\(\int \frac {(c+d x^2)^{3/2} (e+f x^2)}{(a+b x^2)^{5/2}} \, dx\) [34]

Optimal result

Integrand size = 30, antiderivative size = 293 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {d (b e-4 a f) x \sqrt {c+d x^2}}{3 a b^2 \sqrt {a+b x^2}}+\frac {(b e-a f) x \left (c+d x^2\right )^{3/2}}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\left (2 b^2 c e-8 a^2 d f+a b (2 d e+c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} b^{5/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d (b e-4 a f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{5/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/3*d*(-4*a*f+b*e)*x*(d*x^2+c)^(1/2)/a/b^2/(b*x^2+a)^(1/2)+1/3*(-a*f+b*e) 
*x*(d*x^2+c)^(3/2)/a/b/(b*x^2+a)^(3/2)+1/3*(2*b^2*c*e-8*a^2*d*f+a*b*(c*f+2 
*d*e))*(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))/a^(3/2)/b^(5/2)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1 
/2)-1/3*d*(-4*a*f+b*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^ 
(1/2)),(1-a*d/b/c)^(1/2))/a^(1/2)/b^(5/2)/(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 = 6.32 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.01 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\left (\frac {b}{a}\right )^{3/2} \left (\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (-4 a^3 d f+2 b^3 c e x^2+a^2 b d \left (e-5 f x^2\right )+a b^2 \left (3 c e+2 d e x^2+c f x^2\right )\right )-i c \left (-2 b^2 c e+8 a^2 d f-a b (2 d e+c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c \left (-2 b^2 c e+4 a^2 d f-a b (d e+c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b^4 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 351, 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.

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {x \sqrt {c+d x^2} (b c (a f+2 b e)+a d (b e-4 a f))}{a b \sqrt {a+b x^2}}-\frac {d \left (\left (-8 a^2 d f+a b (c f+2 d e)+2 b^2 c e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (b e-4 a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\frac {x \sqrt {c+d x^2} (b c (a f+2 b e)+a d (b e-4 a f))}{a b \sqrt {a+b x^2}}-\frac {d \left (\left (-8 a^2 d f+a b (c f+2 d e)+2 b^2 c e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b e-4 a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a b}}{3 a b}+\frac {x \left (c+d x^2\right )^{3/2} (b e-a f)}{3 a b \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 313

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

Input:

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

Output:

((b*e - a*f)*x*(c + d*x^2)^(3/2))/(3*a*b*(a + b*x^2)^(3/2)) + (((a*d*(b*e 
- 4*a*f) + b*c*(2*b*e + a*f))*x*Sqrt[c + d*x^2])/(a*b*Sqrt[a + b*x^2]) - ( 
d*((2*b^2*c*e - 8*a^2*d*f + a*b*(2*d*e + c*f))*((x*Sqrt[a + b*x^2])/(b*Sqr 
t[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))]*S 
qrt[c + d*x^2])) + (c^(3/2)*(b*e - 4*a*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])))/(a*b))/(3*a*b)
 

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. \(564\) vs. \(2(264)=528\).

Time = 7.76 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.93

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a \,b^{4} \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {\left (b d \,x^{2}+b c \right ) \left (5 a^{2} d f -a b c f -2 a b d e -2 c e \,b^{2}\right ) x}{3 a^{2} b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (-\frac {d \left (2 a d f -2 b c f -b d e \right )}{b^{3}}+\frac {\left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) d}{3 b^{3} a}+\frac {\left (5 a^{2} d f -a b c f -2 a b d e -2 c e \,b^{2}\right ) \left (a d -b c \right )}{3 b^{3} a^{2}}+\frac {c \left (5 a^{2} d f -a b c f -2 a b d e -2 c e \,b^{2}\right )}{3 b^{2} a^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {d^{2} f}{b^{2}}+\frac {\left (5 a^{2} d f -a b c f -2 a b d e -2 c e \,b^{2}\right ) d}{3 b^{2} a^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(565\)
default	\(\text {Expression too large to display}\)	\(1231\)

Input:

int((d*x^2+c)^(3/2)*(f*x^2+e)/(b*x^2+a)^(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^2*d*f- 
a*b*c*f-a*b*d*e+b^2*c*e)/a/b^4*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+ 
a/b)^2-1/3*(b*d*x^2+b*c)*(5*a^2*d*f-a*b*c*f-2*a*b*d*e-2*b^2*c*e)/a^2/b^3*x 
/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+(-d*(2*a*d*f-2*b*c*f-b*d*e)/b^3+1/3*(a^2* 
d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^3*d/a+1/3*(5*a^2*d*f-a*b*c*f-2*a*b*d*e-2*b^ 
2*c*e)/b^3*(a*d-b*c)/a^2+1/3/b^2*c*(5*a^2*d*f-a*b*c*f-2*a*b*d*e-2*b^2*c*e) 
/a^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))-(d^2*f 
/b^2+1/3*(5*a^2*d*f-a*b*c*f-2*a*b*d*e-2*b^2*c*e)/b^2*d/a^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. 564 vs. \(2 (264) = 528\).

