\(\int (1-b x^2) (1-b^2 x^4)^{5/2} \, dx\) [233]

Optimal result

Integrand size = 23, antiderivative size = 120 \[ \int \left (1-b x^2\right ) \left (1-b^2 x^4\right )^{5/2} \, dx=\frac {4 x \left (195-77 b x^2\right ) \sqrt {1-b^2 x^4}}{3003}+\frac {10 x \left (117-77 b x^2\right ) \left (1-b^2 x^4\right )^{3/2}}{9009}+\frac {1}{143} x \left (13-11 b x^2\right ) \left (1-b^2 x^4\right )^{5/2}-\frac {8 E\left (\left .\arcsin \left (\sqrt {b} x\right )\right |-1\right )}{39 \sqrt {b}}+\frac {2176 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {b} x\right ),-1\right )}{3003 \sqrt {b}} \] Output:

4/3003*x*(-77*b*x^2+195)*(-b^2*x^4+1)^(1/2)+10/9009*x*(-77*b*x^2+117)*(-b^ 
2*x^4+1)^(3/2)+1/143*x*(-11*b*x^2+13)*(-b^2*x^4+1)^(5/2)-8/39*EllipticE(b^ 
(1/2)*x,I)/b^(1/2)+2176/3003*EllipticF(b^(1/2)*x,I)/b^(1/2)
 

Mathematica [C] (verified)

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

Time = 9.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.38 \[ \int \left (1-b x^2\right ) \left (1-b^2 x^4\right )^{5/2} \, dx=x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},b^2 x^4\right )-\frac {1}{3} b x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},b^2 x^4\right ) \] Input:

Integrate[(1 - b*x^2)*(1 - b^2*x^4)^(5/2),x]
 

Output:

x*Hypergeometric2F1[-5/2, 1/4, 5/4, b^2*x^4] - (b*x^3*Hypergeometric2F1[-5 
/2, 3/4, 7/4, b^2*x^4])/3
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.88, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {1388, 318, 27, 403, 25, 27, 403, 27, 403, 27, 403, 27, 403, 27, 399, 284, 327, 762}

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.

\(\displaystyle \int \left (1-b x^2\right ) \left (1-b^2 x^4\right )^{5/2} \, dx\)

\(\Big \downarrow \) 1388

\(\displaystyle \int \left (1-b x^2\right )^{7/2} \left (b x^2+1\right )^{5/2}dx\)

