\(\int \frac {(a+b x^2)^{3/2} (e+f x^2)}{(c+d x^2)^{5/2}} \, dx\) [12]

Optimal result

Integrand size = 30, antiderivative size = 291 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(d e-c f) x \left (a+b x^2\right )^{3/2}}{3 c d \left (c+d x^2\right )^{3/2}}-\frac {b (d e-4 c f) x \sqrt {a+b x^2}}{3 c d^2 \sqrt {c+d x^2}}+\frac {(2 b c (d e-4 c f)+a d (2 d e+c f)) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b (d e-4 c f) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 \sqrt {c} d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/3*(-c*f+d*e)*x*(b*x^2+a)^(3/2)/c/d/(d*x^2+c)^(3/2)-1/3*b*(-4*c*f+d*e)*x* 
(b*x^2+a)^(1/2)/c/d^2/(d*x^2+c)^(1/2)+1/3*(2*b*c*(-4*c*f+d*e)+a*d*(c*f+2*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^(3/2)/d^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2 
)-1/3*b*(-4*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^(1/2)/d^(5/2)/(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 = 6.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (a d^2 \left (3 c e+2 d e x^2+c f x^2\right )+b c \left (-4 c^2 f+2 d^2 e x^2+c d \left (e-5 f x^2\right )\right )\right )+i b c (2 b c (d e-4 c f)+a d (2 d e+c f)) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c (2 b c (d e-4 c f)+a d (d e+5 c f)) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} c^2 d^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \] Input:

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

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(a*d^2*(3*c*e + 2*d*e*x^2 + c*f*x^2) + b*c*(-4* 
c^2*f + 2*d^2*e*x^2 + c*d*(e - 5*f*x^2))) + I*b*c*(2*b*c*(d*e - 4*c*f) + a 
*d*(2*d*e + c*f))*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Elli 
pticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(2*b*c*(d*e - 4*c*f) + 
a*d*(d*e + 5*c*f))*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Ell 
ipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*c^2*d^3*Sqrt[a + 
 b*x^2]*(c + d*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {401, 25, 401, 27, 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.

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {x \sqrt {a+b x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{c d \sqrt {c+d x^2}}-\frac {b \left ((a d (c f+2 d e)+2 b c (d e-4 c f)) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (d e-4 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}}{3 c d}+\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{3 c d \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\frac {x \sqrt {a+b x^2} (a d (c f+2 d e)+b c (d e-4 c f))}{c d \sqrt {c+d x^2}}-\frac {b \left ((a d (c f+2 d e)+2 b c (d e-4 c f)) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (d e-4 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c d}}{3 c d}+\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{3 c d \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 313

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

Input:

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

Output:

((d*e - c*f)*x*(a + b*x^2)^(3/2))/(3*c*d*(c + d*x^2)^(3/2)) + (((b*c*(d*e 
- 4*c*f) + a*d*(2*d*e + c*f))*x*Sqrt[a + b*x^2])/(c*d*Sqrt[c + d*x^2]) - ( 
b*((2*b*c*(d*e - 4*c*f) + a*d*(2*d*e + c*f))*((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))]*Sqr 
t[c + d*x^2])) + (c^(3/2)*(d*e - 4*c*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[( 
Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c 
 + d*x^2))]*Sqrt[c + d*x^2])))/(c*d))/(3*c*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 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 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]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(558\) vs. \(2(262)=524\).

Time = 7.74 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.92

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 c \,d^{4} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (b d \,x^{2}+a d \right ) \left (a c d f +2 a \,d^{2} e -5 b \,c^{2} f +2 b c d e \right ) x}{3 c^{2} d^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {b \left (2 a d f -2 b c f +b d e \right )}{d^{3}}-\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) b}{3 d^{3} c}+\frac {\left (a c d f +2 a \,d^{2} e -5 b \,c^{2} f +2 b c d e \right ) \left (a d -b c \right )}{3 d^{3} c^{2}}-\frac {a \left (a c d f +2 a \,d^{2} e -5 b \,c^{2} f +2 b c d e \right )}{3 d^{2} c^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {b^{2} f}{d^{2}}-\frac {\left (a c d f +2 a \,d^{2} e -5 b \,c^{2} f +2 b c d e \right ) b}{3 d^{2} c^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(559\)
default	\(\text {Expression too large to display}\)	\(1251\)

Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)/(d*x^2+c)^(5/2),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)*(-1/3*(a*c*d*f 
-a*d^2*e-b*c^2*f+b*c*d*e)/c/d^4*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2 
+c/d)^2+1/3*(b*d*x^2+a*d)*(a*c*d*f+2*a*d^2*e-5*b*c^2*f+2*b*c*d*e)/c^2/d^3* 
x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(b*(2*a*d*f-2*b*c*f+b*d*e)/d^3-1/3*(a*c* 
d*f-a*d^2*e-b*c^2*f+b*c*d*e)/d^3*b/c+1/3*(a*c*d*f+2*a*d^2*e-5*b*c^2*f+2*b* 
c*d*e)/d^3*(a*d-b*c)/c^2-1/3*a/d^2*(a*c*d*f+2*a*d^2*e-5*b*c^2*f+2*b*c*d*e) 
/c^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^2*f 
/d^2-1/3*(a*c*d*f+2*a*d^2*e-5*b*c^2*f+2*b*c*d*e)/d^2*b/c^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))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (262) = 524\).

