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

Optimal result

Integrand size = 33, antiderivative size = 248 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {B x \sqrt {a+b x^2}}{b d \sqrt {c+d x^2}}-\frac {\sqrt {c} (2 b B c-A b d-a B d) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b d^{3/2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} (B c-A d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{d^{3/2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

B*x*(b*x^2+a)^(1/2)/b/d/(d*x^2+c)^(1/2)-c^(1/2)*(-A*b*d-B*a*d+2*B*b*c)*(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))/b/d^(3/2)/(-a*d+b*c)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+c 
^(1/2)*(-A*d+B*c)*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2) 
),(1-b*c/a/d)^(1/2))/d^(3/2)/(-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 = 7.84 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.89 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d (B c-A d) x \left (a+b x^2\right )-i c (-2 b B c+A b d+a B d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-b c+a d) (2 B c-A d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} d^2 (-b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*d*(B*c - A*d)*x*(a + b*x^2) - I*c*(-2*b*B*c + A*b*d + a*B*d)*Sq 
rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a 
*d)/(b*c)] + I*(-(b*c) + a*d)*(2*B*c - A*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + ( 
d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*d^2*( 
-(b*c) + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {440, 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.

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

\(\Big \downarrow \) 440

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

-(((B*c - A*d)*x*Sqrt[a + b*x^2])/(d*(b*c - a*d)*Sqrt[c + d*x^2])) + ((2*b 
*B*c - A*b*d - a*B*d)*((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))]*Sqrt[c + d*x^2])) + (Sqrt[ 
c]*(B*c - A*d)*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]))/(d*(b*c - a*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 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 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]
 

rule 440

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

Maple [A] (verified)

Time = 8.04 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.43

method	result	size
elliptic	\(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \left (-\frac {\left (b d \,x^{2}+a d \right ) x \left (A d -B c \right )}{d^{2} \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {a \left (A d -B c \right ) \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{d \left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}-\frac {\left (\frac {B}{d}+\frac {b \left (A d -B c \right )}{d \left (a d -b c \right )}\right ) c \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}\)	\(354\)
default	\(\frac {\left (-A \sqrt {-\frac {b}{a}}\, b \,d^{2} x^{3}+B \sqrt {-\frac {b}{a}}\, b c d \,x^{3}+A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,d^{2}-A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d +A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d -2 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d +2 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}+B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d -2 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}-A \sqrt {-\frac {b}{a}}\, a \,d^{2} x +B \sqrt {-\frac {b}{a}}\, a c d x \right ) \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{\sqrt {-\frac {b}{a}}\, d^{2} \left (a d -b c \right ) \left (d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )}\)	\(508\)

Input:

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

Output:

((d*x^2+c)*(b*x^2+a))^(1/2)/(d*x^2+c)^(1/2)/(b*x^2+a)^(1/2)*(-(b*d*x^2+a*d 
)/d^2/(a*d-b*c)*x*(A*d-B*c)/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+a/d/(a*d-b*c)* 
(A*d-B*c)/(-b/a)^(1/2)*(1+1/a*x^2*b)^(1/2)*(1+1/c*x^2*d)^(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/d+b/d*(A*d-B*c)/(a*d-b*c))*c/(-b/a)^(1/2)*(1+1/a*x^2*b)^(1/2)*(1+1/c* 
x^2*d)^(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 [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.32 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (2 \, B b c^{3} d - {\left (B a + A b\right )} c^{2} d^{2}\right )} x^{3} + {\left (2 \, B b c^{4} - {\left (B a + A b\right )} c^{3} d\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (2 \, B b c^{3} d + B a c d^{3} - A a d^{4} - {\left (B a + A b\right )} c^{2} d^{2}\right )} x^{3} + {\left (2 \, B b c^{4} + B a c^{2} d^{2} - A a c d^{3} - {\left (B a + A b\right )} c^{3} d\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, B b c^{3} d - {\left (B a + A b\right )} c^{2} d^{2} + {\left (B b c^{2} d^{2} - B a c d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (b^{2} c^{2} d^{4} - a b c d^{5}\right )} x^{3} + {\left (b^{2} c^{3} d^{3} - a b c^{2} d^{4}\right )} x} \] Input:

integrate(x^2*(B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),x, algorithm="fric 
as")
 

Output:

-(((2*B*b*c^3*d - (B*a + A*b)*c^2*d^2)*x^3 + (2*B*b*c^4 - (B*a + A*b)*c^3* 
d)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ( 
(2*B*b*c^3*d + B*a*c*d^3 - A*a*d^4 - (B*a + A*b)*c^2*d^2)*x^3 + (2*B*b*c^4 
 + B*a*c^2*d^2 - A*a*c*d^3 - (B*a + A*b)*c^3*d)*x)*sqrt(b*d)*sqrt(-c/d)*el 
liptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (2*B*b*c^3*d - (B*a + A*b)*c^2 
*d^2 + (B*b*c^2*d^2 - B*a*c*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/((b 
^2*c^2*d^4 - a*b*c*d^5)*x^3 + (b^2*c^3*d^3 - a*b*c^2*d^4)*x)
 

Sympy [F]

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

integrate(x**2*(B*x**2+A)/(b*x**2+a)**(1/2)/(d*x**2+c)**(3/2),x)
 

Output:

Integral(x**2*(A + B*x**2)/(sqrt(a + b*x**2)*(c + d*x**2)**(3/2)), x)
 

Maxima [F]

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

integrate(x^2*(B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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