\(\int \frac {x^4 (a+b x^2)^{5/2}}{\sqrt {c+d x^2}} \, dx\) [1195]

Optimal result

Integrand size = 26, antiderivative size = 553 \[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {\left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) x \sqrt {c+d x^2}}{315 b d^5 \sqrt {a+b x^2}}-\frac {\left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 b d^4}+\frac {\left (48 b^2 c^2-115 a b c d+75 a^2 d^2\right ) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{315 d^3}-\frac {4 b (2 b c-3 a d) x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{63 d^2}+\frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\sqrt {a} \left (128 b^4 c^4-328 a b^3 c^3 d+243 a^2 b^2 c^2 d^2-25 a^3 b c d^3-10 a^4 d^4\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{315 b^{3/2} d^5 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (64 b^3 c^3-156 a b^2 c^2 d+105 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{315 b^{3/2} d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/315*(-10*a^4*d^4-25*a^3*b*c*d^3+243*a^2*b^2*c^2*d^2-328*a*b^3*c^3*d+128* 
b^4*c^4)*x*(d*x^2+c)^(1/2)/b/d^5/(b*x^2+a)^(1/2)-1/315*(-5*a^3*d^3+105*a^2 
*b*c*d^2-156*a*b^2*c^2*d+64*b^3*c^3)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d 
^4+1/315*(75*a^2*d^2-115*a*b*c*d+48*b^2*c^2)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c) 
^(1/2)/d^3-4/63*b*(-3*a*d+2*b*c)*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^2+1 
/9*b*x^5*(b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/d-1/315*a^(1/2)*(-10*a^4*d^4-25*a 
^3*b*c*d^3+243*a^2*b^2*c^2*d^2-328*a*b^3*c^3*d+128*b^4*c^4)*(d*x^2+c)^(1/2 
)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(3/2) 
/d^5/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/315*a^(3/2)*(-5*a^3 
*d^3+105*a^2*b*c*d^2-156*a*b^2*c^2*d+64*b^3*c^3)*(d*x^2+c)^(1/2)*InverseJa 
cobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(3/2)/d^4/(b*x^2+a)^ 
(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.16 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.69 \[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (5 a^3 d^3+15 a^2 b d^2 \left (-7 c+5 d x^2\right )+a b^2 d \left (156 c^2-115 c d x^2+95 d^2 x^4\right )+b^3 \left (-64 c^3+48 c^2 d x^2-40 c d^2 x^4+35 d^3 x^6\right )\right )+i c \left (-128 b^4 c^4+328 a b^3 c^3 d-243 a^2 b^2 c^2 d^2+25 a^3 b c d^3+10 a^4 d^4\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (-128 b^4 c^4+392 a b^3 c^3 d-399 a^2 b^2 c^2 d^2+130 a^3 b c d^3+5 a^4 d^4\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{315 b \sqrt {\frac {b}{a}} d^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(x^4*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(5*a^3*d^3 + 15*a^2*b*d^2*(-7*c + 5 
*d*x^2) + a*b^2*d*(156*c^2 - 115*c*d*x^2 + 95*d^2*x^4) + b^3*(-64*c^3 + 48 
*c^2*d*x^2 - 40*c*d^2*x^4 + 35*d^3*x^6)) + I*c*(-128*b^4*c^4 + 328*a*b^3*c 
^3*d - 243*a^2*b^2*c^2*d^2 + 25*a^3*b*c*d^3 + 10*a^4*d^4)*Sqrt[1 + (b*x^2) 
/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I 
*c*(-128*b^4*c^4 + 392*a*b^3*c^3*d - 399*a^2*b^2*c^2*d^2 + 130*a^3*b*c*d^3 
 + 5*a^4*d^4)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[ 
Sqrt[b/a]*x], (a*d)/(b*c)])/(315*b*Sqrt[b/a]*d^5*Sqrt[a + b*x^2]*Sqrt[c + 
d*x^2])
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {379, 25, 443, 25, 444, 27, 444, 406, 320, 388, 313}

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.

