\(\int \frac {(a+b x^2)^{5/2} (e+f x^2)^2}{(c+d x^2)^{9/2}} \, dx\) [73]

Optimal result

Integrand size = 32, antiderivative size = 791 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=-\frac {\left (2 a b c d (d e-c f)^2-a^2 d^2 (d e-c f)^2-b^2 c^2 \left (d^2 e^2-2 c d e f+8 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{7 c d^4 \left (c+d x^2\right )^{7/2}}+\frac {b^2 f^2 x^7 \sqrt {a+b x^2}}{d \left (c+d x^2\right )^{7/2}}+\frac {\left (2 a^2 d^2 \left (3 d^2 e^2+c d e f-4 c^2 f^2\right )+a b c d \left (3 d^2 e^2-34 c d e f+31 c^2 f^2\right )-b^2 c^2 \left (9 d^2 e^2-32 c d e f+128 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{35 c^2 d^4 \left (c+d x^2\right )^{5/2}}+\frac {\left (a b c d \left (13 d^2 e^2+16 c d e f-99 c^2 f^2\right )+a^2 d^2 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+2 b^2 c^2 \left (4 d^2 e^2-57 c d e f+228 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{105 c^3 d^4 \left (c+d x^2\right )^{3/2}}+\frac {\left (8 b^3 c^3 \left (d^2 e^2+12 c d e f-48 c^2 f^2\right )+a^2 b c d^2 \left (16 d^2 e^2-18 c d e f-33 c^2 f^2\right )-2 a^3 d^3 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+a b^2 c^2 d \left (9 d^2 e^2-32 c d e f+408 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 c^{7/2} d^{9/2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \left (4 b^2 c^2 \left (d^2 e^2+12 c d e f-48 c^2 f^2\right )-a^2 d^2 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+5 a b c d \left (d^2 e^2-2 c d e f+36 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 c^{5/2} d^{9/2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

-1/7*(2*a*b*c*d*(-c*f+d*e)^2-a^2*d^2*(-c*f+d*e)^2-b^2*c^2*(8*c^2*f^2-2*c*d 
*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)/c/d^4/(d*x^2+c)^(7/2)+b^2*f^2*x^7*(b*x^2+ 
a)^(1/2)/d/(d*x^2+c)^(7/2)+1/35*(2*a^2*d^2*(-4*c^2*f^2+c*d*e*f+3*d^2*e^2)+ 
a*b*c*d*(31*c^2*f^2-34*c*d*e*f+3*d^2*e^2)-b^2*c^2*(128*c^2*f^2-32*c*d*e*f+ 
9*d^2*e^2))*x*(b*x^2+a)^(1/2)/c^2/d^4/(d*x^2+c)^(5/2)+1/105*(a*b*c*d*(-99* 
c^2*f^2+16*c*d*e*f+13*d^2*e^2)+a^2*d^2*(3*c^2*f^2+8*c*d*e*f+24*d^2*e^2)+2* 
b^2*c^2*(228*c^2*f^2-57*c*d*e*f+4*d^2*e^2))*x*(b*x^2+a)^(1/2)/c^3/d^4/(d*x 
^2+c)^(3/2)+1/105*(8*b^3*c^3*(-48*c^2*f^2+12*c*d*e*f+d^2*e^2)+a^2*b*c*d^2* 
(-33*c^2*f^2-18*c*d*e*f+16*d^2*e^2)-2*a^3*d^3*(3*c^2*f^2+8*c*d*e*f+24*d^2* 
e^2)+a*b^2*c^2*d*(408*c^2*f^2-32*c*d*e*f+9*d^2*e^2))*(b*x^2+a)^(1/2)*Ellip 
ticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/c^(7/2)/d^(9/2 
)/(-a*d+b*c)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/105*b*(4*b^ 
2*c^2*(-48*c^2*f^2+12*c*d*e*f+d^2*e^2)-a^2*d^2*(3*c^2*f^2+8*c*d*e*f+24*d^2 
*e^2)+5*a*b*c*d*(36*c^2*f^2-2*c*d*e*f+d^2*e^2))*(b*x^2+a)^(1/2)*InverseJac 
obiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/c^(5/2)/d^(9/2)/(-a*d+b 
*c)/(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 = 8.77 (sec) , antiderivative size = 724, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (15 c^3 (b c-a d)^3 (d e-c f)^2-3 c^2 (b c-a d)^2 (d e-c f) (b c (9 d e-23 c f)+2 a d (3 d e+4 c f)) \left (c+d x^2\right )+c (b c-a d) \left (a b c d \left (13 d^2 e^2+16 c d e f-99 c^2 f^2\right )+a^2 d^2 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+b^2 c^2 \left (8 d^2 e^2-114 c d e f+141 c^2 f^2\right )\right ) \left (c+d x^2\right )^2-\left (a b^2 c^2 d \left (-9 d^2 e^2+32 c d e f-303 c^2 f^2\right )+2 a^3 d^3 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+a^2 b c d^2 \left (-16 d^2 e^2+18 c d e f+33 c^2 f^2\right )+b^3 c^3 \left (-8 d^2 e^2-96 c d e f+279 c^2 f^2\right )\right ) \left (c+d x^2\right )^3\right )-i b c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^3 \sqrt {1+\frac {d x^2}{c}} \left (\left (a b^2 c^2 d \left (-9 d^2 e^2+32 c d e f-408 c^2 f^2\right )+2 a^3 d^3 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+a^2 b c d^2 \left (-16 d^2 e^2+18 c d e f+33 c^2 f^2\right )+8 b^3 c^3 \left (-d^2 e^2-12 c d e f+48 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (-a^2 d^2 \left (24 d^2 e^2+8 c d e f+3 c^2 f^2\right )+8 b^2 c^2 \left (-d^2 e^2-12 c d e f+48 c^2 f^2\right )-a b c d \left (13 d^2 e^2+16 c d e f+216 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{105 \sqrt {\frac {b}{a}} c^4 d^5 (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \] Input:

Integrate[((a + b*x^2)^(5/2)*(e + f*x^2)^2)/(c + d*x^2)^(9/2),x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(15*c^3*(b*c - a*d)^3*(d*e - c*f)^2 - 3*c^2*(b* 
c - a*d)^2*(d*e - c*f)*(b*c*(9*d*e - 23*c*f) + 2*a*d*(3*d*e + 4*c*f))*(c + 
 d*x^2) + c*(b*c - a*d)*(a*b*c*d*(13*d^2*e^2 + 16*c*d*e*f - 99*c^2*f^2) + 
a^2*d^2*(24*d^2*e^2 + 8*c*d*e*f + 3*c^2*f^2) + b^2*c^2*(8*d^2*e^2 - 114*c* 
d*e*f + 141*c^2*f^2))*(c + d*x^2)^2 - (a*b^2*c^2*d*(-9*d^2*e^2 + 32*c*d*e* 
f - 303*c^2*f^2) + 2*a^3*d^3*(24*d^2*e^2 + 8*c*d*e*f + 3*c^2*f^2) + a^2*b* 
c*d^2*(-16*d^2*e^2 + 18*c*d*e*f + 33*c^2*f^2) + b^3*c^3*(-8*d^2*e^2 - 96*c 
*d*e*f + 279*c^2*f^2))*(c + d*x^2)^3) - I*b*c*Sqrt[1 + (b*x^2)/a]*(c + d*x 
^2)^3*Sqrt[1 + (d*x^2)/c]*((a*b^2*c^2*d*(-9*d^2*e^2 + 32*c*d*e*f - 408*c^2 
*f^2) + 2*a^3*d^3*(24*d^2*e^2 + 8*c*d*e*f + 3*c^2*f^2) + a^2*b*c*d^2*(-16* 
d^2*e^2 + 18*c*d*e*f + 33*c^2*f^2) + 8*b^3*c^3*(-(d^2*e^2) - 12*c*d*e*f + 
48*c^2*f^2))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (b*c - a*d)* 
(-(a^2*d^2*(24*d^2*e^2 + 8*c*d*e*f + 3*c^2*f^2)) + 8*b^2*c^2*(-(d^2*e^2) - 
 12*c*d*e*f + 48*c^2*f^2) - a*b*c*d*(13*d^2*e^2 + 16*c*d*e*f + 216*c^2*f^2 
))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(105*Sqrt[b/a]*c^4*d^5 
*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^(7/2))
 

