\(\int x^9 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx\) [1292]

Optimal result

Integrand size = 27, antiderivative size = 138 \[ \int x^9 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=-\frac {a^4 x^2 \sqrt {a-b x^2} \sqrt {a+b x^2}}{32 b^4}-\frac {a^2 x^6 \sqrt {a-b x^2} \sqrt {a+b x^2}}{48 b^2}+\frac {1}{12} x^{10} \sqrt {a-b x^2} \sqrt {a+b x^2}+\frac {a^6 \arctan \left (\frac {\sqrt {a+b x^2}}{\sqrt {a-b x^2}}\right )}{16 b^5} \] Output:

-1/32*a^4*x^2*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/b^4-1/48*a^2*x^6*(-b*x^2+a) 
^(1/2)*(b*x^2+a)^(1/2)/b^2+1/12*x^10*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)+1/16 
*a^6*arctan((b*x^2+a)^(1/2)/(-b*x^2+a)^(1/2))/b^5
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.66 \[ \int x^9 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\frac {b x^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \left (-3 a^4-2 a^2 b^2 x^4+8 b^4 x^8\right )-6 a^6 \arctan \left (\frac {\sqrt {a-b x^2}}{\sqrt {a+b x^2}}\right )}{96 b^5} \] Input:

Integrate[x^9*Sqrt[a - b*x^2]*Sqrt[a + b*x^2],x]
 

Output:

(b*x^2*Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*(-3*a^4 - 2*a^2*b^2*x^4 + 8*b^4*x^8 
) - 6*a^6*ArcTan[Sqrt[a - b*x^2]/Sqrt[a + b*x^2]])/(96*b^5)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {338, 111, 27, 101, 25, 27, 40, 45, 218}

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.

\(\displaystyle \int x^9 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx\)

\(\Big \downarrow \) 338

\(\displaystyle \frac {1}{2} \int x^8 \sqrt {a-b x^2} \sqrt {b x^2+a}dx^2\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {1}{2} \left (-\frac {\int -3 a^2 x^4 \sqrt {a-b x^2} \sqrt {b x^2+a}dx^2}{6 b^2}-\frac {x^6 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{6 b^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {a^2 \int x^4 \sqrt {a-b x^2} \sqrt {b x^2+a}dx^2}{2 b^2}-\frac {x^6 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{6 b^2}\right )\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {1}{2} \left (\frac {a^2 \left (-\frac {\int -a^2 \sqrt {a-b x^2} \sqrt {b x^2+a}dx^2}{4 b^2}-\frac {x^2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{4 b^2}\right )}{2 b^2}-\frac {x^6 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{6 b^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (\frac {a^2 \left (\frac {\int a^2 \sqrt {a-b x^2} \sqrt {b x^2+a}dx^2}{4 b^2}-\frac {x^2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{4 b^2}\right )}{2 b^2}-\frac {x^6 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{6 b^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {a^2 \left (\frac {a^2 \int \sqrt {a-b x^2} \sqrt {b x^2+a}dx^2}{4 b^2}-\frac {x^2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{4 b^2}\right )}{2 b^2}-\frac {x^6 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{6 b^2}\right )\)

\(\Big \downarrow \) 40

\(\displaystyle \frac {1}{2} \left (\frac {a^2 \left (\frac {a^2 \left (\frac {1}{2} a^2 \int \frac {1}{\sqrt {a-b x^2} \sqrt {b x^2+a}}dx^2+\frac {1}{2} x^2 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )}{4 b^2}-\frac {x^2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{4 b^2}\right )}{2 b^2}-\frac {x^6 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{6 b^2}\right )\)

\(\Big \downarrow \) 45

\(\displaystyle \frac {1}{2} \left (\frac {a^2 \left (\frac {a^2 \left (a^2 \int \frac {1}{-b x^4-b}d\frac {\sqrt {a-b x^2}}{\sqrt {b x^2+a}}+\frac {1}{2} x^2 \sqrt {a-b x^2} \sqrt {a+b x^2}\right )}{4 b^2}-\frac {x^2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{4 b^2}\right )}{2 b^2}-\frac {x^6 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{6 b^2}\right )\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (\frac {a^2 \left (\frac {a^2 \left (\frac {1}{2} x^2 \sqrt {a-b x^2} \sqrt {a+b x^2}-\frac {a^2 \arctan \left (\frac {\sqrt {a-b x^2}}{\sqrt {a+b x^2}}\right )}{b}\right )}{4 b^2}-\frac {x^2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{4 b^2}\right )}{2 b^2}-\frac {x^6 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{6 b^2}\right )\)

