Integrand size = 15, antiderivative size = 154 \[ \int x^2 \left (a+b x^2\right )^{9/2} \, dx=\frac {21 a^5 x \sqrt {a+b x^2}}{1024 b}+\frac {21}{512} a^4 x^3 \sqrt {a+b x^2}+\frac {7}{128} a^3 x^3 \left (a+b x^2\right )^{3/2}+\frac {21}{320} a^2 x^3 \left (a+b x^2\right )^{5/2}+\frac {3}{40} a x^3 \left (a+b x^2\right )^{7/2}+\frac {1}{12} x^3 \left (a+b x^2\right )^{9/2}-\frac {21 a^6 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{3/2}} \] Output:
21/1024*a^5*x*(b*x^2+a)^(1/2)/b+21/512*a^4*x^3*(b*x^2+a)^(1/2)+7/128*a^3*x ^3*(b*x^2+a)^(3/2)+21/320*a^2*x^3*(b*x^2+a)^(5/2)+3/40*a*x^3*(b*x^2+a)^(7/ 2)+1/12*x^3*(b*x^2+a)^(9/2)-21/1024*a^6*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2)) /b^(3/2)
Time = 0.49 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.75 \[ \int x^2 \left (a+b x^2\right )^{9/2} \, dx=\frac {\sqrt {a+b x^2} \left (315 a^5 x+4910 a^4 b x^3+11432 a^3 b^2 x^5+12144 a^2 b^3 x^7+6272 a b^4 x^9+1280 b^5 x^{11}\right )}{15360 b}-\frac {21 a^6 \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{512 b^{3/2}} \] Input:
Integrate[x^2*(a + b*x^2)^(9/2),x]
Output:
(Sqrt[a + b*x^2]*(315*a^5*x + 4910*a^4*b*x^3 + 11432*a^3*b^2*x^5 + 12144*a ^2*b^3*x^7 + 6272*a*b^4*x^9 + 1280*b^5*x^11))/(15360*b) - (21*a^6*ArcTanh[ (Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/(512*b^(3/2))
Time = 0.23 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {248, 248, 248, 248, 248, 262, 224, 219}
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.
| \(\displaystyle \int x^2 \left (a+b x^2\right )^{9/2} \, dx\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {3}{4} a \int x^2 \left (b x^2+a\right )^{7/2}dx+\frac {1}{12} x^3 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {3}{4} a \left (\frac {7}{10} a \int x^2 \left (b x^2+a\right )^{5/2}dx+\frac {1}{10} x^3 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{12} x^3 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {3}{4} a \left (\frac {7}{10} a \left (\frac {5}{8} a \int x^2 \left (b x^2+a\right )^{3/2}dx+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{10} x^3 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{12} x^3 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {3}{4} a \left (\frac {7}{10} a \left (\frac {5}{8} a \left (\frac {1}{2} a \int x^2 \sqrt {b x^2+a}dx+\frac {1}{6} x^3 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{10} x^3 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{12} x^3 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {3}{4} a \left (\frac {7}{10} a \left (\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \int \frac {x^2}{\sqrt {b x^2+a}}dx+\frac {1}{4} x^3 \sqrt {a+b x^2}\right )+\frac {1}{6} x^3 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{10} x^3 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{12} x^3 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {3}{4} a \left (\frac {7}{10} a \left (\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{\sqrt {b x^2+a}}dx}{2 b}\right )+\frac {1}{4} x^3 \sqrt {a+b x^2}\right )+\frac {1}{6} x^3 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{10} x^3 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{12} x^3 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {3}{4} a \left (\frac {7}{10} a \left (\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{2 b}\right )+\frac {1}{4} x^3 \sqrt {a+b x^2}\right )+\frac {1}{6} x^3 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{10} x^3 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{12} x^3 \left (a+b x^2\right )^{9/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{4} a \left (\frac {7}{10} a \left (\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {x \sqrt {a+b x^2}}{2 b}-\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\right )+\frac {1}{4} x^3 \sqrt {a+b x^2}\right )+\frac {1}{6} x^3 \left (a+b x^2\right )^{3/2}\right )+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}\right )+\frac {1}{10} x^3 \left (a+b x^2\right )^{7/2}\right )+\frac {1}{12} x^3 \left (a+b x^2\right )^{9/2}\) |