Rubi [A] (verified)

Time = 2.03 (sec) , antiderivative size = 1366, normalized size of antiderivative = 1.73, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 433

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (\frac {2 d a^2}{c}+9 b a-\frac {16 b^2 c}{d}\right ) f^2 \sqrt {b x^2+a} x^3}{35 c d^2 \left (d x^2+c\right )^{3/2}}-\frac {(8 b c-3 a d) f^2 \left (b x^2+a\right )^{3/2} x^3}{35 c d^2 \left (d x^2+c\right )^{5/2}}-\frac {f^2 \left (b x^2+a\right )^{5/2} x^3}{7 d \left (d x^2+c\right )^{7/2}}-\frac {b \left (64 b^2 c^2-60 a b d c+a^2 d^2\right ) f^2 \sqrt {b x^2+a} x}{35 c d^4 (b c-a d) \sqrt {d x^2+c}}+\frac {\left (128 b^3 c^3-136 a b^2 d c^2+11 a^2 b d^2 c+2 a^3 d^3\right ) f^2 \sqrt {b x^2+a} x}{35 c^2 d^4 (b c-a d) \sqrt {d x^2+c}}+\frac {\left (8 b^2 c^2+13 a b d c+24 a^2 d^2\right ) e^2 \sqrt {b x^2+a} x}{105 c^3 d^2 \left (d x^2+c\right )^{3/2}}+\frac {2 \left (\frac {4 d a^2}{c}+5 b a-\frac {24 b^2 c}{d}\right ) e f \sqrt {b x^2+a} x}{105 c d^2 \left (d x^2+c\right )^{3/2}}-\frac {2 (6 b c-a d) e f \left (b x^2+a\right )^{3/2} x}{35 c d^2 \left (d x^2+c\right )^{5/2}}-\frac {2 (b c-a d) (2 b c+3 a d) e^2 \sqrt {b x^2+a} x}{35 c^2 d^2 \left (d x^2+c\right )^{5/2}}-\frac {2 e f \left (b x^2+a\right )^{5/2} x}{7 d \left (d x^2+c\right )^{7/2}}-\frac {(b c-a d) e^2 \left (b x^2+a\right )^{3/2} x}{7 c d \left (d x^2+c\right )^{7/2}}+\frac {\left (8 b^3 c^3+9 a b^2 d c^2+16 a^2 b d^2 c-48 a^3 d^3\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 )}{105 c^{7/2} d^{5/2} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\left (128 b^3 c^3-136 a b^2 d c^2+11 a^2 b d^2 c+2 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 c^{3/2} d^{9/2} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 \left (48 b^3 c^3-16 a b^2 d c^2-9 a^2 b d^2 c-8 a^3 d^3\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 )}{105 c^{5/2} d^{7/2} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \left (4 b^2 c^2+5 a b d c-24 a^2 d^2\right ) 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 )}{105 c^{5/2} d^{5/2} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {b \left (64 b^2 c^2-60 a b d c+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 \sqrt {c} d^{9/2} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 b \left (24 b^2 c^2-5 a b d c-4 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 )}{105 c^{3/2} d^{7/2} (b c-a d) \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)^(9/2),x]
 

Output:

-1/7*((b*c - a*d)*e^2*x*(a + b*x^2)^(3/2))/(c*d*(c + d*x^2)^(7/2)) - (2*e* 
f*x*(a + b*x^2)^(5/2))/(7*d*(c + d*x^2)^(7/2)) - (f^2*x^3*(a + b*x^2)^(5/2 
))/(7*d*(c + d*x^2)^(7/2)) - (2*(b*c - a*d)*(2*b*c + 3*a*d)*e^2*x*Sqrt[a + 
 b*x^2])/(35*c^2*d^2*(c + d*x^2)^(5/2)) - (2*(6*b*c - a*d)*e*f*x*(a + b*x^ 
2)^(3/2))/(35*c*d^2*(c + d*x^2)^(5/2)) - ((8*b*c - 3*a*d)*f^2*x^3*(a + b*x 
^2)^(3/2))/(35*c*d^2*(c + d*x^2)^(5/2)) + ((8*b^2*c^2 + 13*a*b*c*d + 24*a^ 
2*d^2)*e^2*x*Sqrt[a + b*x^2])/(105*c^3*d^2*(c + d*x^2)^(3/2)) + (2*(5*a*b 
- (24*b^2*c)/d + (4*a^2*d)/c)*e*f*x*Sqrt[a + b*x^2])/(105*c*d^2*(c + d*x^2 
)^(3/2)) + ((9*a*b - (16*b^2*c)/d + (2*a^2*d)/c)*f^2*x^3*Sqrt[a + b*x^2])/ 
(35*c*d^2*(c + d*x^2)^(3/2)) - (b*(64*b^2*c^2 - 60*a*b*c*d + a^2*d^2)*f^2* 
x*Sqrt[a + b*x^2])/(35*c*d^4*(b*c - a*d)*Sqrt[c + d*x^2]) + ((128*b^3*c^3 
- 136*a*b^2*c^2*d + 11*a^2*b*c*d^2 + 2*a^3*d^3)*f^2*x*Sqrt[a + b*x^2])/(35 
*c^2*d^4*(b*c - a*d)*Sqrt[c + d*x^2]) + ((8*b^3*c^3 + 9*a*b^2*c^2*d + 16*a 
^2*b*c*d^2 - 48*a^3*d^3)*e^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/ 
Sqrt[c]], 1 - (b*c)/(a*d)])/(105*c^(7/2)*d^(5/2)*(b*c - a*d)*Sqrt[(c*(a + 
b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*(48*b^3*c^3 - 16*a*b^2*c^2* 
d - 9*a^2*b*c*d^2 - 8*a^3*d^3)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[ 
d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*c^(5/2)*d^(7/2)*(b*c - a*d)*Sqrt[(c 
*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - ((128*b^3*c^3 - 136*a*b^ 
2*c^2*d + 11*a^2*b*c*d^2 + 2*a^3*d^3)*f^2*Sqrt[a + b*x^2]*EllipticE[Arc...
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1572\) vs. \(2(752)=1504\).

Time = 12.36 (sec) , antiderivative size = 1573, normalized size of antiderivative = 1.99

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1573\)
default	\(\text {Expression too large to display}\)	\(7943\)

Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/7*(a^2*c^2* 
d^2*f^2-2*a^2*c*d^3*e*f+a^2*d^4*e^2-2*a*b*c^3*d*f^2+4*a*b*c^2*d^2*e*f-2*a* 
b*c*d^3*e^2+b^2*c^4*f^2-2*b^2*c^3*d*e*f+b^2*c^2*d^2*e^2)/c/d^8*x*(b*d*x^4+ 
a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^4-1/35*(8*a^2*c^2*d^2*f^2-2*a^2*c*d^3 
*e*f-6*a^2*d^4*e^2-31*a*b*c^3*d*f^2+34*a*b*c^2*d^2*e*f-3*a*b*c*d^3*e^2+23* 
b^2*c^4*f^2-32*b^2*c^3*d*e*f+9*b^2*c^2*d^2*e^2)/c^2/d^7*x*(b*d*x^4+a*d*x^2 
+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^3+1/105*(3*a^2*c^2*d^2*f^2+8*a^2*c*d^3*e*f+2 
4*a^2*d^4*e^2-99*a*b*c^3*d*f^2+16*a*b*c^2*d^2*e*f+13*a*b*c*d^3*e^2+141*b^2 
*c^4*f^2-114*b^2*c^3*d*e*f+8*b^2*c^2*d^2*e^2)/c^3/d^6*x*(b*d*x^4+a*d*x^2+b 
*c*x^2+a*c)^(1/2)/(x^2+c/d)^2+1/105*(b*d*x^2+a*d)/c^4/d^5/(a*d-b*c)*x*(6*a 
^3*c^2*d^3*f^2+16*a^3*c*d^4*e*f+48*a^3*d^5*e^2+33*a^2*b*c^3*d^2*f^2+18*a^2 
*b*c^2*d^3*e*f-16*a^2*b*c*d^4*e^2-303*a*b^2*c^4*d*f^2+32*a*b^2*c^3*d^2*e*f 
-9*a*b^2*c^2*d^3*e^2+279*b^3*c^5*f^2-96*b^3*c^4*d*e*f-8*b^3*c^3*d^2*e^2)/( 
(x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(b^2*f*(3*a*d*f-4*b*c*f+2*b*d*e)/d^5+1/105* 
b*(3*a^2*c^2*d^2*f^2+8*a^2*c*d^3*e*f+24*a^2*d^4*e^2-99*a*b*c^3*d*f^2+16*a* 
b*c^2*d^2*e*f+13*a*b*c*d^3*e^2+141*b^2*c^4*f^2-114*b^2*c^3*d*e*f+8*b^2*c^2 
*d^2*e^2)/c^3/d^5+1/105/d^5*(6*a^3*c^2*d^3*f^2+16*a^3*c*d^4*e*f+48*a^3*d^5 
*e^2+33*a^2*b*c^3*d^2*f^2+18*a^2*b*c^2*d^3*e*f-16*a^2*b*c*d^4*e^2-303*a*b^ 
2*c^4*d*f^2+32*a*b^2*c^3*d^2*e*f-9*a*b^2*c^2*d^3*e^2+279*b^3*c^5*f^2-96*b^ 
3*c^4*d*e*f-8*b^3*c^3*d^2*e^2)/c^4-1/105*a/d^4/c^4/(a*d-b*c)*(6*a^3*c^2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2837 vs. \(2 (752) = 1504\).

Time = 0.21 (sec) , antiderivative size = 2837, normalized size of antiderivative = 3.59 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(9/2),x, algorithm="fricas 
")
 

Output:

1/105*((((8*b^3*c^4*d^6 + 9*a*b^2*c^3*d^7 + 16*a^2*b*c^2*d^8 - 48*a^3*c*d^ 
9)*e^2 + 2*(48*b^3*c^5*d^5 - 16*a*b^2*c^4*d^6 - 9*a^2*b*c^3*d^7 - 8*a^3*c^ 
2*d^8)*e*f - 3*(128*b^3*c^6*d^4 - 136*a*b^2*c^5*d^5 + 11*a^2*b*c^4*d^6 + 2 
*a^3*c^3*d^7)*f^2)*x^9 + 4*((8*b^3*c^5*d^5 + 9*a*b^2*c^4*d^6 + 16*a^2*b*c^ 
3*d^7 - 48*a^3*c^2*d^8)*e^2 + 2*(48*b^3*c^6*d^4 - 16*a*b^2*c^5*d^5 - 9*a^2 
*b*c^4*d^6 - 8*a^3*c^3*d^7)*e*f - 3*(128*b^3*c^7*d^3 - 136*a*b^2*c^6*d^4 + 
 11*a^2*b*c^5*d^5 + 2*a^3*c^4*d^6)*f^2)*x^7 + 6*((8*b^3*c^6*d^4 + 9*a*b^2* 
c^5*d^5 + 16*a^2*b*c^4*d^6 - 48*a^3*c^3*d^7)*e^2 + 2*(48*b^3*c^7*d^3 - 16* 
a*b^2*c^6*d^4 - 9*a^2*b*c^5*d^5 - 8*a^3*c^4*d^6)*e*f - 3*(128*b^3*c^8*d^2 
- 136*a*b^2*c^7*d^3 + 11*a^2*b*c^6*d^4 + 2*a^3*c^5*d^5)*f^2)*x^5 + 4*((8*b 
^3*c^7*d^3 + 9*a*b^2*c^6*d^4 + 16*a^2*b*c^5*d^5 - 48*a^3*c^4*d^6)*e^2 + 2* 
(48*b^3*c^8*d^2 - 16*a*b^2*c^7*d^3 - 9*a^2*b*c^6*d^4 - 8*a^3*c^5*d^5)*e*f 
- 3*(128*b^3*c^9*d - 136*a*b^2*c^8*d^2 + 11*a^2*b*c^7*d^3 + 2*a^3*c^6*d^4) 
*f^2)*x^3 + ((8*b^3*c^8*d^2 + 9*a*b^2*c^7*d^3 + 16*a^2*b*c^6*d^4 - 48*a^3* 
c^5*d^5)*e^2 + 2*(48*b^3*c^9*d - 16*a*b^2*c^8*d^2 - 9*a^2*b*c^7*d^3 - 8*a^ 
3*c^6*d^4)*e*f - 3*(128*b^3*c^10 - 136*a*b^2*c^9*d + 11*a^2*b*c^8*d^2 + 2* 
a^3*c^7*d^3)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), 
 a*d/(b*c)) - (((8*b^3*c^4*d^6 + 9*a*b^2*c^3*d^7 - 24*a^3*d^10 + 4*(4*a^2* 
b + a*b^2)*c^2*d^8 - (48*a^3 - 5*a^2*b)*c*d^9)*e^2 + 2*(48*b^3*c^5*d^5 - 1 
6*a*b^2*c^4*d^6 - 4*a^3*c*d^9 - 3*(3*a^2*b - 8*a*b^2)*c^3*d^7 - (8*a^3 ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)*(f*x**2+e)**2/(d*x**2+c)**(9/2),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(9/2),x, algorithm="maxima 
")
 