Input:

Int[x^9*Sqrt[a - b*x^2]*Sqrt[a + b*x^2],x]
 

Output:

(-1/6*(x^6*(a - b*x^2)^(3/2)*(a + b*x^2)^(3/2))/b^2 + (a^2*(-1/4*(x^2*(a - 
 b*x^2)^(3/2)*(a + b*x^2)^(3/2))/b^2 + (a^2*((x^2*Sqrt[a - b*x^2]*Sqrt[a + 
 b*x^2])/2 - (a^2*ArcTan[Sqrt[a - b*x^2]/Sqrt[a + b*x^2]])/b))/(4*b^2)))/( 
2*b^2))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x* 
(a + b*x)^m*((c + d*x)^m/(2*m + 1)), x] + Simp[2*a*c*(m/(2*m + 1))   Int[(a 
 + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ 
b*c + a*d, 0] && IGtQ[m + 1/2, 0]
 

rule 45

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] &&  !GtQ[c, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 111

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^p, x], x, 
x^2], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c + a*d, 0] && IntegerQ[(m - 
 1)/2]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93

method	result	size
risch	\(-\frac {x^{2} \left (-8 b^{4} x^{8}+2 a^{2} b^{2} x^{4}+3 a^{4}\right ) \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}{96 b^{4}}+\frac {a^{6} \arctan \left (\frac {\sqrt {b^{2}}\, x^{2}}{\sqrt {-b^{2} x^{4}+a^{2}}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (b \,x^{2}+a \right )}}{32 b^{4} \sqrt {b^{2}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(128\)
elliptic	\(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (\frac {x^{10} \sqrt {-b^{2} x^{4}+a^{2}}}{12}-\frac {a^{2} x^{6} \sqrt {-b^{2} x^{4}+a^{2}}}{48 b^{2}}-\frac {a^{4} x^{2} \sqrt {-b^{2} x^{4}+a^{2}}}{32 b^{4}}+\frac {a^{6} \arctan \left (\frac {\sqrt {b^{2}}\, x^{2}}{\sqrt {-b^{2} x^{4}+a^{2}}}\right )}{32 b^{4} \sqrt {b^{2}}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(142\)
default	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (-8 x^{10} \sqrt {-b^{2} x^{4}+a^{2}}\, b^{5} \operatorname {csgn}\left (b \right )+2 a^{2} x^{6} \sqrt {-b^{2} x^{4}+a^{2}}\, b^{3} \operatorname {csgn}\left (b \right )+3 a^{4} x^{2} \sqrt {-b^{2} x^{4}+a^{2}}\, \operatorname {csgn}\left (b \right ) b -3 a^{6} \arctan \left (\frac {\operatorname {csgn}\left (b \right ) b \,x^{2}}{\sqrt {-b^{2} x^{4}+a^{2}}}\right )\right ) \operatorname {csgn}\left (b \right )}{96 \sqrt {-b^{2} x^{4}+a^{2}}\, b^{5}}\)	\(145\)

Input:

int(x^9*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/96*x^2*(-8*b^4*x^8+2*a^2*b^2*x^4+3*a^4)/b^4*(-b*x^2+a)^(1/2)*(b*x^2+a)^ 
(1/2)+1/32/b^4*a^6/(b^2)^(1/2)*arctan((b^2)^(1/2)*x^2/(-b^2*x^4+a^2)^(1/2) 
)*((-b*x^2+a)*(b*x^2+a))^(1/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.67 \[ \int x^9 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=-\frac {6 \, a^{6} \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} - a}{b x^{2}}\right ) - {\left (8 \, b^{5} x^{10} - 2 \, a^{2} b^{3} x^{6} - 3 \, a^{4} b x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{96 \, b^{5}} \] Input:

integrate(x^9*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

-1/96*(6*a^6*arctan((sqrt(b*x^2 + a)*sqrt(-b*x^2 + a) - a)/(b*x^2)) - (8*b 
^5*x^10 - 2*a^2*b^3*x^6 - 3*a^4*b*x^2)*sqrt(b*x^2 + a)*sqrt(-b*x^2 + a))/b 
^5
 