Time = 0.12 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.92 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (2 \, {\left (b^{4} c^{2} + a b^{3} c d\right )} e + {\left (a b^{3} c^{2} - 8 \, a^{2} b^{2} c d\right )} f\right )} x^{5} + 2 \, {\left (2 \, {\left (a b^{3} c^{2} + a^{2} b^{2} c d\right )} e + {\left (a^{2} b^{2} c^{2} - 8 \, a^{3} b c d\right )} f\right )} x^{3} + {\left (2 \, {\left (a^{2} b^{2} c^{2} + a^{3} b c d\right )} e + {\left (a^{3} b c^{2} - 8 \, a^{4} c d\right )} f\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left ({\left (2 \, b^{4} c^{2} + 2 \, a b^{3} c d + a b^{3} d^{2}\right )} e + {\left (a b^{3} c^{2} - 8 \, a^{2} b^{2} c d - 4 \, a^{2} b^{2} d^{2}\right )} f\right )} x^{5} + 2 \, {\left ({\left (2 \, a b^{3} c^{2} + 2 \, a^{2} b^{2} c d + a^{2} b^{2} d^{2}\right )} e + {\left (a^{2} b^{2} c^{2} - 8 \, a^{3} b c d - 4 \, a^{3} b d^{2}\right )} f\right )} x^{3} + {\left ({\left (2 \, a^{2} b^{2} c^{2} + 2 \, a^{3} b c d + a^{3} b d^{2}\right )} e + {\left (a^{3} b c^{2} - 8 \, a^{4} c d - 4 \, a^{4} d^{2}\right )} f\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (3 \, a^{2} b^{2} d^{2} f x^{4} - {\left ({\left (a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} e + 2 \, {\left (a^{2} b^{2} c d - 6 \, a^{3} b d^{2}\right )} f\right )} x^{2} - 2 \, {\left (a^{2} b^{2} c d + a^{3} b d^{2}\right )} e - {\left (a^{3} b c d - 8 \, a^{4} d^{2}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a^{2} b^{5} d x^{5} + 2 \, a^{3} b^{4} d x^{3} + a^{4} b^{3} d x\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*d*f*x + 2*sqrt(c + d*x**2)*sqr 
t(a + b*x**2)*a*d**2*f*x**3 + sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c**2*f*x 
 + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*d*e*x - 2*sqrt(c + d*x**2)*sqrt 
(a + b*x**2)*b*c*d*f*x**3 - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4) 
/(a**4*c*d + a**4*d**2*x**2 - a**3*b*c**2 + 2*a**3*b*c*d*x**2 + 3*a**3*b*d 
**2*x**4 - 3*a**2*b**2*c**2*x**2 + 3*a**2*b**2*d**2*x**6 - 3*a*b**3*c**2*x 
**4 - 2*a*b**3*c*d*x**6 + a*b**3*d**2*x**8 - b**4*c**2*x**6 - b**4*c*d*x** 
8),x)*a**5*d**4*f + 17*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**4* 
c*d + a**4*d**2*x**2 - a**3*b*c**2 + 2*a**3*b*c*d*x**2 + 3*a**3*b*d**2*x** 
4 - 3*a**2*b**2*c**2*x**2 + 3*a**2*b**2*d**2*x**6 - 3*a*b**3*c**2*x**4 - 2 
*a*b**3*c*d*x**6 + a*b**3*d**2*x**8 - b**4*c**2*x**6 - b**4*c*d*x**8),x)*a 
**4*b*c*d**3*f + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**4*c*d 
+ a**4*d**2*x**2 - a**3*b*c**2 + 2*a**3*b*c*d*x**2 + 3*a**3*b*d**2*x**4 - 
3*a**2*b**2*c**2*x**2 + 3*a**2*b**2*d**2*x**6 - 3*a*b**3*c**2*x**4 - 2*a*b 
**3*c*d*x**6 + a*b**3*d**2*x**8 - b**4*c**2*x**6 - b**4*c*d*x**8),x)*a**4* 
b*d**4*e - 16*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**4*c*d + a** 
4*d**2*x**2 - a**3*b*c**2 + 2*a**3*b*c*d*x**2 + 3*a**3*b*d**2*x**4 - 3*a** 
2*b**2*c**2*x**2 + 3*a**2*b**2*d**2*x**6 - 3*a*b**3*c**2*x**4 - 2*a*b**3*c 
*d*x**6 + a*b**3*d**2*x**8 - b**4*c**2*x**6 - b**4*c*d*x**8),x)*a**4*b*d** 
4*f*x**2 - 12*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**4*c*d + ...