\(\displaystyle \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 379

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 443

\(\displaystyle \frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\frac {\int -\frac {x^4 \left (b \left (48 b^2 c^2-115 a b d c+75 a^2 d^2\right ) x^2+a \left (40 b^2 c^2-95 a b d c+63 a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{7 d}+\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{7 d}}{9 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{7 d}-\frac {\int \frac {x^4 \left (b \left (48 b^2 c^2-115 a b d c+75 a^2 d^2\right ) x^2+a \left (40 b^2 c^2-95 a b d c+63 a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{7 d}}{9 d}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{7 d}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{5 d}-\frac {\int \frac {3 b x^2 \left (\left (64 b^3 c^3-156 a b^2 d c^2+105 a^2 b d^2 c-5 a^3 d^3\right ) x^2+a c \left (48 b^2 c^2-115 a b d c+75 a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}}{7 d}}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{7 d}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{5 d}-\frac {3 \int \frac {x^2 \left (\left (64 b^3 c^3-156 a b^2 d c^2+105 a^2 b d^2 c-5 a^3 d^3\right ) x^2+a c \left (48 b^2 c^2-115 a b d c+75 a^2 d^2\right )\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}}{7 d}}{9 d}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{7 d}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{5 d}-\frac {3 \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right )}{3 b d}-\frac {\int \frac {\left (128 b^4 c^4-328 a b^3 d c^3+243 a^2 b^2 d^2 c^2-25 a^3 b d^3 c-10 a^4 d^4\right ) x^2+a c \left (64 b^3 c^3-156 a b^2 d c^2+105 a^2 b d^2 c-5 a^3 d^3\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 d}}{7 d}}{9 d}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{7 d}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{5 d}-\frac {3 \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right )}{3 b d}-\frac {a c \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}\right )}{5 d}}{7 d}}{9 d}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{7 d}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{5 d}-\frac {3 \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right )}{3 b d}-\frac {\left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}\right )}{5 d}}{7 d}}{9 d}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{7 d}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{5 d}-\frac {3 \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right )}{3 b d}-\frac {\left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}\right )}{5 d}}{7 d}}{9 d}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {b x^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{9 d}-\frac {\frac {4 b x^5 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{7 d}-\frac {\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (75 a^2 d^2-115 a b c d+48 b^2 c^2\right )}{5 d}-\frac {3 \left (\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right )}{3 b d}-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (-5 a^3 d^3+105 a^2 b c d^2-156 a b^2 c^2 d+64 b^3 c^3\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (-10 a^4 d^4-25 a^3 b c d^3+243 a^2 b^2 c^2 d^2-328 a b^3 c^3 d+128 b^4 c^4\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 b d}\right )}{5 d}}{7 d}}{9 d}\)

Input:

Int[(x^4*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]
 

Output:

(b*x^5*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(9*d) - ((4*b*(2*b*c - 3*a*d)*x^ 
5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7*d) - (((48*b^2*c^2 - 115*a*b*c*d + 7 
5*a^2*d^2)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d) - (3*(((64*b^3*c^3 - 
 156*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x^2]*Sqrt[c + 
 d*x^2])/(3*b*d) - ((128*b^4*c^4 - 328*a*b^3*c^3*d + 243*a^2*b^2*c^2*d^2 - 
 25*a^3*b*c*d^3 - 10*a^4*d^4)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - ( 
Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/( 
a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) 
+ (c^(3/2)*(64*b^3*c^3 - 156*a*b^2*c^2*d + 105*a^2*b*c*d^2 - 5*a^3*d^3)*Sq 
rt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sq 
rt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*b*d)))/(5 
*d))/(7*d))/(9*d)
 

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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 379

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

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

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

rule 443

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

rule 444

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

Maple [A] (verified)

Time = 17.37 (sec) , antiderivative size = 738, normalized size of antiderivative = 1.33