Sympy [F]

\[ \int x^9 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\int x^{9} \sqrt {a - b x^{2}} \sqrt {a + b x^{2}}\, dx \] Input:

integrate(x**9*(-b*x**2+a)**(1/2)*(b*x**2+a)**(1/2),x)
 

Output:

Integral(x**9*sqrt(a - b*x**2)*sqrt(a + b*x**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.65 \[ \int x^9 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=-\frac {{\left (-b^{2} x^{4} + a^{2}\right )}^{\frac {3}{2}} x^{6}}{12 \, b^{2}} + \frac {\sqrt {-b^{2} x^{4} + a^{2}} a^{4} x^{2}}{32 \, b^{4}} + \frac {a^{6} \arcsin \left (\frac {b x^{2}}{a}\right )}{32 \, b^{5}} - \frac {{\left (-b^{2} x^{4} + a^{2}\right )}^{\frac {3}{2}} a^{2} x^{2}}{16 \, b^{4}} \] Input:

integrate(x^9*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

-1/12*(-b^2*x^4 + a^2)^(3/2)*x^6/b^2 + 1/32*sqrt(-b^2*x^4 + a^2)*a^4*x^2/b 
^4 + 1/32*a^6*arcsin(b*x^2/a)/b^5 - 1/16*(-b^2*x^4 + a^2)^(3/2)*a^2*x^2/b^ 
4
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (114) = 228\).

Time = 0.21 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.80 \[ \int x^9 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\frac {{\left (\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a} {\left ({\left (b x^{2} + a\right )} {\left (2 \, {\left (b x^{2} + a\right )} {\left ({\left (b x^{2} + a\right )} {\left (4 \, {\left (b x^{2} + a\right )} {\left (\frac {5 \, {\left (b x^{2} + a\right )}}{b^{5}} - \frac {31 \, a}{b^{5}}\right )} + \frac {321 \, a^{2}}{b^{5}}\right )} - \frac {451 \, a^{3}}{b^{5}}\right )} + \frac {745 \, a^{4}}{b^{5}}\right )} - \frac {405 \, a^{5}}{b^{5}}\right )} - \frac {150 \, a^{6} \arcsin \left (\frac {\sqrt {2} \sqrt {b x^{2} + a}}{2 \, \sqrt {a}}\right )}{b^{5}}\right )} b + \frac {2 \, {\left (90 \, a^{5} \arcsin \left (\frac {\sqrt {2} \sqrt {b x^{2} + a}}{2 \, \sqrt {a}}\right ) + {\left (195 \, a^{4} - {\left (295 \, a^{3} - 2 \, {\left (b x^{2} + a\right )} {\left (3 \, {\left (4 \, b x^{2} - 17 \, a\right )} {\left (b x^{2} + a\right )} + 133 \, a^{2}\right )}\right )} {\left (b x^{2} + a\right )}\right )} \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}\right )} a}{b^{4}}}{480 \, b} \] Input:

integrate(x^9*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

1/480*((sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)*((b*x^2 + a)*(2*(b*x^2 + a)*((b*x 
^2 + a)*(4*(b*x^2 + a)*(5*(b*x^2 + a)/b^5 - 31*a/b^5) + 321*a^2/b^5) - 451 
*a^3/b^5) + 745*a^4/b^5) - 405*a^5/b^5) - 150*a^6*arcsin(1/2*sqrt(2)*sqrt( 
b*x^2 + a)/sqrt(a))/b^5)*b + 2*(90*a^5*arcsin(1/2*sqrt(2)*sqrt(b*x^2 + a)/ 
sqrt(a)) + (195*a^4 - (295*a^3 - 2*(b*x^2 + a)*(3*(4*b*x^2 - 17*a)*(b*x^2 
+ a) + 133*a^2))*(b*x^2 + a))*sqrt(b*x^2 + a)*sqrt(-b*x^2 + a))*a/b^4)/b
 

Mupad [B] (verification not implemented)