Input:
Int[x^2*(a + b*x^2)^(9/2),x]
Output:
(x^3*(a + b*x^2)^(9/2))/12 + (3*a*((x^3*(a + b*x^2)^(7/2))/10 + (7*a*((x^3 *(a + b*x^2)^(5/2))/8 + (5*a*((x^3*(a + b*x^2)^(3/2))/6 + (a*((x^3*Sqrt[a + b*x^2])/4 + (a*((x*Sqrt[a + b*x^2])/(2*b) - (a*ArcTanh[(Sqrt[b]*x)/Sqrt[ a + b*x^2]])/(2*b^(3/2))))/4))/2))/8))/10))/4
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Time = 0.41 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.62
| method | result | size |
| risch | \(\frac {x \left (1280 b^{5} x^{10}+6272 a \,b^{4} x^{8}+12144 a^{2} b^{3} x^{6}+11432 a^{3} b^{2} x^{4}+4910 a^{4} b \,x^{2}+315 a^{5}\right ) \sqrt {b \,x^{2}+a}}{15360 b}-\frac {21 a^{6} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{1024 b^{\frac {3}{2}}}\) | \(95\) |
| default | \(\frac {x \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{12 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {9}{2}}}{10}+\frac {9 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8}+\frac {7 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8}\right )}{10}\right )}{12 b}\) | \(122\) |
| pseudoelliptic | \(\frac {1280 \sqrt {b \,x^{2}+a}\, b^{\frac {11}{2}} x^{11}+6272 a \,b^{\frac {9}{2}} x^{9} \sqrt {b \,x^{2}+a}+12144 a^{2} b^{\frac {7}{2}} x^{7} \sqrt {b \,x^{2}+a}+11432 a^{3} b^{\frac {5}{2}} x^{5} \sqrt {b \,x^{2}+a}+4910 a^{4} b^{\frac {3}{2}} x^{3} \sqrt {b \,x^{2}+a}+315 a^{5} x \sqrt {b \,x^{2}+a}\, \sqrt {b}-315 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) a^{6}}{15360 b^{\frac {3}{2}}}\) | \(142\) |
Input:
int(x^2*(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/15360*x*(1280*b^5*x^10+6272*a*b^4*x^8+12144*a^2*b^3*x^6+11432*a^3*b^2*x^ 4+4910*a^4*b*x^2+315*a^5)*(b*x^2+a)^(1/2)/b-21/1024/b^(3/2)*a^6*ln(b^(1/2) *x+(b*x^2+a)^(1/2))
Time = 0.11 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.37 \[ \int x^2 \left (a+b x^2\right )^{9/2} \, dx=\left [\frac {315 \, a^{6} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (1280 \, b^{6} x^{11} + 6272 \, a b^{5} x^{9} + 12144 \, a^{2} b^{4} x^{7} + 11432 \, a^{3} b^{3} x^{5} + 4910 \, a^{4} b^{2} x^{3} + 315 \, a^{5} b x\right )} \sqrt {b x^{2} + a}}{30720 \, b^{2}}, \frac {315 \, a^{6} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (1280 \, b^{6} x^{11} + 6272 \, a b^{5} x^{9} + 12144 \, a^{2} b^{4} x^{7} + 11432 \, a^{3} b^{3} x^{5} + 4910 \, a^{4} b^{2} x^{3} + 315 \, a^{5} b x\right )} \sqrt {b x^{2} + a}}{15360 \, b^{2}}\right ] \] Input:
integrate(x^2*(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
[1/30720*(315*a^6*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(1280*b^6*x^11 + 6272*a*b^5*x^9 + 12144*a^2*b^4*x^7 + 11432*a^3*b^3*x^ 5 + 4910*a^4*b^2*x^3 + 315*a^5*b*x)*sqrt(b*x^2 + a))/b^2, 1/15360*(315*a^6 *sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (1280*b^6*x^11 + 6272*a*b^5 *x^9 + 12144*a^2*b^4*x^7 + 11432*a^3*b^3*x^5 + 4910*a^4*b^2*x^3 + 315*a^5* b*x)*sqrt(b*x^2 + a))/b^2]