\(\Big \downarrow \) 318

\(\displaystyle -\frac {\int -2 b \left (1-b x^2\right )^{7/2} \sqrt {b x^2+1} \left (10 b x^2+7\right )dx}{13 b}-\frac {1}{13} x \left (b x^2+1\right )^{3/2} \left (1-b x^2\right )^{9/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{13} \int \left (1-b x^2\right )^{7/2} \sqrt {b x^2+1} \left (10 b x^2+7\right )dx-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {2}{13} \left (-\frac {\int -\frac {b \left (1-b x^2\right )^{7/2} \left (107 b x^2+87\right )}{\sqrt {b x^2+1}}dx}{11 b}-\frac {10}{11} x \sqrt {b x^2+1} \left (1-b x^2\right )^{9/2}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2}{13} \left (\frac {\int \frac {b \left (1-b x^2\right )^{7/2} \left (107 b x^2+87\right )}{\sqrt {b x^2+1}}dx}{11 b}-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \int \frac {\left (1-b x^2\right )^{7/2} \left (107 b x^2+87\right )}{\sqrt {b x^2+1}}dx-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {\int \frac {2 b \left (1-b x^2\right )^{5/2} \left (411 b x^2+338\right )}{\sqrt {b x^2+1}}dx}{9 b}+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {\left (1-b x^2\right )^{5/2} \left (411 b x^2+338\right )}{\sqrt {b x^2+1}}dx+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \left (\frac {\int \frac {5 b \left (1-b x^2\right )^{3/2} \left (431 b x^2+391\right )}{\sqrt {b x^2+1}}dx}{7 b}+\frac {411}{7} x \sqrt {b x^2+1} \left (1-b x^2\right )^{5/2}\right )+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \left (\frac {5}{7} \int \frac {\left (1-b x^2\right )^{3/2} \left (431 b x^2+391\right )}{\sqrt {b x^2+1}}dx+\frac {411}{7} x \sqrt {b x^2+1} \left (1-b x^2\right )^{5/2}\right )+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \left (\frac {5}{7} \left (\frac {\int \frac {6 b \sqrt {1-b x^2} \left (177 b x^2+254\right )}{\sqrt {b x^2+1}}dx}{5 b}+\frac {431}{5} x \sqrt {b x^2+1} \left (1-b x^2\right )^{3/2}\right )+\frac {411}{7} x \sqrt {b x^2+1} \left (1-b x^2\right )^{5/2}\right )+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \left (\frac {5}{7} \left (\frac {6}{5} \int \frac {\sqrt {1-b x^2} \left (177 b x^2+254\right )}{\sqrt {b x^2+1}}dx+\frac {431}{5} x \sqrt {b x^2+1} \left (1-b x^2\right )^{3/2}\right )+\frac {411}{7} x \sqrt {b x^2+1} \left (1-b x^2\right )^{5/2}\right )+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \left (\frac {5}{7} \left (\frac {6}{5} \left (\frac {\int \frac {3 b \left (195-77 b x^2\right )}{\sqrt {1-b x^2} \sqrt {b x^2+1}}dx}{3 b}+59 x \sqrt {1-b x^2} \sqrt {b x^2+1}\right )+\frac {431}{5} x \sqrt {b x^2+1} \left (1-b x^2\right )^{3/2}\right )+\frac {411}{7} x \sqrt {b x^2+1} \left (1-b x^2\right )^{5/2}\right )+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \left (\frac {5}{7} \left (\frac {6}{5} \left (\int \frac {195-77 b x^2}{\sqrt {1-b x^2} \sqrt {b x^2+1}}dx+59 x \sqrt {1-b x^2} \sqrt {b x^2+1}\right )+\frac {431}{5} x \sqrt {b x^2+1} \left (1-b x^2\right )^{3/2}\right )+\frac {411}{7} x \sqrt {b x^2+1} \left (1-b x^2\right )^{5/2}\right )+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \left (\frac {5}{7} \left (\frac {6}{5} \left (272 \int \frac {1}{\sqrt {1-b x^2} \sqrt {b x^2+1}}dx-77 \int \frac {\sqrt {b x^2+1}}{\sqrt {1-b x^2}}dx+59 x \sqrt {1-b x^2} \sqrt {b x^2+1}\right )+\frac {431}{5} x \sqrt {b x^2+1} \left (1-b x^2\right )^{3/2}\right )+\frac {411}{7} x \sqrt {b x^2+1} \left (1-b x^2\right )^{5/2}\right )+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 284

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \left (\frac {5}{7} \left (\frac {6}{5} \left (272 \int \frac {1}{\sqrt {1-b^2 x^4}}dx-77 \int \frac {\sqrt {b x^2+1}}{\sqrt {1-b x^2}}dx+59 x \sqrt {1-b x^2} \sqrt {b x^2+1}\right )+\frac {431}{5} x \sqrt {b x^2+1} \left (1-b x^2\right )^{3/2}\right )+\frac {411}{7} x \sqrt {b x^2+1} \left (1-b x^2\right )^{5/2}\right )+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \left (\frac {5}{7} \left (\frac {6}{5} \left (272 \int \frac {1}{\sqrt {1-b^2 x^4}}dx-\frac {77 E\left (\left .\arcsin \left (\sqrt {b} x\right )\right |-1\right )}{\sqrt {b}}+59 x \sqrt {1-b x^2} \sqrt {b x^2+1}\right )+\frac {431}{5} x \sqrt {b x^2+1} \left (1-b x^2\right )^{3/2}\right )+\frac {411}{7} x \sqrt {b x^2+1} \left (1-b x^2\right )^{5/2}\right )+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {2}{13} \left (\frac {1}{11} \left (\frac {2}{9} \left (\frac {5}{7} \left (\frac {6}{5} \left (\frac {272 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {b} x\right ),-1\right )}{\sqrt {b}}-\frac {77 E\left (\left .\arcsin \left (\sqrt {b} x\right )\right |-1\right )}{\sqrt {b}}+59 x \sqrt {1-b x^2} \sqrt {b x^2+1}\right )+\frac {431}{5} x \sqrt {b x^2+1} \left (1-b x^2\right )^{3/2}\right )+\frac {411}{7} x \sqrt {b x^2+1} \left (1-b x^2\right )^{5/2}\right )+\frac {107}{9} x \sqrt {b x^2+1} \left (1-b x^2\right )^{7/2}\right )-\frac {10}{11} x \left (1-b x^2\right )^{9/2} \sqrt {b x^2+1}\right )-\frac {1}{13} x \left (1-b x^2\right )^{9/2} \left (b x^2+1\right )^{3/2}\)

