\(\int \frac {x^{10}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx\) [1329]

Optimal result

Integrand size = 27, antiderivative size = 208 \[ \int \frac {x^{10}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=-\frac {7 a^2 x^3 \sqrt {a-b x^2} \sqrt {a+b x^2}}{45 b^4}-\frac {x^7 \sqrt {a-b x^2} \sqrt {a+b x^2}}{9 b^2}+\frac {7 a^{11/2} \sqrt {1-\frac {b^2 x^4}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |-1\right )}{15 b^{11/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}-\frac {7 a^{11/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{15 b^{11/2} \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Output:

-7/45*a^2*x^3*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/b^4-1/9*x^7*(-b*x^2+a)^(1/2 
)*(b*x^2+a)^(1/2)/b^2+7/15*a^(11/2)*(1-b^2*x^4/a^2)^(1/2)*EllipticE(b^(1/2 
)*x/a^(1/2),I)/b^(11/2)/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)-7/15*a^(11/2)*(1- 
b^2*x^4/a^2)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),I)/b^(11/2)/(-b*x^2+a)^(1/2 
)/(b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 1.78 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.33 \[ \int \frac {x^{10}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\frac {x^{11} \sqrt {1-\frac {b^2 x^4}{a^2}} \, _2F_1\left (\frac {1}{2},\frac {11}{4};\frac {15}{4};\frac {b^2 x^4}{a^2}\right )}{11 \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Input:

Integrate[x^10/(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]),x]
 

Output:

(x^11*Sqrt[1 - (b^2*x^4)/a^2]*HypergeometricPFQ[{1/2, 11/4}, {15/4}, (b^2* 
x^4)/a^2])/(11*Sqrt[a - b*x^2]*Sqrt[a + b*x^2])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {344, 843, 843, 836, 27, 765, 762, 1390, 1389, 327}

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.

\(\displaystyle \int \frac {x^{10}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx\)

\(\Big \downarrow \) 344

\(\displaystyle \frac {\sqrt {a^2-b^2 x^4} \int \frac {x^{10}}{\sqrt {a^2-b^2 x^4}}dx}{\sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {\sqrt {a^2-b^2 x^4} \left (\frac {7 a^2 \int \frac {x^6}{\sqrt {a^2-b^2 x^4}}dx}{9 b^2}-\frac {x^7 \sqrt {a^2-b^2 x^4}}{9 b^2}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 843

\(\displaystyle \frac {\sqrt {a^2-b^2 x^4} \left (\frac {7 a^2 \left (\frac {3 a^2 \int \frac {x^2}{\sqrt {a^2-b^2 x^4}}dx}{5 b^2}-\frac {x^3 \sqrt {a^2-b^2 x^4}}{5 b^2}\right )}{9 b^2}-\frac {x^7 \sqrt {a^2-b^2 x^4}}{9 b^2}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {\sqrt {a^2-b^2 x^4} \left (\frac {7 a^2 \left (\frac {3 a^2 \left (\frac {a \int \frac {b x^2+a}{a \sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{b}\right )}{5 b^2}-\frac {x^3 \sqrt {a^2-b^2 x^4}}{5 b^2}\right )}{9 b^2}-\frac {x^7 \sqrt {a^2-b^2 x^4}}{9 b^2}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2-b^2 x^4} \left (\frac {7 a^2 \left (\frac {3 a^2 \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \int \frac {1}{\sqrt {a^2-b^2 x^4}}dx}{b}\right )}{5 b^2}-\frac {x^3 \sqrt {a^2-b^2 x^4}}{5 b^2}\right )}{9 b^2}-\frac {x^7 \sqrt {a^2-b^2 x^4}}{9 b^2}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {a^2-b^2 x^4} \left (\frac {7 a^2 \left (\frac {3 a^2 \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {1}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{b \sqrt {a^2-b^2 x^4}}\right )}{5 b^2}-\frac {x^3 \sqrt {a^2-b^2 x^4}}{5 b^2}\right )}{9 b^2}-\frac {x^7 \sqrt {a^2-b^2 x^4}}{9 b^2}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {a^2-b^2 x^4} \left (\frac {7 a^2 \left (\frac {3 a^2 \left (\frac {\int \frac {b x^2+a}{\sqrt {a^2-b^2 x^4}}dx}{b}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 b^2}-\frac {x^3 \sqrt {a^2-b^2 x^4}}{5 b^2}\right )}{9 b^2}-\frac {x^7 \sqrt {a^2-b^2 x^4}}{9 b^2}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {\sqrt {a^2-b^2 x^4} \left (\frac {7 a^2 \left (\frac {3 a^2 \left (\frac {\sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {b x^2+a}{\sqrt {1-\frac {b^2 x^4}{a^2}}}dx}{b \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 b^2}-\frac {x^3 \sqrt {a^2-b^2 x^4}}{5 b^2}\right )}{9 b^2}-\frac {x^7 \sqrt {a^2-b^2 x^4}}{9 b^2}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {\sqrt {a^2-b^2 x^4} \left (\frac {7 a^2 \left (\frac {3 a^2 \left (\frac {a \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{b \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 b^2}-\frac {x^3 \sqrt {a^2-b^2 x^4}}{5 b^2}\right )}{9 b^2}-\frac {x^7 \sqrt {a^2-b^2 x^4}}{9 b^2}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {a^2-b^2 x^4} \left (\frac {7 a^2 \left (\frac {3 a^2 \left (\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} E\left (\left .\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}-\frac {a^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-1\right )}{b^{3/2} \sqrt {a^2-b^2 x^4}}\right )}{5 b^2}-\frac {x^3 \sqrt {a^2-b^2 x^4}}{5 b^2}\right )}{9 b^2}-\frac {x^7 \sqrt {a^2-b^2 x^4}}{9 b^2}\right )}{\sqrt {a-b x^2} \sqrt {a+b x^2}}\)