Time = 48.29 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.32 \[ \int x^2 \left (a+b x^2\right )^{9/2} \, dx=\frac {21 a^{\frac {11}{2}} x}{1024 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {1045 a^{\frac {9}{2}} x^{3}}{3072 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {8171 a^{\frac {7}{2}} b x^{5}}{7680 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2947 a^{\frac {5}{2}} b^{2} x^{7}}{1920 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {1151 a^{\frac {3}{2}} b^{3} x^{9}}{960 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {59 \sqrt {a} b^{4} x^{11}}{120 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {21 a^{6} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{1024 b^{\frac {3}{2}}} + \frac {b^{5} x^{13}}{12 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \] Input:
integrate(x**2*(b*x**2+a)**(9/2),x)
Output:
21*a**(11/2)*x/(1024*b*sqrt(1 + b*x**2/a)) + 1045*a**(9/2)*x**3/(3072*sqrt (1 + b*x**2/a)) + 8171*a**(7/2)*b*x**5/(7680*sqrt(1 + b*x**2/a)) + 2947*a* *(5/2)*b**2*x**7/(1920*sqrt(1 + b*x**2/a)) + 1151*a**(3/2)*b**3*x**9/(960* sqrt(1 + b*x**2/a)) + 59*sqrt(a)*b**4*x**11/(120*sqrt(1 + b*x**2/a)) - 21* a**6*asinh(sqrt(b)*x/sqrt(a))/(1024*b**(3/2)) + b**5*x**13/(12*sqrt(a)*sqr t(1 + b*x**2/a))
Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.79 \[ \int x^2 \left (a+b x^2\right )^{9/2} \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} x}{12 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {9}{2}} a x}{120 \, b} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x}{320 \, b} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3} x}{640 \, b} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4} x}{512 \, b} - \frac {21 \, \sqrt {b x^{2} + a} a^{5} x}{1024 \, b} - \frac {21 \, a^{6} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{1024 \, b^{\frac {3}{2}}} \] Input:
integrate(x^2*(b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
1/12*(b*x^2 + a)^(11/2)*x/b - 1/120*(b*x^2 + a)^(9/2)*a*x/b - 3/320*(b*x^2 + a)^(7/2)*a^2*x/b - 7/640*(b*x^2 + a)^(5/2)*a^3*x/b - 7/512*(b*x^2 + a)^ (3/2)*a^4*x/b - 21/1024*sqrt(b*x^2 + a)*a^5*x/b - 21/1024*a^6*arcsinh(b*x/ sqrt(a*b))/b^(3/2)
Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.68 \[ \int x^2 \left (a+b x^2\right )^{9/2} \, dx=\frac {21 \, a^{6} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{1024 \, b^{\frac {3}{2}}} + \frac {1}{15360} \, {\left (\frac {315 \, a^{5}}{b} + 2 \, {\left (2455 \, a^{4} + 4 \, {\left (1429 \, a^{3} b + 2 \, {\left (759 \, a^{2} b^{2} + 8 \, {\left (10 \, b^{4} x^{2} + 49 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} x \] Input:
integrate(x^2*(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
21/1024*a^6*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2) + 1/15360*(315* a^5/b + 2*(2455*a^4 + 4*(1429*a^3*b + 2*(759*a^2*b^2 + 8*(10*b^4*x^2 + 49* a*b^3)*x^2)*x^2)*x^2)*x^2)*sqrt(b*x^2 + a)*x
Timed out. \[ \int x^2 \left (a+b x^2\right )^{9/2} \, dx=\int x^2\,{\left (b\,x^2+a\right )}^{9/2} \,d x \] Input:
int(x^2*(a + b*x^2)^(9/2),x)
Output:
int(x^2*(a + b*x^2)^(9/2), x)
Time = 0.22 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.89 \[ \int x^2 \left (a+b x^2\right )^{9/2} \, dx=\frac {315 \sqrt {b \,x^{2}+a}\, a^{5} b x +4910 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} x^{3}+11432 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} x^{5}+12144 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} x^{7}+6272 \sqrt {b \,x^{2}+a}\, a \,b^{5} x^{9}+1280 \sqrt {b \,x^{2}+a}\, b^{6} x^{11}-315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{6}}{15360 b^{2}} \] Input:
int(x^2*(b*x^2+a)^(9/2),x)
Output:
(315*sqrt(a + b*x**2)*a**5*b*x + 4910*sqrt(a + b*x**2)*a**4*b**2*x**3 + 11 432*sqrt(a + b*x**2)*a**3*b**3*x**5 + 12144*sqrt(a + b*x**2)*a**2*b**4*x** 7 + 6272*sqrt(a + b*x**2)*a*b**5*x**9 + 1280*sqrt(a + b*x**2)*b**6*x**11 - 315*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**6)/(15360*b**2 )