Input:

Int[(1 - b*x^2)*(1 - b^2*x^4)^(5/2),x]
 

Output:

-1/13*(x*(1 - b*x^2)^(9/2)*(1 + b*x^2)^(3/2)) + (2*((-10*x*(1 - b*x^2)^(9/ 
2)*Sqrt[1 + b*x^2])/11 + ((107*x*(1 - b*x^2)^(7/2)*Sqrt[1 + b*x^2])/9 + (2 
*((411*x*(1 - b*x^2)^(5/2)*Sqrt[1 + b*x^2])/7 + (5*((431*x*(1 - b*x^2)^(3/ 
2)*Sqrt[1 + b*x^2])/5 + (6*(59*x*Sqrt[1 - b*x^2]*Sqrt[1 + b*x^2] - (77*Ell 
ipticE[ArcSin[Sqrt[b]*x], -1])/Sqrt[b] + (272*EllipticF[ArcSin[Sqrt[b]*x], 
 -1])/Sqrt[b]))/5))/7))/9)/11))/13
 

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 284

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> I 
nt[(a*c + b*d*x^4)^p, x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c + a*d, 0] 
&& (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0]))
 

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, 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 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 403

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

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 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

Maple [C] (verified)

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

Time = 1.86 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.30

method	result	size
meijerg	\(x \operatorname {hypergeom}\left (\left [-\frac {5}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], b^{2} x^{4}\right )-\frac {b \,x^{3} \operatorname {hypergeom}\left (\left [-\frac {5}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], b^{2} x^{4}\right )}{3}\)	\(36\)
risch	\(\frac {x \left (693 x^{10} b^{5}-819 b^{4} x^{8}-2156 b^{3} x^{6}+2808 b^{2} x^{4}+2387 b \,x^{2}-4329\right ) \left (b^{2} x^{4}-1\right )}{9009 \sqrt {-b^{2} x^{4}+1}}+\frac {40 \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}\, \operatorname {EllipticF}\left (\sqrt {b}\, x , i\right )}{77 \sqrt {b}\, \sqrt {-b^{2} x^{4}+1}}+\frac {8 \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (\sqrt {b}\, x , i\right )-\operatorname {EllipticE}\left (\sqrt {b}\, x , i\right )\right )}{39 \sqrt {b}\, \sqrt {-b^{2} x^{4}+1}}\)	\(165\)
elliptic	\(-\frac {b^{5} x^{11} \sqrt {-b^{2} x^{4}+1}}{13}+\frac {b^{4} x^{9} \sqrt {-b^{2} x^{4}+1}}{11}+\frac {28 b^{3} x^{7} \sqrt {-b^{2} x^{4}+1}}{117}-\frac {24 b^{2} x^{5} \sqrt {-b^{2} x^{4}+1}}{77}-\frac {31 b \,x^{3} \sqrt {-b^{2} x^{4}+1}}{117}+\frac {37 x \sqrt {-b^{2} x^{4}+1}}{77}+\frac {40 \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}\, \operatorname {EllipticF}\left (\sqrt {b}\, x , i\right )}{77 \sqrt {b}\, \sqrt {-b^{2} x^{4}+1}}+\frac {8 \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (\sqrt {b}\, x , i\right )-\operatorname {EllipticE}\left (\sqrt {b}\, x , i\right )\right )}{39 \sqrt {b}\, \sqrt {-b^{2} x^{4}+1}}\)	\(214\)
default	\(-b \left (\frac {b^{4} x^{11} \sqrt {-b^{2} x^{4}+1}}{13}-\frac {28 b^{2} x^{7} \sqrt {-b^{2} x^{4}+1}}{117}+\frac {31 x^{3} \sqrt {-b^{2} x^{4}+1}}{117}-\frac {8 \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (\sqrt {b}\, x , i\right )-\operatorname {EllipticE}\left (\sqrt {b}\, x , i\right )\right )}{39 b^{\frac {3}{2}} \sqrt {-b^{2} x^{4}+1}}\right )+\frac {b^{4} x^{9} \sqrt {-b^{2} x^{4}+1}}{11}-\frac {24 b^{2} x^{5} \sqrt {-b^{2} x^{4}+1}}{77}+\frac {37 x \sqrt {-b^{2} x^{4}+1}}{77}+\frac {40 \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}\, \operatorname {EllipticF}\left (\sqrt {b}\, x , i\right )}{77 \sqrt {b}\, \sqrt {-b^{2} x^{4}+1}}\)	\(217\)

