\(\int \frac {\sqrt {c+d x^2} (e+f x^2)}{(a+b x^2)^{3/2}} \, dx\) [29]

Optimal result

Integrand size = 30, antiderivative size = 206 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {f x \sqrt {c+d x^2}}{b \sqrt {a+b x^2}}+\frac {(b e-2 a f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} b^{3/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} f \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{b^{3/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

f*x*(d*x^2+c)^(1/2)/b/(b*x^2+a)^(1/2)+(-2*a*f+b*e)*(d*x^2+c)^(1/2)*Ellipti 
cE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(1/2)/b^(3/2)/ 
(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+a^(1/2)*f*(d*x^2+c)^(1/2)* 
InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(3/2)/(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 = 3.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {-i c (-b e+2 a f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+(b e-a f) \left (\sqrt {\frac {b}{a}} x \left (c+d x^2\right )-i c \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{a^2 \left (\frac {b}{a}\right )^{3/2} \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {401, 25, 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.

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

((b*e - a*f)*x*Sqrt[c + d*x^2])/(a*b*Sqrt[a + b*x^2]) + (-(d*(b*e - 2*a*f) 
*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*Ellip 
ticE[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))]*Sqrt[c + d*x^2]))) + (c^(3/2)*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)
 

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 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 [A] (verified)

Time = 5.70 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.59

method	result	size
default	\(\frac {\left (-\sqrt {-\frac {b}{a}}\, a d f \,x^{3}+\sqrt {-\frac {b}{a}}\, b d e \,x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c f +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c e +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c f -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c e -\sqrt {-\frac {b}{a}}\, a c f x +\sqrt {-\frac {b}{a}}\, b c e x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{b \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) a \sqrt {-\frac {b}{a}}}\)	\(328\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+b c \right ) \left (a f -b e \right ) x}{a \,b^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (-\frac {a d f -b c f -b d e}{b^{2}}+\frac {\left (a f -b e \right ) \left (a d -b c \right )}{b^{2} a}+\frac {c \left (a f -b e \right )}{b a}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {d f}{b}+\frac {\left (a f -b e \right ) d}{b a}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(379\)

Input:

int((d*x^2+c)^(1/2)*(f*x^2+e)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (b^{2} c e - 2 \, a b c f\right )} x^{3} + {\left (a b c e - 2 \, a^{2} c f\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (b^{2} c e - {\left (2 \, a b c + a b d\right )} f\right )} x^{3} + {\left (a b c e - {\left (2 \, a^{2} c + a^{2} d\right )} f\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (a b d f x^{2} - a b d e + 2 \, a^{2} d f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{a b^{3} d x^{3} + a^{2} b^{2} d x} \] Input:

integrate((d*x^2+c)^(1/2)*(f*x^2+e)/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

(((b^2*c*e - 2*a*b*c*f)*x^3 + (a*b*c*e - 2*a^2*c*f)*x)*sqrt(b*d)*sqrt(-c/d 
)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((b^2*c*e - (2*a*b*c + a*b 
*d)*f)*x^3 + (a*b*c*e - (2*a^2*c + a^2*d)*f)*x)*sqrt(b*d)*sqrt(-c/d)*ellip 
tic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (a*b*d*f*x^2 - a*b*d*e + 2*a^2*d* 
f)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a*b^3*d*x^3 + a^2*b^2*d*x)
 

Sympy [F]

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

integrate((d*x**2+c)**(1/2)*(f*x**2+e)/(b*x**2+a)**(3/2),x)
 

Output:

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

Maxima [F]

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

integrate((d*x^2+c)^(1/2)*(f*x^2+e)/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((d*x^2+c)^(1/2)*(f*x^2+e)/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, c f x +\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, d e x +2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} d^{2} f -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b c d f -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b \,d^{2} e +2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b \,d^{2} f \,x^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} c d f \,x^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} d^{2} e \,x^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} c^{2} f +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} c d e -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b \,c^{2} f \,x^{2}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b c d e \,x^{2}}{2 a d \left (b \,x^{2}+a \right )} \] Input:

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

Output:

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