Input:

Int[x^10/(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]),x]
 

Output:

(Sqrt[a^2 - b^2*x^4]*(-1/9*(x^7*Sqrt[a^2 - b^2*x^4])/b^2 + (7*a^2*(-1/5*(x 
^3*Sqrt[a^2 - b^2*x^4])/b^2 + (3*a^2*((a^(3/2)*Sqrt[1 - (b^2*x^4)/a^2]*Ell 
ipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -1])/(b^(3/2)*Sqrt[a^2 - b^2*x^4]) - ( 
a^(3/2)*Sqrt[1 - (b^2*x^4)/a^2]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -1] 
)/(b^(3/2)*Sqrt[a^2 - b^2*x^4])))/(5*b^2)))/(9*b^2)))/(Sqrt[a - b*x^2]*Sqr 
t[a + b*x^2])
 

Defintions of rubi rules used

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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 344

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c 
+ b*d*x^4)^FracPart[p])   Int[(e*x)^m*(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a 
, b, c, d, e, m, p}, x] && EqQ[b*c + a*d, 0] &&  !IntegerQ[p]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 843

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n 
 - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ 
a*c^n*((m - n + 1)/(b*(m + n*p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^p, x] 
, x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* 
p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

Maple [A] (verified)

Time = 4.03 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {x^{3} \left (5 b^{2} x^{4}+7 a^{2}\right ) \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}{45 b^{4}}-\frac {7 a^{5} \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (b \,x^{2}+a \right )}}{15 b^{5} \sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(161\)
elliptic	\(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (-\frac {x^{7} \sqrt {-b^{2} x^{4}+a^{2}}}{9 b^{2}}-\frac {7 a^{2} x^{3} \sqrt {-b^{2} x^{4}+a^{2}}}{45 b^{4}}-\frac {7 a^{5} \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right )}{15 b^{5} \sqrt {\frac {b}{a}}\, \sqrt {-b^{2} x^{4}+a^{2}}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}}\)	\(164\)
default	\(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {b \,x^{2}+a}\, \left (-5 \sqrt {\frac {b}{a}}\, b^{5} x^{11}-2 \sqrt {\frac {b}{a}}\, a^{2} b^{3} x^{7}+7 \sqrt {\frac {b}{a}}\, a^{4} b \,x^{3}+21 a^{5} \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, i\right )-21 a^{5} \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, i\right )\right )}{45 \left (-b^{2} x^{4}+a^{2}\right ) b^{5} \sqrt {\frac {b}{a}}}\)	\(184\)

Input:

int(x^10/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/45*x^3*(5*b^2*x^4+7*a^2)/b^4*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)-7/15*a^5/ 
b^5/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+b*x^2/a)^(1/2)/(-b^2*x^4+a^2)^(1/2)*( 
EllipticF(x*(b/a)^(1/2),I)-EllipticE(x*(b/a)^(1/2),I))*((-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 = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.59 \[ \int \frac {x^{10}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=-\frac {21 \, \sqrt {-b^{2}} a^{5} x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-1) - 21 \, \sqrt {-b^{2}} a^{5} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-1) + {\left (5 \, b^{5} x^{8} + 7 \, a^{2} b^{3} x^{4} + 21 \, a^{4} b\right )} \sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}}{45 \, b^{7} x} \] Input:

integrate(x^10/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

-1/45*(21*sqrt(-b^2)*a^5*x*sqrt(a/b)*elliptic_e(arcsin(sqrt(a/b)/x), -1) - 
 21*sqrt(-b^2)*a^5*x*sqrt(a/b)*elliptic_f(arcsin(sqrt(a/b)/x), -1) + (5*b^ 
5*x^8 + 7*a^2*b^3*x^4 + 21*a^4*b)*sqrt(b*x^2 + a)*sqrt(-b*x^2 + a))/(b^7*x 
)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{10}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\text {Timed out} \] Input:

integrate(x**10/(-b*x**2+a)**(1/2)/(b*x**2+a)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^{10}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\int { \frac {x^{10}}{\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}} \,d x } \] Input:

integrate(x^10/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^10/(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)), x)
 

Giac [F]

\[ \int \frac {x^{10}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\int { \frac {x^{10}}{\sqrt {b x^{2} + a} \sqrt {-b x^{2} + a}} \,d x } \] Input:

integrate(x^10/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^10/(sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{10}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\int \frac {x^{10}}{\sqrt {b\,x^2+a}\,\sqrt {a-b\,x^2}} \,d x \] Input:

int(x^10/((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2)),x)
 

Output:

int(x^10/((a + b*x^2)^(1/2)*(a - b*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^{10}}{\sqrt {a-b x^2} \sqrt {a+b x^2}} \, dx=\frac {-7 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a^{2} x^{3}-5 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, b^{2} x^{7}+21 \left (\int \frac {\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b^{2} x^{4}+a^{2}}d x \right ) a^{4}}{45 b^{4}} \] Input:

int(x^10/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2),x)
 

Output:

( - 7*sqrt(a + b*x**2)*sqrt(a - b*x**2)*a**2*x**3 - 5*sqrt(a + b*x**2)*sqr 
t(a - b*x**2)*b**2*x**7 + 21*int((sqrt(a + b*x**2)*sqrt(a - b*x**2)*x**2)/ 
(a**2 - b**2*x**4),x)*a**4)/(45*b**4)