Input:

int((-b*x^2+1)*(-b^2*x^4+1)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

x*hypergeom([-5/2,1/4],[5/4],b^2*x^4)-1/3*b*x^3*hypergeom([-5/2,3/4],[7/4] 
,b^2*x^4)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.05 \[ \int \left (1-b x^2\right ) \left (1-b^2 x^4\right )^{5/2} \, dx=\frac {\frac {24 \, \sqrt {-b^{2}} {\left (195 \, b - 77\right )} x F(\arcsin \left (\frac {1}{\sqrt {b} x}\right )\,|\,-1)}{\sqrt {b}} + \frac {1848 \, \sqrt {-b^{2}} x E(\arcsin \left (\frac {1}{\sqrt {b} x}\right )\,|\,-1)}{\sqrt {b}} - {\left (693 \, b^{7} x^{12} - 819 \, b^{6} x^{10} - 2156 \, b^{5} x^{8} + 2808 \, b^{4} x^{6} + 2387 \, b^{3} x^{4} - 4329 \, b^{2} x^{2} - 1848 \, b\right )} \sqrt {-b^{2} x^{4} + 1}}{9009 \, b^{2} x} \] Input:

integrate((-b*x^2+1)*(-b^2*x^4+1)^(5/2),x, algorithm="fricas")
 

Output:

1/9009*(24*sqrt(-b^2)*(195*b - 77)*x*elliptic_f(arcsin(1/(sqrt(b)*x)), -1) 
/sqrt(b) + 1848*sqrt(-b^2)*x*elliptic_e(arcsin(1/(sqrt(b)*x)), -1)/sqrt(b) 
 - (693*b^7*x^12 - 819*b^6*x^10 - 2156*b^5*x^8 + 2808*b^4*x^6 + 2387*b^3*x 
^4 - 4329*b^2*x^2 - 1848*b)*sqrt(-b^2*x^4 + 1))/(b^2*x)
 

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (107) = 214\).