Time = 0.11 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (2 \, {\left (b c^{2} d^{3} + a c d^{4}\right )} e - {\left (8 \, b c^{3} d^{2} - a c^{2} d^{3}\right )} f\right )} x^{5} + 2 \, {\left (2 \, {\left (b c^{3} d^{2} + a c^{2} d^{3}\right )} e - {\left (8 \, b c^{4} d - a c^{3} d^{2}\right )} f\right )} x^{3} + {\left (2 \, {\left (b c^{4} d + a c^{3} d^{2}\right )} e - {\left (8 \, b c^{5} - a c^{4} d\right )} f\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left ({\left (2 \, b c^{2} d^{3} + 2 \, a c d^{4} + a d^{5}\right )} e - {\left (8 \, b c^{3} d^{2} - a c^{2} d^{3} + 4 \, a c d^{4}\right )} f\right )} x^{5} + 2 \, {\left ({\left (2 \, b c^{3} d^{2} + 2 \, a c^{2} d^{3} + a c d^{4}\right )} e - {\left (8 \, b c^{4} d - a c^{3} d^{2} + 4 \, a c^{2} d^{3}\right )} f\right )} x^{3} + {\left ({\left (2 \, b c^{4} d + 2 \, a c^{3} d^{2} + a c^{2} d^{3}\right )} e - {\left (8 \, b c^{5} - a c^{4} d + 4 \, a c^{3} d^{2}\right )} f\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (3 \, b c^{2} d^{3} f x^{4} - {\left ({\left (3 \, b c^{2} d^{3} + a c d^{4}\right )} e - 2 \, {\left (6 \, b c^{3} d^{2} - a c^{2} d^{3}\right )} f\right )} x^{2} - 2 \, {\left (b c^{3} d^{2} + a c^{2} d^{3}\right )} e + {\left (8 \, b c^{4} d - a c^{3} d^{2}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (c^{2} d^{6} x^{5} + 2 \, c^{3} d^{5} x^{3} + c^{4} d^{4} x\right )}} \] Input:

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

Output:

1/3*(((2*(b*c^2*d^3 + a*c*d^4)*e - (8*b*c^3*d^2 - a*c^2*d^3)*f)*x^5 + 2*(2 
*(b*c^3*d^2 + a*c^2*d^3)*e - (8*b*c^4*d - a*c^3*d^2)*f)*x^3 + (2*(b*c^4*d 
+ a*c^3*d^2)*e - (8*b*c^5 - a*c^4*d)*f)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e 
(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (((2*b*c^2*d^3 + 2*a*c*d^4 + a*d^5)*e 
- (8*b*c^3*d^2 - a*c^2*d^3 + 4*a*c*d^4)*f)*x^5 + 2*((2*b*c^3*d^2 + 2*a*c^2 
*d^3 + a*c*d^4)*e - (8*b*c^4*d - a*c^3*d^2 + 4*a*c^2*d^3)*f)*x^3 + ((2*b*c 
^4*d + 2*a*c^3*d^2 + a*c^2*d^3)*e - (8*b*c^5 - a*c^4*d + 4*a*c^3*d^2)*f)*x 
)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (3*b* 
c^2*d^3*f*x^4 - ((3*b*c^2*d^3 + a*c*d^4)*e - 2*(6*b*c^3*d^2 - a*c^2*d^3)*f 
)*x^2 - 2*(b*c^3*d^2 + a*c^2*d^3)*e + (8*b*c^4*d - a*c^3*d^2)*f)*sqrt(b*x^ 
2 + a)*sqrt(d*x^2 + c))/(c^2*d^6*x^5 + 2*c^3*d^5*x^3 + c^4*d^4*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d*f*x + 3*sqrt(c + d*x**2)*sqrt 
(a + b*x**2)*a*b*c*f*x - 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d*e*x + 2 
*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d*f*x**3 - 2*sqrt(c + d*x**2)*sqrt( 
a + b*x**2)*b**2*c*f*x**3 + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4) 
/(a**2*c**3*d + 3*a**2*c**2*d**2*x**2 + 3*a**2*c*d**3*x**4 + a**2*d**4*x** 
6 - a*b*c**4 - 2*a*b*c**3*d*x**2 + 2*a*b*c*d**3*x**6 + a*b*d**4*x**8 - b** 
2*c**4*x**2 - 3*b**2*c**3*d*x**4 - 3*b**2*c**2*d**2*x**6 - b**2*c*d**3*x** 
8),x)*a**3*b*c**2*d**3*f + 6*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/ 
(a**2*c**3*d + 3*a**2*c**2*d**2*x**2 + 3*a**2*c*d**3*x**4 + a**2*d**4*x**6 
 - a*b*c**4 - 2*a*b*c**3*d*x**2 + 2*a*b*c*d**3*x**6 + a*b*d**4*x**8 - b**2 
*c**4*x**2 - 3*b**2*c**3*d*x**4 - 3*b**2*c**2*d**2*x**6 - b**2*c*d**3*x**8 
),x)*a**3*b*c*d**4*f*x**2 + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4) 
/(a**2*c**3*d + 3*a**2*c**2*d**2*x**2 + 3*a**2*c*d**3*x**4 + a**2*d**4*x** 
6 - a*b*c**4 - 2*a*b*c**3*d*x**2 + 2*a*b*c*d**3*x**6 + a*b*d**4*x**8 - b** 
2*c**4*x**2 - 3*b**2*c**3*d*x**4 - 3*b**2*c**2*d**2*x**6 - b**2*c*d**3*x** 
8),x)*a**3*b*d**5*f*x**4 - 12*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4) 
/(a**2*c**3*d + 3*a**2*c**2*d**2*x**2 + 3*a**2*c*d**3*x**4 + a**2*d**4*x** 
6 - a*b*c**4 - 2*a*b*c**3*d*x**2 + 2*a*b*c*d**3*x**6 + a*b*d**4*x**8 - b** 
2*c**4*x**2 - 3*b**2*c**3*d*x**4 - 3*b**2*c**2*d**2*x**6 - b**2*c*d**3*x** 
8),x)*a**2*b**2*c**3*d**2*f - 24*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)...