Time = 11.87 (sec) , antiderivative size = 569, normalized size of antiderivative = 4.12 \[ \int x^9 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\frac {a^6\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}-\sqrt {a}}{\sqrt {a-b\,x^2}-\sqrt {a}}\right )}{8\,b^5}-\frac {\frac {35\,a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^3}{24\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^3}-\frac {757\,a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^5}{8\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^5}+\frac {7339\,a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^7}{8\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^7}-\frac {41929\,a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^9}{12\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^9}+\frac {25661\,a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{11}}{4\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^{11}}-\frac {25661\,a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{13}}{4\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^{13}}+\frac {41929\,a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{15}}{12\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^{15}}-\frac {7339\,a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{17}}{8\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^{17}}+\frac {757\,a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{19}}{8\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^{19}}-\frac {35\,a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{21}}{24\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^{21}}-\frac {a^6\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{23}}{8\,{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^{23}}+\frac {a^6\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{8\,\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}}{b^5\,{\left (\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{{\left (\sqrt {a-b\,x^2}-\sqrt {a}\right )}^2}+1\right )}^{12}} \] Input:

int(x^9*(a + b*x^2)^(1/2)*(a - b*x^2)^(1/2),x)
 

Output:

(a^6*atan(((a + b*x^2)^(1/2) - a^(1/2))/((a - b*x^2)^(1/2) - a^(1/2))))/(8 
*b^5) - ((35*a^6*((a + b*x^2)^(1/2) - a^(1/2))^3)/(24*((a - b*x^2)^(1/2) - 
 a^(1/2))^3) - (757*a^6*((a + b*x^2)^(1/2) - a^(1/2))^5)/(8*((a - b*x^2)^( 
1/2) - a^(1/2))^5) + (7339*a^6*((a + b*x^2)^(1/2) - a^(1/2))^7)/(8*((a - b 
*x^2)^(1/2) - a^(1/2))^7) - (41929*a^6*((a + b*x^2)^(1/2) - a^(1/2))^9)/(1 
2*((a - b*x^2)^(1/2) - a^(1/2))^9) + (25661*a^6*((a + b*x^2)^(1/2) - a^(1/ 
2))^11)/(4*((a - b*x^2)^(1/2) - a^(1/2))^11) - (25661*a^6*((a + b*x^2)^(1/ 
2) - a^(1/2))^13)/(4*((a - b*x^2)^(1/2) - a^(1/2))^13) + (41929*a^6*((a + 
b*x^2)^(1/2) - a^(1/2))^15)/(12*((a - b*x^2)^(1/2) - a^(1/2))^15) - (7339* 
a^6*((a + b*x^2)^(1/2) - a^(1/2))^17)/(8*((a - b*x^2)^(1/2) - a^(1/2))^17) 
 + (757*a^6*((a + b*x^2)^(1/2) - a^(1/2))^19)/(8*((a - b*x^2)^(1/2) - a^(1 
/2))^19) - (35*a^6*((a + b*x^2)^(1/2) - a^(1/2))^21)/(24*((a - b*x^2)^(1/2 
) - a^(1/2))^21) - (a^6*((a + b*x^2)^(1/2) - a^(1/2))^23)/(8*((a - b*x^2)^ 
(1/2) - a^(1/2))^23) + (a^6*((a + b*x^2)^(1/2) - a^(1/2)))/(8*((a - b*x^2) 
^(1/2) - a^(1/2))))/(b^5*(((a + b*x^2)^(1/2) - a^(1/2))^2/((a - b*x^2)^(1/ 
2) - a^(1/2))^2 + 1)^12)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.86 \[ \int x^9 \sqrt {a-b x^2} \sqrt {a+b x^2} \, dx=\frac {6 \mathit {atan} \left (\frac {\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{2}+a}\right ) a^{6}-3 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a^{4} b \,x^{2}-2 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a^{2} b^{3} x^{6}+8 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, b^{5} x^{10}}{96 b^{5}} \] Input:

int(x^9*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2),x)
 

Output:

(6*atan((sqrt(a + b*x**2)*sqrt(a - b*x**2))/(a - b*x**2))*a**6 - 3*sqrt(a 
+ b*x**2)*sqrt(a - b*x**2)*a**4*b*x**2 - 2*sqrt(a + b*x**2)*sqrt(a - b*x** 
2)*a**2*b**3*x**6 + 8*sqrt(a + b*x**2)*sqrt(a - b*x**2)*b**5*x**10)/(96*b* 
*5)