method	result	size
risch	\(\frac {x \left (35 b^{3} d^{3} x^{6}+95 a \,b^{2} d^{3} x^{4}-40 b^{3} c \,d^{2} x^{4}+75 x^{2} a^{2} b \,d^{3}-115 x^{2} a \,b^{2} c \,d^{2}+48 x^{2} b^{3} c^{2} d +5 a^{3} d^{3}-105 a^{2} b c \,d^{2}+156 a \,b^{2} c^{2} d -64 b^{3} c^{3}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{315 b \,d^{4}}-\frac {\left (-\frac {\left (10 d^{4} a^{4}+25 a^{3} b c \,d^{3}-243 a^{2} b^{2} c^{2} d^{2}+328 a \,b^{3} c^{3} d -128 c^{4} b^{4}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}-\frac {64 a \,b^{3} c^{4} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {5 a^{4} c \,d^{3} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {156 a^{2} b^{2} c^{3} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {105 a^{3} b \,c^{2} d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{315 b \,d^{4} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(738\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {b^{2} x^{7} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{9 d}+\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 b d}+\frac {\left (3 a^{2} b -\frac {7 b^{2} a c}{9 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) \left (6 a d +6 b c \right )}{7 b d}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (a^{3}-\frac {5 \left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) a c}{7 b d}-\frac {\left (3 a^{2} b -\frac {7 b^{2} a c}{9 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) \left (6 a d +6 b c \right )}{7 b d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}-\frac {\left (a^{3}-\frac {5 \left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) a c}{7 b d}-\frac {\left (3 a^{2} b -\frac {7 b^{2} a c}{9 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) \left (6 a d +6 b c \right )}{7 b d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {3 \left (3 a^{2} b -\frac {7 b^{2} a c}{9 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) \left (6 a d +6 b c \right )}{7 b d}\right ) a c}{5 b d}-\frac {\left (a^{3}-\frac {5 \left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) a c}{7 b d}-\frac {\left (3 a^{2} b -\frac {7 b^{2} a c}{9 d}-\frac {\left (3 a \,b^{2}-\frac {b^{2} \left (8 a d +8 b c \right )}{9 d}\right ) \left (6 a d +6 b c \right )}{7 b d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(901\)
default	\(\text {Expression too large to display}\)	\(1047\)

Input:

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

Output:

1/315/b*x*(35*b^3*d^3*x^6+95*a*b^2*d^3*x^4-40*b^3*c*d^2*x^4+75*a^2*b*d^3*x 
^2-115*a*b^2*c*d^2*x^2+48*b^3*c^2*d*x^2+5*a^3*d^3-105*a^2*b*c*d^2+156*a*b^ 
2*c^2*d-64*b^3*c^3)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^4-1/315/b/d^4*(-(10* 
a^4*d^4+25*a^3*b*c*d^3-243*a^2*b^2*c^2*d^2+328*a*b^3*c^3*d-128*b^4*c^4)*c/ 
(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+ 
a*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE 
(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))-64*a*b^3*c^4/(-b/a)^(1/2)*(1+b* 
x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*Ellipti 
cF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+5*a^4*c*d^3/(-b/a)^(1/2)*(1+b* 
x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*Ellipti 
cF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+156*a^2*b^2*c^3*d/(-b/a)^(1/2) 
*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*E 
llipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-105*a^3*b*c^2*d^2/(-b/a) 
^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^( 
1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))*((b*x^2+a)*(d*x^2 
+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=-\frac {{\left (128 \, b^{4} c^{5} - 328 \, a b^{3} c^{4} d + 243 \, a^{2} b^{2} c^{3} d^{2} - 25 \, a^{3} b c^{2} d^{3} - 10 \, a^{4} c d^{4}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (128 \, b^{4} c^{5} - 328 \, a b^{3} c^{4} d - 5 \, a^{4} d^{5} + {\left (243 \, a^{2} b^{2} + 64 \, a b^{3}\right )} c^{3} d^{2} - {\left (25 \, a^{3} b + 156 \, a^{2} b^{2}\right )} c^{2} d^{3} - 5 \, {\left (2 \, a^{4} - 21 \, a^{3} b\right )} c d^{4}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (35 \, b^{4} d^{5} x^{8} + 128 \, b^{4} c^{4} d - 328 \, a b^{3} c^{3} d^{2} + 243 \, a^{2} b^{2} c^{2} d^{3} - 25 \, a^{3} b c d^{4} - 10 \, a^{4} d^{5} - 5 \, {\left (8 \, b^{4} c d^{4} - 19 \, a b^{3} d^{5}\right )} x^{6} + {\left (48 \, b^{4} c^{2} d^{3} - 115 \, a b^{3} c d^{4} + 75 \, a^{2} b^{2} d^{5}\right )} x^{4} - {\left (64 \, b^{4} c^{3} d^{2} - 156 \, a b^{3} c^{2} d^{3} + 105 \, a^{2} b^{2} c d^{4} - 5 \, a^{3} b d^{5}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{315 \, b^{2} d^{6} x} \] Input:

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

Output:

-1/315*((128*b^4*c^5 - 328*a*b^3*c^4*d + 243*a^2*b^2*c^3*d^2 - 25*a^3*b*c^ 
2*d^3 - 10*a^4*c*d^4)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/ 
x), a*d/(b*c)) - (128*b^4*c^5 - 328*a*b^3*c^4*d - 5*a^4*d^5 + (243*a^2*b^2 
 + 64*a*b^3)*c^3*d^2 - (25*a^3*b + 156*a^2*b^2)*c^2*d^3 - 5*(2*a^4 - 21*a^ 
3*b)*c*d^4)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b 
*c)) - (35*b^4*d^5*x^8 + 128*b^4*c^4*d - 328*a*b^3*c^3*d^2 + 243*a^2*b^2*c 
^2*d^3 - 25*a^3*b*c*d^4 - 10*a^4*d^5 - 5*(8*b^4*c*d^4 - 19*a*b^3*d^5)*x^6 
+ (48*b^4*c^2*d^3 - 115*a*b^3*c*d^4 + 75*a^2*b^2*d^5)*x^4 - (64*b^4*c^3*d^ 
2 - 156*a*b^3*c^2*d^3 + 105*a^2*b^2*c*d^4 - 5*a^3*b*d^5)*x^2)*sqrt(b*x^2 + 
 a)*sqrt(d*x^2 + c))/(b^2*d^6*x)
 

Sympy [F]

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

integrate(x**4*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)
 

Output:

Integral(x**4*(a + b*x**2)**(5/2)/sqrt(c + d*x**2), x)
 

Maxima [F]

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

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

Output:

integrate((b*x^2 + a)^(5/2)*x^4/sqrt(d*x^2 + c), x)
 

Giac [F]

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

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

Output:

integrate((b*x^2 + a)^(5/2)*x^4/sqrt(d*x^2 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^4\,{\left (b\,x^2+a\right )}^{5/2}}{\sqrt {d\,x^2+c}} \,d x \] Input:

int((x^4*(a + b*x^2)^(5/2))/(c + d*x^2)^(1/2),x)
 

Output:

int((x^4*(a + b*x^2)^(5/2))/(c + d*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^4 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{3} d^{3} x -105 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b c \,d^{2} x +75 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a^{2} b \,d^{3} x^{3}+156 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} c^{2} d x -115 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} c \,d^{2} x^{3}+95 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} d^{3} x^{5}-64 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{3} c^{3} x +48 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{3} c^{2} d \,x^{3}-40 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{3} c \,d^{2} x^{5}+35 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b^{3} d^{3} x^{7}-10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{4} d^{4}-25 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} b c \,d^{3}+243 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b^{2} c^{2} d^{2}-328 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{3} c^{3} d +128 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{4} c^{4}-5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{4} c \,d^{3}+105 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{3} b \,c^{2} d^{2}-156 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} b^{2} c^{3} d +64 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a \,b^{3} c^{4}}{315 b \,d^{4}} \] Input:

int(x^4*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)
 

Output:

(5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**3*x - 105*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*a**2*b*c*d**2*x + 75*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2 
*b*d**3*x**3 + 156*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*d*x - 115 
*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*x**3 + 95*sqrt(c + d*x**2 
)*sqrt(a + b*x**2)*a*b**2*d**3*x**5 - 64*sqrt(c + d*x**2)*sqrt(a + b*x**2) 
*b**3*c**3*x + 48*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c**2*d*x**3 - 40* 
sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*x**5 + 35*sqrt(c + d*x**2)*s 
qrt(a + b*x**2)*b**3*d**3*x**7 - 10*int((sqrt(c + d*x**2)*sqrt(a + b*x**2) 
*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**4*d**4 - 25*int((sqrt( 
c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4), 
x)*a**3*b*c*d**3 + 243*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + 
 a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*b**2*c**2*d**2 - 328*int((sqrt(c 
+ d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x) 
*a*b**3*c**3*d + 128*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a 
*d*x**2 + b*c*x**2 + b*d*x**4),x)*b**4*c**4 - 5*int((sqrt(c + d*x**2)*sqrt 
(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**4*c*d**3 + 105* 
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x 
**4),x)*a**3*b*c**2*d**2 - 156*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a* 
c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*b**2*c**3*d + 64*int((sqrt(c + 
 d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*...