\(\int \frac {(a+b x^2)^{5/2} (e+f x^2)}{(c+d x^2)^{9/2}} \, dx\) [20]

Optimal result

Integrand size = 30, antiderivative size = 490 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {(d e-c f) x \left (a+b x^2\right )^{5/2}}{7 c d \left (c+d x^2\right )^{7/2}}+\frac {(a d (6 d e+c f)-b c (d e+6 c f)) x \left (a+b x^2\right )^{3/2}}{35 c^2 d^2 \left (c+d x^2\right )^{5/2}}-\frac {\left (5 a b c d (d e-c f)-4 a^2 d^2 (6 d e+c f)+4 b^2 c^2 (d e+6 c f)\right ) x \sqrt {a+b x^2}}{105 c^3 d^3 \left (c+d x^2\right )^{3/2}}+\frac {\left (a b^2 c^2 d (9 d e-16 c f)+a^2 b c d^2 (16 d e-9 c f)-8 a^3 d^3 (6 d e+c f)+8 b^3 c^3 (d e+6 c f)\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 c^{7/2} d^{7/2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \left (5 a b c d (d e-c f)-4 a^2 d^2 (6 d e+c f)+4 b^2 c^2 (d e+6 c f)\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 c^{5/2} d^{7/2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/7*(-c*f+d*e)*x*(b*x^2+a)^(5/2)/c/d/(d*x^2+c)^(7/2)+1/35*(a*d*(c*f+6*d*e) 
-b*c*(6*c*f+d*e))*x*(b*x^2+a)^(3/2)/c^2/d^2/(d*x^2+c)^(5/2)-1/105*(5*a*b*c 
*d*(-c*f+d*e)-4*a^2*d^2*(c*f+6*d*e)+4*b^2*c^2*(6*c*f+d*e))*x*(b*x^2+a)^(1/ 
2)/c^3/d^3/(d*x^2+c)^(3/2)+1/105*(a*b^2*c^2*d*(-16*c*f+9*d*e)+a^2*b*c*d^2* 
(-9*c*f+16*d*e)-8*a^3*d^3*(c*f+6*d*e)+8*b^3*c^3*(6*c*f+d*e))*(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))/c^(7/2 
)/d^(7/2)/(-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*(5*a*b*c*d*(-c*f+d*e)-4*a^2*d^2*(c*f+6*d*e)+4*b^2*c^2*(6*c*f+d*e))*(b*x 
^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/c 
^(5/2)/d^(7/2)/(-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 = 7.62 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (15 c^3 (b c-a d)^3 (d e-c f)-3 c^2 (b c-a d)^2 (b c (9 d e-16 c f)+a d (6 d e+c f)) \left (c+d x^2\right )+c (b c-a d) \left (b^2 c^2 (8 d e-57 c f)+4 a^2 d^2 (6 d e+c f)+a b c d (13 d e+8 c f)\right ) \left (c+d x^2\right )^2+\left (a b^2 c^2 d (9 d e-16 c f)+a^2 b c d^2 (16 d e-9 c f)-8 a^3 d^3 (6 d e+c f)+8 b^3 c^3 (d e+6 c f)\right ) \left (c+d x^2\right )^3\right )+i b c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^3 \sqrt {1+\frac {d x^2}{c}} \left (\left (a b^2 c^2 d (9 d e-16 c f)+a^2 b c d^2 (16 d e-9 c f)-8 a^3 d^3 (6 d e+c f)+8 b^3 c^3 (d e+6 c f)\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (4 a^2 d^2 (6 d e+c f)+8 b^2 c^2 (d e+6 c f)+a b c d (13 d e+8 c f)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{105 \sqrt {\frac {b}{a}} c^4 d^4 (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^{7/2}} \] Input:

Integrate[((a + b*x^2)^(5/2)*(e + f*x^2))/(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) - 3*c^2*(b*c 
- a*d)^2*(b*c*(9*d*e - 16*c*f) + a*d*(6*d*e + c*f))*(c + d*x^2) + c*(b*c - 
 a*d)*(b^2*c^2*(8*d*e - 57*c*f) + 4*a^2*d^2*(6*d*e + c*f) + a*b*c*d*(13*d* 
e + 8*c*f))*(c + d*x^2)^2 + (a*b^2*c^2*d*(9*d*e - 16*c*f) + a^2*b*c*d^2*(1 
6*d*e - 9*c*f) - 8*a^3*d^3*(6*d*e + c*f) + 8*b^3*c^3*(d*e + 6*c*f))*(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*e - 16*c*f) + a^2*b*c*d^2*(16*d*e - 9*c*f) - 8*a^3*d^3*(6* 
d*e + c*f) + 8*b^3*c^3*(d*e + 6*c*f))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a 
*d)/(b*c)] - (b*c - a*d)*(4*a^2*d^2*(6*d*e + c*f) + 8*b^2*c^2*(d*e + 6*c*f 
) + a*b*c*d*(13*d*e + 8*c*f))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c 
)]))/(105*Sqrt[b/a]*c^4*d^4*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^(7/2))
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {401, 25, 401, 25, 401, 25, 400, 313, 320}

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.

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

\(\Big \downarrow \) 401

\(\displaystyle \frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{7 c d \left (c+d x^2\right )^{7/2}}-\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (b (d e+6 c f) x^2+a (6 d e+c f)\right )}{\left (d x^2+c\right )^{7/2}}dx}{7 c d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (b (d e+6 c f) x^2+a (6 d e+c f)\right )}{\left (d x^2+c\right )^{7/2}}dx}{7 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{7 c d \left (c+d x^2\right )^{7/2}}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+6 d e)-b c (6 c f+d e))}{5 c d \left (c+d x^2\right )^{5/2}}-\frac {\int -\frac {\sqrt {b x^2+a} \left (b (a d (6 d e+c f)+4 b c (d e+6 c f)) x^2+a (4 a d (6 d e+c f)+b c (d e+6 c f))\right )}{\left (d x^2+c\right )^{5/2}}dx}{5 c d}}{7 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{7 c d \left (c+d x^2\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\sqrt {b x^2+a} \left (b (a d (6 d e+c f)+4 b c (d e+6 c f)) x^2+a (4 a d (6 d e+c f)+b c (d e+6 c f))\right )}{\left (d x^2+c\right )^{5/2}}dx}{5 c d}+\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+6 d e)-b c (6 c f+d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{7 c d \left (c+d x^2\right )^{7/2}}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {\frac {-\frac {\int -\frac {b \left (8 b^2 (d e+6 c f) c^2+a b d (13 d e+8 c f) c+4 a^2 d^2 (6 d e+c f)\right ) x^2+a \left (4 b^2 (d e+6 c f) c^2+a b d (8 d e+13 c f) c+8 a^2 d^2 (6 d e+c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} \left (-\frac {4 a^2 d (c f+6 d e)}{c}+5 a b (d e-c f)+\frac {4 b^2 c (6 c f+d e)}{d}\right )}{3 \left (c+d x^2\right )^{3/2}}}{5 c d}+\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+6 d e)-b c (6 c f+d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{7 c d \left (c+d x^2\right )^{7/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {b \left (8 b^2 (d e+6 c f) c^2+a b d (13 d e+8 c f) c+4 a^2 d^2 (6 d e+c f)\right ) x^2+a \left (4 b^2 (d e+6 c f) c^2+a b d (8 d e+13 c f) c+8 a^2 d^2 (6 d e+c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c d}-\frac {x \sqrt {a+b x^2} \left (-\frac {4 a^2 d (c f+6 d e)}{c}+5 a b (d e-c f)+\frac {4 b^2 c (6 c f+d e)}{d}\right )}{3 \left (c+d x^2\right )^{3/2}}}{5 c d}+\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+6 d e)-b c (6 c f+d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{7 c d \left (c+d x^2\right )^{7/2}}\)

