\(\int \frac {(a+b x^2)^{5/2} (e+f x^2)^2}{(c+d x^2)^{3/2}} \, dx\) [70]

Optimal result

Integrand size = 32, antiderivative size = 653 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\left (15 a^3 d^3 f^2+a^2 b d^2 f (322 d e-219 c f)-4 b^3 c \left (35 d^2 e^2-84 c d e f+48 c^2 f^2\right )+a b^2 d \left (245 d^2 e^2-658 c d e f+396 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{105 b d^4 \sqrt {c+d x^2}}+\frac {\left (45 a^2 d^2 f^2+a b d f (154 d e-93 c f)+b^2 \left (35 d^2 e^2-84 c d e f+48 c^2 f^2\right )\right ) x^3 \sqrt {a+b x^2}}{105 d^3 \sqrt {c+d x^2}}+\frac {b f (14 b d e-8 b c f+15 a d f) x^5 \sqrt {a+b x^2}}{35 d^2 \sqrt {c+d x^2}}+\frac {b^2 f^2 x^7 \sqrt {a+b x^2}}{7 d \sqrt {c+d x^2}}-\frac {\left (15 a^3 c d^3 f^2-8 b^3 c^2 \left (35 d^2 e^2-84 c d e f+48 c^2 f^2\right )-a^2 b d^2 \left (105 d^2 e^2-532 c d e f+369 c^2 f^2\right )+a b^2 c d \left (455 d^2 e^2-1232 c d e f+744 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 b \sqrt {c} d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {2 \sqrt {c} \left (15 a^2 d^2 f (7 d e-5 c f)-2 b^2 c \left (35 d^2 e^2-84 c d e f+48 c^2 f^2\right )+a b d \left (105 d^2 e^2-287 c d e f+174 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 d^{9/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/105*(15*a^3*d^3*f^2+a^2*b*d^2*f*(-219*c*f+322*d*e)-4*b^3*c*(48*c^2*f^2-8 
4*c*d*e*f+35*d^2*e^2)+a*b^2*d*(396*c^2*f^2-658*c*d*e*f+245*d^2*e^2))*x*(b* 
x^2+a)^(1/2)/b/d^4/(d*x^2+c)^(1/2)+1/105*(45*a^2*d^2*f^2+a*b*d*f*(-93*c*f+ 
154*d*e)+b^2*(48*c^2*f^2-84*c*d*e*f+35*d^2*e^2))*x^3*(b*x^2+a)^(1/2)/d^3/( 
d*x^2+c)^(1/2)+1/35*b*f*(15*a*d*f-8*b*c*f+14*b*d*e)*x^5*(b*x^2+a)^(1/2)/d^ 
2/(d*x^2+c)^(1/2)+1/7*b^2*f^2*x^7*(b*x^2+a)^(1/2)/d/(d*x^2+c)^(1/2)-1/105* 
(15*a^3*c*d^3*f^2-8*b^3*c^2*(48*c^2*f^2-84*c*d*e*f+35*d^2*e^2)-a^2*b*d^2*( 
369*c^2*f^2-532*c*d*e*f+105*d^2*e^2)+a*b^2*c*d*(744*c^2*f^2-1232*c*d*e*f+4 
55*d^2*e^2))*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2) 
,(1-b*c/a/d)^(1/2))/b/c^(1/2)/d^(9/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x 
^2+c)^(1/2)+2/105*c^(1/2)*(15*a^2*d^2*f*(-5*c*f+7*d*e)-2*b^2*c*(48*c^2*f^2 
-84*c*d*e*f+35*d^2*e^2)+a*b*d*(174*c^2*f^2-287*c*d*e*f+105*d^2*e^2))*(b*x^ 
2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/d^ 
(9/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.70 (sec) , antiderivative size = 545, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (15 a^2 d^2 \left (7 d^2 e^2+10 c^2 f^2+c d f \left (-14 e+3 f x^2\right )\right )+b^2 c \left (192 c^3 f^2+48 c^2 d f \left (-7 e+f x^2\right )+4 c d^2 \left (35 e^2-21 e f x^2-6 f^2 x^4\right )+d^3 x^2 \left (35 e^2+42 e f x^2+15 f^2 x^4\right )\right )+a b c d \left (-348 c^2 f^2+c d f \left (574 e-93 f x^2\right )+d^2 \left (-210 e^2+154 e f x^2+45 f^2 x^4\right )\right )\right )-i c \left (15 a^3 c d^3 f^2+a^2 b d^2 \left (-105 d^2 e^2+532 c d e f-369 c^2 f^2\right )-8 b^3 c^2 \left (35 d^2 e^2-84 c d e f+48 c^2 f^2\right )+a b^2 c d \left (455 d^2 e^2-1232 c d e f+744 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) \left (15 a^2 d^2 f (-14 d e+11 c f)+a b d \left (-315 d^2 e^2+896 c d e f-552 c^2 f^2\right )+8 b^2 c \left (35 d^2 e^2-84 c d e f+48 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 \sqrt {\frac {b}{a}} c d^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[((a + b*x^2)^(5/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(15*a^2*d^2*(7*d^2*e^2 + 10*c^2*f^2 + c*d*f*(-1 
4*e + 3*f*x^2)) + b^2*c*(192*c^3*f^2 + 48*c^2*d*f*(-7*e + f*x^2) + 4*c*d^2 
*(35*e^2 - 21*e*f*x^2 - 6*f^2*x^4) + d^3*x^2*(35*e^2 + 42*e*f*x^2 + 15*f^2 
*x^4)) + a*b*c*d*(-348*c^2*f^2 + c*d*f*(574*e - 93*f*x^2) + d^2*(-210*e^2 
+ 154*e*f*x^2 + 45*f^2*x^4))) - I*c*(15*a^3*c*d^3*f^2 + a^2*b*d^2*(-105*d^ 
2*e^2 + 532*c*d*e*f - 369*c^2*f^2) - 8*b^3*c^2*(35*d^2*e^2 - 84*c*d*e*f + 
48*c^2*f^2) + a*b^2*c*d*(455*d^2*e^2 - 1232*c*d*e*f + 744*c^2*f^2))*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*(-(b*c) + a*d)*(15*a^2*d^2*f*(-14*d*e + 11*c*f) + a*b*d*(-315 
*d^2*e^2 + 896*c*d*e*f - 552*c^2*f^2) + 8*b^2*c*(35*d^2*e^2 - 84*c*d*e*f + 
 48*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[ 
Sqrt[b/a]*x], (a*d)/(b*c)])/(105*Sqrt[b/a]*c*d^5*Sqrt[a + b*x^2]*Sqrt[c + 
d*x^2])
 

Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 1231, normalized size of antiderivative = 1.89, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {433, 2009}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 433

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {8 b f^2 \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x^3}{7 d^2}-\frac {3 b (16 b c-15 a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 d^3}-\frac {f^2 \left (b x^2+a\right )^{5/2} x^3}{d \sqrt {d x^2+c}}+\frac {12 b e f \left (b x^2+a\right )^{3/2} \sqrt {d x^2+c} x}{5 d^2}+\frac {b (4 b c-3 a d) e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 c d^2}+\frac {2 \left (32 b^2 c^2-58 a b d c+25 a^2 d^2\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{35 d^4}-\frac {2 b (24 b c-23 a d) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 d^3}-\frac {2 e f \left (b x^2+a\right )^{5/2} x}{d \sqrt {d x^2+c}}-\frac {(b c-a d) e^2 \left (b x^2+a\right )^{3/2} x}{c d \sqrt {d x^2+c}}-\frac {\left (8 b^2 c^2-13 a b d c+3 a^2 d^2\right ) e^2 \sqrt {b x^2+a} x}{3 c d^2 \sqrt {d x^2+c}}-\frac {\left (128 b^3 c^3-248 a b^2 d c^2+123 a^2 b d^2 c-5 a^3 d^3\right ) f^2 \sqrt {b x^2+a} x}{35 b d^4 \sqrt {d x^2+c}}+\frac {4 \left (24 b^2 c^2-44 a b d c+19 a^2 d^2\right ) e f \sqrt {b x^2+a} x}{15 d^3 \sqrt {d x^2+c}}+\frac {\left (8 b^2 c^2-13 a b d c+3 a^2 d^2\right ) e^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 \sqrt {c} d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {c} \left (128 b^3 c^3-248 a b^2 d c^2+123 a^2 b d^2 c-5 a^3 d^3\right ) f^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{35 b d^{9/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {4 \sqrt {c} \left (24 b^2 c^2-44 a b d c+19 a^2 d^2\right ) e f \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 d^{7/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 b \sqrt {c} (2 b c-3 a d) e^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 c^{3/2} \left (32 b^2 c^2-58 a b d c+25 a^2 d^2\right ) f^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{35 d^{9/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 \sqrt {c} \left (24 b^2 c^2-41 a b d c+15 a^2 d^2\right ) e f \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{7/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\)

Input:

Int[((a + b*x^2)^(5/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x]
 

Output:

-1/3*((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*e^2*x*Sqrt[a + b*x^2])/(c*d^2*S 
qrt[c + d*x^2]) + (4*(24*b^2*c^2 - 44*a*b*c*d + 19*a^2*d^2)*e*f*x*Sqrt[a + 
 b*x^2])/(15*d^3*Sqrt[c + d*x^2]) - ((128*b^3*c^3 - 248*a*b^2*c^2*d + 123* 
a^2*b*c*d^2 - 5*a^3*d^3)*f^2*x*Sqrt[a + b*x^2])/(35*b*d^4*Sqrt[c + d*x^2]) 
 - ((b*c - a*d)*e^2*x*(a + b*x^2)^(3/2))/(c*d*Sqrt[c + d*x^2]) - (2*e*f*x* 
(a + b*x^2)^(5/2))/(d*Sqrt[c + d*x^2]) - (f^2*x^3*(a + b*x^2)^(5/2))/(d*Sq 
rt[c + d*x^2]) + (b*(4*b*c - 3*a*d)*e^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) 
/(3*c*d^2) - (2*b*(24*b*c - 23*a*d)*e*f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) 
/(15*d^3) + (2*(32*b^2*c^2 - 58*a*b*c*d + 25*a^2*d^2)*f^2*x*Sqrt[a + b*x^2 
]*Sqrt[c + d*x^2])/(35*d^4) - (3*b*(16*b*c - 15*a*d)*f^2*x^3*Sqrt[a + b*x^ 
2]*Sqrt[c + d*x^2])/(35*d^3) + (12*b*e*f*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^ 
2])/(5*d^2) + (8*b*f^2*x^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(7*d^2) + (( 
8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*e^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[( 
Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]*d^(5/2)*Sqrt[(c*(a + b*x 
^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (4*Sqrt[c]*(24*b^2*c^2 - 44*a*b*c 
*d + 19*a^2*d^2)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]] 
, 1 - (b*c)/(a*d)])/(15*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt 
[c + d*x^2]) + (Sqrt[c]*(128*b^3*c^3 - 248*a*b^2*c^2*d + 123*a^2*b*c*d^2 - 
 5*a^3*d^3)*f^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - 
 (b*c)/(a*d)])/(35*b*d^(9/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt...
 

Defintions of rubi rules used

rule 433

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 28.41 (sec) , antiderivative size = 1083, normalized size of antiderivative = 1.66

method	result	size
risch	\(\text {Expression too large to display}\)	\(1083\)
elliptic	\(\text {Expression too large to display}\)	\(1639\)
default	\(\text {Expression too large to display}\)	\(2029\)

Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/105*x*(15*b^2*d^2*f^2*x^4+45*a*b*d^2*f^2*x^2-39*b^2*c*d*f^2*x^2+42*b^2*d 
^2*e*f*x^2+45*a^2*d^2*f^2-138*a*b*c*d*f^2+154*a*b*d^2*e*f+87*b^2*c^2*f^2-1 
26*b^2*c*d*e*f+35*b^2*d^2*e^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d^4+1/105/d 
^4*(-(15*a^3*d^3*f^2-264*a^2*b*c*d^2*f^2+322*a^2*b*d^3*e*f+534*a*b^2*c^2*d 
*f^2-812*a*b^2*c*d^2*e*f+245*a*b^2*d^3*e^2-279*b^3*c^3*f^2+462*b^3*c^2*d*e 
*f-175*b^3*c*d^2*e^2)*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)))+105*(a^ 
3*c^2*d^3*f^2-2*a^3*c*d^4*e*f+a^3*d^5*e^2-3*a^2*b*c^3*d^2*f^2+6*a^2*b*c^2* 
d^3*e*f-3*a^2*b*c*d^4*e^2+3*a*b^2*c^4*d*f^2-6*a*b^2*c^3*d^2*e*f+3*a*b^2*c^ 
2*d^3*e^2-b^3*c^5*f^2+2*b^3*c^4*d*e*f-b^3*c^3*d^2*e^2)/d*((b*d*x^2+a*d)/c/ 
(a*d-b*c)*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(1/c-1/(a*d-b*c)/c*a*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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+b/(a*d-b*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)*(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))))-(150*a^3*c*d^3*f^2-210*a^3*d^4*e*f-453 
*a^2*b*c^2*d^2*f^2+784*a^2*b*c*d^3*e*f-315*a^2*b*d^4*e^2+402*a*b^2*c^3*d*f 
^2-756*a*b^2*c^2*d^2*e*f+350*a*b^2*c*d^3*e^2-105*b^3*c^4*f^2+210*b^3*c^3*d 
*e*f-105*b^3*c^2*d^2*e^2)/d/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^...
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 1184, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="fricas 
")
 

Output:

1/105*(((35*(8*b^3*c^3*d^3 - 13*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5)*e^2 - 28*(2 
4*b^3*c^4*d^2 - 44*a*b^2*c^3*d^3 + 19*a^2*b*c^2*d^4)*e*f + 3*(128*b^3*c^5* 
d - 248*a*b^2*c^4*d^2 + 123*a^2*b*c^3*d^3 - 5*a^3*c^2*d^4)*f^2)*x^3 + (35* 
(8*b^3*c^4*d^2 - 13*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4)*e^2 - 28*(24*b^3*c^5* 
d - 44*a*b^2*c^4*d^2 + 19*a^2*b*c^3*d^3)*e*f + 3*(128*b^3*c^6 - 248*a*b^2* 
c^5*d + 123*a^2*b*c^4*d^2 - 5*a^3*c^3*d^3)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*el 
liptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((35*(8*b^3*c^3*d^3 - 13*a*b^2 
*c^2*d^4 - 6*a^2*b*d^6 + (3*a^2*b + 4*a*b^2)*c*d^5)*e^2 - 14*(48*b^3*c^4*d 
^2 - 88*a*b^2*c^3*d^3 - 41*a^2*b*c*d^5 + 15*a^3*d^6 + 2*(19*a^2*b + 12*a*b 
^2)*c^2*d^4)*e*f + 3*(128*b^3*c^5*d - 248*a*b^2*c^4*d^2 + 50*a^3*c*d^5 + ( 
123*a^2*b + 64*a*b^2)*c^3*d^3 - (5*a^3 + 116*a^2*b)*c^2*d^4)*f^2)*x^3 + (3 
5*(8*b^3*c^4*d^2 - 13*a*b^2*c^3*d^3 - 6*a^2*b*c*d^5 + (3*a^2*b + 4*a*b^2)* 
c^2*d^4)*e^2 - 14*(48*b^3*c^5*d - 88*a*b^2*c^4*d^2 - 41*a^2*b*c^2*d^4 + 15 
*a^3*c*d^5 + 2*(19*a^2*b + 12*a*b^2)*c^3*d^3)*e*f + 3*(128*b^3*c^6 - 248*a 
*b^2*c^5*d + 50*a^3*c^2*d^4 + (123*a^2*b + 64*a*b^2)*c^4*d^2 - (5*a^3 + 11 
6*a^2*b)*c^3*d^3)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d 
)/x), a*d/(b*c)) + (15*b^3*c*d^5*f^2*x^8 + 3*(14*b^3*c*d^5*e*f - (8*b^3*c^ 
2*d^4 - 15*a*b^2*c*d^5)*f^2)*x^6 + (35*b^3*c*d^5*e^2 - 14*(6*b^3*c^2*d^4 - 
 11*a*b^2*c*d^5)*e*f + 3*(16*b^3*c^3*d^3 - 31*a*b^2*c^2*d^4 + 15*a^2*b*c*d 
^5)*f^2)*x^4 - 35*(8*b^3*c^3*d^3 - 13*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5)*e^...
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((b*x**2+a)**(5/2)*(f*x**2+e)**2/(d*x**2+c)**(3/2),x)
 

Output:

Integral((a + b*x**2)**(5/2)*(e + f*x**2)**2/(c + d*x**2)**(3/2), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="maxima 
")
 

Output:

integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)^2/(d*x^2 + c)^(3/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)^2/(d*x^2 + c)^(3/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,{\left (f\,x^2+e\right )}^2}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x)
 

Output:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x)
 

Output:

( - 135*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*c*d**2*f**2*x + 210*sqrt(c 
+ d*x**2)*sqrt(a + b*x**2)*a**3*d**3*e*f*x + 279*sqrt(c + d*x**2)*sqrt(a + 
 b*x**2)*a**2*b*c**2*d*f**2*x - 462*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2 
*b*c*d**2*e*f*x + 90*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c*d**2*f**2* 
x**3 + 315*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d**3*e**2*x - 144*sqrt 
(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**3*f**2*x + 252*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*a*b**2*c**2*d*e*f*x - 186*sqrt(c + d*x**2)*sqrt(a + b*x**2) 
*a*b**2*c**2*d*f**2*x**3 - 105*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c* 
d**2*e**2*x + 308*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*e*f*x**3 
 + 90*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*f**2*x**5 + 96*sqrt( 
c + d*x**2)*sqrt(a + b*x**2)*b**3*c**3*f**2*x**3 - 168*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*b**3*c**2*d*e*f*x**3 - 48*sqrt(c + d*x**2)*sqrt(a + b*x**2) 
*b**3*c**2*d*f**2*x**5 + 70*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2* 
e**2*x**3 + 84*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e*f*x**5 + 30 
*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*f**2*x**7 + 165*int((sqrt(c 
 + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b 
*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**3*b*c**2*d**3*f**2 - 210*in 
t((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2 
*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**3*b*c*d**4*e*f + 1 
65*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 ...