Time = 2.47 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.97 \[ \int \left (1-b x^2\right ) \left (1-b^2 x^4\right )^{5/2} \, dx=- \frac {b^{5} x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {b^{2} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} + \frac {b^{4} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {b^{2} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} + \frac {b^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {b^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} - \frac {b^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {b^{2} x^{4} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} - \frac {b x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {b^{2} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {b^{2} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \] Input:

integrate((-b*x**2+1)*(-b**2*x**4+1)**(5/2),x)
 

Output:

-b**5*x**11*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), b**2*x**4*exp_polar(2 
*I*pi))/(4*gamma(15/4)) + b**4*x**9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), 
 b**2*x**4*exp_polar(2*I*pi))/(4*gamma(13/4)) + b**3*x**7*gamma(7/4)*hyper 
((-1/2, 7/4), (11/4,), b**2*x**4*exp_polar(2*I*pi))/(2*gamma(11/4)) - b**2 
*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b**2*x**4*exp_polar(2*I*pi))/( 
2*gamma(9/4)) - b*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b**2*x**4*exp 
_polar(2*I*pi))/(4*gamma(7/4)) + x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b 
**2*x**4*exp_polar(2*I*pi))/(4*gamma(5/4))
 

Maxima [F]

\[ \int \left (1-b x^2\right ) \left (1-b^2 x^4\right )^{5/2} \, dx=\int { -{\left (-b^{2} x^{4} + 1\right )}^{\frac {5}{2}} {\left (b x^{2} - 1\right )} \,d x } \] Input:

integrate((-b*x^2+1)*(-b^2*x^4+1)^(5/2),x, algorithm="maxima")
 

Output:

-integrate((-b^2*x^4 + 1)^(5/2)*(b*x^2 - 1), x)
 

Giac [F]

\[ \int \left (1-b x^2\right ) \left (1-b^2 x^4\right )^{5/2} \, dx=\int { -{\left (-b^{2} x^{4} + 1\right )}^{\frac {5}{2}} {\left (b x^{2} - 1\right )} \,d x } \] Input:

integrate((-b*x^2+1)*(-b^2*x^4+1)^(5/2),x, algorithm="giac")
 

Output:

integrate(-(-b^2*x^4 + 1)^(5/2)*(b*x^2 - 1), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \left (1-b x^2\right ) \left (1-b^2 x^4\right )^{5/2} \, dx=-\int {\left (1-b^2\,x^4\right )}^{5/2}\,\left (b\,x^2-1\right ) \,d x \] Input:

int(-(1 - b^2*x^4)^(5/2)*(b*x^2 - 1),x)
 

Output:

-int((1 - b^2*x^4)^(5/2)*(b*x^2 - 1), x)
 

Reduce [F]

\[ \int \left (1-b x^2\right ) \left (1-b^2 x^4\right )^{5/2} \, dx=-\frac {\sqrt {-b^{2} x^{4}+1}\, b^{5} x^{11}}{13}+\frac {\sqrt {-b^{2} x^{4}+1}\, b^{4} x^{9}}{11}+\frac {28 \sqrt {-b^{2} x^{4}+1}\, b^{3} x^{7}}{117}-\frac {24 \sqrt {-b^{2} x^{4}+1}\, b^{2} x^{5}}{77}-\frac {31 \sqrt {-b^{2} x^{4}+1}\, b \,x^{3}}{117}+\frac {37 \sqrt {-b^{2} x^{4}+1}\, x}{77}-\frac {40 \left (\int \frac {\sqrt {-b^{2} x^{4}+1}}{b^{2} x^{4}-1}d x \right )}{77}+\frac {8 \left (\int \frac {\sqrt {-b^{2} x^{4}+1}\, x^{2}}{b^{2} x^{4}-1}d x \right ) b}{39} \] Input:

int((-b*x^2+1)*(-b^2*x^4+1)^(5/2),x)
 

Output:

( - 693*sqrt( - b**2*x**4 + 1)*b**5*x**11 + 819*sqrt( - b**2*x**4 + 1)*b** 
4*x**9 + 2156*sqrt( - b**2*x**4 + 1)*b**3*x**7 - 2808*sqrt( - b**2*x**4 + 
1)*b**2*x**5 - 2387*sqrt( - b**2*x**4 + 1)*b*x**3 + 4329*sqrt( - b**2*x**4 
 + 1)*x - 4680*int(sqrt( - b**2*x**4 + 1)/(b**2*x**4 - 1),x) + 1848*int((s 
qrt( - b**2*x**4 + 1)*x**2)/(b**2*x**4 - 1),x)*b)/9009