\(\Big \downarrow \) 400

\(\displaystyle \frac {\frac {\frac {\frac {\left (-8 a^3 d^3 (c f+6 d e)+a^2 b c d^2 (16 d e-9 c f)+a b^2 c^2 d (9 d e-16 c f)+8 b^3 c^3 (6 c f+d e)\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}-\frac {a b \left (-4 a^2 d^2 (c f+6 d e)+5 a b c d (d e-c f)+4 b^2 c^2 (6 c f+d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c d}-\frac {x \sqrt {a+b x^2} \left (-\frac {4 a^2 d (c f+6 d e)}{c}+5 a b (d e-c f)+\frac {4 b^2 c (6 c f+d e)}{d}\right )}{3 \left (c+d x^2\right )^{3/2}}}{5 c d}+\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+6 d e)-b c (6 c f+d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{7 c d \left (c+d x^2\right )^{7/2}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {\frac {\frac {\sqrt {a+b x^2} \left (-8 a^3 d^3 (c f+6 d e)+a^2 b c d^2 (16 d e-9 c f)+a b^2 c^2 d (9 d e-16 c f)+8 b^3 c^3 (6 c f+d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a b \left (-4 a^2 d^2 (c f+6 d e)+5 a b c d (d e-c f)+4 b^2 c^2 (6 c f+d e)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c d}-\frac {x \sqrt {a+b x^2} \left (-\frac {4 a^2 d (c f+6 d e)}{c}+5 a b (d e-c f)+\frac {4 b^2 c (6 c f+d e)}{d}\right )}{3 \left (c+d x^2\right )^{3/2}}}{5 c d}+\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+6 d e)-b c (6 c f+d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{7 c d \left (c+d x^2\right )^{7/2}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {\frac {\frac {\sqrt {a+b x^2} \left (-8 a^3 d^3 (c f+6 d e)+a^2 b c d^2 (16 d e-9 c f)+a b^2 c^2 d (9 d e-16 c f)+8 b^3 c^3 (6 c f+d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (-4 a^2 d^2 (c f+6 d e)+5 a b c d (d e-c f)+4 b^2 c^2 (6 c f+d e)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c d}-\frac {x \sqrt {a+b x^2} \left (-\frac {4 a^2 d (c f+6 d e)}{c}+5 a b (d e-c f)+\frac {4 b^2 c (6 c f+d e)}{d}\right )}{3 \left (c+d x^2\right )^{3/2}}}{5 c d}+\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+6 d e)-b c (6 c f+d e))}{5 c d \left (c+d x^2\right )^{5/2}}}{7 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{7 c d \left (c+d x^2\right )^{7/2}}\)

Input:

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

Output:

((d*e - c*f)*x*(a + b*x^2)^(5/2))/(7*c*d*(c + d*x^2)^(7/2)) + (((a*d*(6*d* 
e + c*f) - b*c*(d*e + 6*c*f))*x*(a + b*x^2)^(3/2))/(5*c*d*(c + d*x^2)^(5/2 
)) + (-1/3*((5*a*b*(d*e - c*f) - (4*a^2*d*(6*d*e + c*f))/c + (4*b^2*c*(d*e 
 + 6*c*f))/d)*x*Sqrt[a + b*x^2])/(c + d*x^2)^(3/2) + (((a*b^2*c^2*d*(9*d*e 
 - 16*c*f) + a^2*b*c*d^2*(16*d*e - 9*c*f) - 8*a^3*d^3*(6*d*e + c*f) + 8*b^ 
3*c^3*(d*e + 6*c*f))*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]] 
, 1 - (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*( 
c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*(5*a*b*c*d*(d*e - c*f) - 4*a^2* 
d^2*(6*d*e + c*f) + 4*b^2*c^2*(d*e + 6*c*f))*Sqrt[a + b*x^2]*EllipticF[Arc 
Tan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*Sqrt[(c*( 
a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*c*d))/(5*c*d))/(7*c*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 400

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ 
(3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(Sqrt[a + b*x^2]* 
Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[Sqrt[a + b*x^ 
2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & 
& PosQ[d/c]
 

rule 401

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1060\) vs. \(2(455)=910\).

Time = 10.48 (sec) , antiderivative size = 1061, normalized size of antiderivative = 2.17

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1061\)
default	\(\text {Expression too large to display}\)	\(5139\)

Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)/(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*d 
^2*f-a^2*d^3*e-2*a*b*c^2*d*f+2*a*b*c*d^2*e+b^2*c^3*f-b^2*c^2*d*e)/c/d^7*x* 
(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^4+1/35*(a^2*c*d^2*f+6*a^2*d^ 
3*e-17*a*b*c^2*d*f+3*a*b*c*d^2*e+16*b^2*c^3*f-9*b^2*c^2*d*e)/c^2/d^6*x*(b* 
d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^3+1/105*(4*a^2*c*d^2*f+24*a^2*d 
^3*e+8*a*b*c^2*d*f+13*a*b*c*d^2*e-57*b^2*c^3*f+8*b^2*c^2*d*e)/c^3/d^5*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^4/ 
(a*d-b*c)*x*(8*a^3*c*d^3*f+48*a^3*d^4*e+9*a^2*b*c^2*d^2*f-16*a^2*b*c*d^3*e 
+16*a*b^2*c^3*d*f-9*a*b^2*c^2*d^2*e-48*b^3*c^4*f-8*b^3*c^3*d*e)/((x^2+c/d) 
*(b*d*x^2+a*d))^(1/2)+(b^3*f/d^4+1/105*b*(4*a^2*c*d^2*f+24*a^2*d^3*e+8*a*b 
*c^2*d*f+13*a*b*c*d^2*e-57*b^2*c^3*f+8*b^2*c^2*d*e)/c^3/d^4+1/105/d^4*(8*a 
^3*c*d^3*f+48*a^3*d^4*e+9*a^2*b*c^2*d^2*f-16*a^2*b*c*d^3*e+16*a*b^2*c^3*d* 
f-9*a*b^2*c^2*d^2*e-48*b^3*c^4*f-8*b^3*c^3*d*e)/c^4-1/105*a/d^3/c^4/(a*d-b 
*c)*(8*a^3*c*d^3*f+48*a^3*d^4*e+9*a^2*b*c^2*d^2*f-16*a^2*b*c*d^3*e+16*a*b^ 
2*c^3*d*f-9*a*b^2*c^2*d^2*e-48*b^3*c^4*f-8*b^3*c^3*d*e))/(-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)*Ellipt 
icF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+1/105/d^4*b*(8*a^3*c*d^3*f+48 
*a^3*d^4*e+9*a^2*b*c^2*d^2*f-16*a^2*b*c*d^3*e+16*a*b^2*c^3*d*f-9*a*b^2*c^2 
*d^2*e-48*b^3*c^4*f-8*b^3*c^3*d*e)/(a*d-b*c)/c^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)*(EllipticF(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1935 vs. \(2 (455) = 910\).

Time = 0.19 (sec) , antiderivative size = 1935, normalized size of antiderivative = 3.95 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**3*d**2*f*x - 3*sqrt(c + d*x**2) 
*sqrt(a + b*x**2)*a**2*b*c*d*f*x - 9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a** 
2*b*d**2*e*x - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*b*d**2*f*x**3 - 18 
*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*c**2*f*x - 3*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*a*b**2*c*d*e*x - 34*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b** 
2*c*d*f*x**3 - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**2*e*x**3 - 18 
*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b**2*d**2*f*x**5 + 12*sqrt(c + d*x**2 
)*sqrt(a + b*x**2)*b**3*c**2*f*x**3 + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)* 
b**3*c*d*e*x**3 + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**3*c*d*f*x**5 + 45 
*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(3*a**2*c**5*d + 15*a**2*c** 
4*d**2*x**2 + 30*a**2*c**3*d**3*x**4 + 30*a**2*c**2*d**4*x**6 + 15*a**2*c* 
d**5*x**8 + 3*a**2*d**6*x**10 - a*b*c**6 - 2*a*b*c**5*d*x**2 + 5*a*b*c**4* 
d**2*x**4 + 20*a*b*c**3*d**3*x**6 + 25*a*b*c**2*d**4*x**8 + 14*a*b*c*d**5* 
x**10 + 3*a*b*d**6*x**12 - b**2*c**6*x**2 - 5*b**2*c**5*d*x**4 - 10*b**2*c 
**4*d**2*x**6 - 10*b**2*c**3*d**3*x**8 - 5*b**2*c**2*d**4*x**10 - b**2*c*d 
**5*x**12),x)*a**4*b*c**4*d**4*f + 180*int((sqrt(c + d*x**2)*sqrt(a + b*x* 
*2)*x**4)/(3*a**2*c**5*d + 15*a**2*c**4*d**2*x**2 + 30*a**2*c**3*d**3*x**4 
 + 30*a**2*c**2*d**4*x**6 + 15*a**2*c*d**5*x**8 + 3*a**2*d**6*x**10 - a*b* 
c**6 - 2*a*b*c**5*d*x**2 + 5*a*b*c**4*d**2*x**4 + 20*a*b*c**3*d**3*x**6 + 
25*a*b*c**2*d**4*x**8 + 14*a*b*c*d**5*x**10 + 3*a*b*d**6*x**12 - b**2*c...