2048 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{7}}{26558675 \, b^{8}} \]
1/25*(b*x^2 + a)^(11/2)*x^14/b - 14/575*(b*x^2 + a)^(11/2)*a*x^12/b^2 + 8/ 575*(b*x^2 + a)^(11/2)*a^2*x^10/b^3 - 16/2185*(b*x^2 + a)^(11/2)*a^3*x^8/b ^4 + 128/37145*(b*x^2 + a)^(11/2)*a^4*x^6/b^5 - 256/185725*(b*x^2 + a)^(11 /2)*a^5*x^4/b^6 + 1024/2414425*(b*x^2 + a)^(11/2)*a^6*x^2/b^7 - 2048/26558 675*(b*x^2 + a)^(11/2)*a^7/b^8
Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.70 \[ \int x^{15} \left (a+b x^2\right )^{9/2} \, dx=\frac {1062347 \, {\left (b x^{2} + a\right )}^{\frac {25}{2}} - 8083075 \, {\left (b x^{2} + a\right )}^{\frac {23}{2}} a + 26558675 \, {\left (b x^{2} + a\right )}^{\frac {21}{2}} a^{2} - 48923875 \, {\left (b x^{2} + a\right )}^{\frac {19}{2}} a^{3} + 54679625 \, {\left (b x^{2} + a\right )}^{\frac {17}{2}} a^{4} - 37182145 \, {\left (b x^{2} + a\right )}^{\frac {15}{2}} a^{5} + 14300825 \, {\left (b x^{2} + a\right )}^{\frac {13}{2}} a^{6} - 2414425 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{7}}{26558675 \, b^{8}} \]
1/26558675*(1062347*(b*x^2 + a)^(25/2) - 8083075*(b*x^2 + a)^(23/2)*a + 26 558675*(b*x^2 + a)^(21/2)*a^2 - 48923875*(b*x^2 + a)^(19/2)*a^3 + 54679625 *(b*x^2 + a)^(17/2)*a^4 - 37182145*(b*x^2 + a)^(15/2)*a^5 + 14300825*(b*x^ 2 + a)^(13/2)*a^6 - 2414425*(b*x^2 + a)^(11/2)*a^7)/b^8
Time = 4.75 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.88 \[ \int x^{15} \left (a+b x^2\right )^{9/2} \, dx=\sqrt {b\,x^2+a}\,\left (\frac {2321\,a^4\,x^{16}}{37145}-\frac {2048\,a^{12}}{26558675\,b^8}+\frac {b^4\,x^{24}}{25}+\frac {478\,a^3\,b\,x^{18}}{2185}+\frac {101\,a\,b^3\,x^{22}}{575}+\frac {3\,a^5\,x^{14}}{185725\,b}-\frac {42\,a^6\,x^{12}}{2414425\,b^2}+\frac {504\,a^7\,x^{10}}{26558675\,b^3}-\frac {112\,a^8\,x^8}{5311735\,b^4}+\frac {128\,a^9\,x^6}{5311735\,b^5}-\frac {768\,a^{10}\,x^4}{26558675\,b^6}+\frac {1024\,a^{11}\,x^2}{26558675\,b^7}+\frac {168\,a^2\,b^2\,x^{20}}{575}\right ) \]
(a + b*x^2)^(1/2)*((2321*a^4*x^16)/37145 - (2048*a^12)/(26558675*b^8) + (b ^4*x^24)/25 + (478*a^3*b*x^18)/2185 + (101*a*b^3*x^22)/575 + (3*a^5*x^14)/ (185725*b) - (42*a^6*x^12)/(2414425*b^2) + (504*a^7*x^10)/(26558675*b^3) - (112*a^8*x^8)/(5311735*b^4) + (128*a^9*x^6)/(5311735*b^5) - (768*a^10*x^4 )/(26558675*b^6) + (1024*a^11*x^2)/(26558675*b^7) + (168*a^2*b^2*x^20)/575 )