Output:

integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)^2/(d*x^2 + c)^(9/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(9/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)^2/(d*x^2 + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,{\left (f\,x^2+e\right )}^2}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^2)/(c + d*x^2)^(9/2),x)
 

Output:

int(((a + b*x^2)^(5/2)*(e + f*x^2)^2)/(c + d*x^2)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{9/2}} \, dx=\text {too large to display} \] Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)^2/(d*x^2+c)^(9/2),x)
 

Output:

(9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*c*d**2*f**2*x - 6*sqrt(c + d*x** 
2)*sqrt(a + b*x**2)*a**3*d**3*e*f*x + 18*sqrt(c + d*x**2)*sqrt(a + b*x**2) 
*a**3*d**3*f**2*x**3 - 81*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c**2*d* 
f**2*x - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c*d**2*e*f*x - 168*sqr 
t(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*c*d**2*f**2*x**3 - 9*sqrt(c + d*x**2 
)*sqrt(a + b*x**2)*a**2*b*d**3*e**2*x - 12*sqrt(c + d*x**2)*sqrt(a + b*x** 
2)*a**2*b*d**3*e*f*x**3 - 54*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d**3 
*f**2*x**5 + 144*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**3*f**2*x - 36 
*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*d*e*f*x + 342*sqrt(c + d*x* 
*2)*sqrt(a + b*x**2)*a*b**2*c**2*d*f**2*x**3 - 3*sqrt(c + d*x**2)*sqrt(a + 
 b*x**2)*a*b**2*c*d**2*e**2*x - 68*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b** 
2*c*d**2*e*f*x**3 + 162*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c*d**2*f* 
*2*x**5 - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*e**2*x**3 - 36*s 
qrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**3*e*f*x**5 + 18*sqrt(c + d*x**2 
)*sqrt(a + b*x**2)*a*b**2*d**3*f**2*x**7 - 96*sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*b**3*c**3*f**2*x**3 + 24*sqrt(c + d*x**2)*sqrt(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 + 
 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e**2*x**3 + 12*sqrt(c + d 
*x**2)*sqrt(a + b*x**2)*b**3*c*d**2*e*f*x**5 - 6*sqrt(c + d*x**2)*sqrt(a + 
 b*x**2)*b**3*c*d**2*f**2*x**7 + 270*int((sqrt(c + d*x**2)*sqrt(a + b*x...