\(\int \frac {x^2 (A+B x^2)}{\sqrt {a-b x^2} (c-d x^2)^{3/2}} \, dx\) [5]

Optimal result

Integrand size = 35, antiderivative size = 264 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=-\frac {(B c+A d) x \sqrt {a-b x^2}}{d (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt {a} (2 b B c+A b d-a B d) \sqrt {1-\frac {b x^2}{a}} \sqrt {c-d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|\frac {a d}{b c}\right )}{\sqrt {b} d^2 (b c-a d) \sqrt {a-b x^2} \sqrt {1-\frac {d x^2}{c}}}-\frac {\sqrt {a} (2 B c+A d) \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} d^2 \sqrt {a-b x^2} \sqrt {c-d x^2}} \] Output:

-(A*d+B*c)*x*(-b*x^2+a)^(1/2)/d/(-a*d+b*c)/(-d*x^2+c)^(1/2)+a^(1/2)*(A*b*d 
-B*a*d+2*B*b*c)*(1-b*x^2/a)^(1/2)*(-d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^( 
1/2),(a*d/b/c)^(1/2))/b^(1/2)/d^2/(-a*d+b*c)/(-b*x^2+a)^(1/2)/(1-d*x^2/c)^ 
(1/2)-a^(1/2)*(A*d+2*B*c)*(1-b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticF(b^ 
(1/2)*x/a^(1/2),(a*d/b/c)^(1/2))/b^(1/2)/d^2/(-b*x^2+a)^(1/2)/(-d*x^2+c)^( 
1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.07 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.88 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {b}{a}} d (B c+A d) x \left (a-b x^2\right )-i c (a B d-b (2 B c+A d)) \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-b c+a d) (2 B c+A d) \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} d^2 (-b c+a d) \sqrt {a-b x^2} \sqrt {c-d x^2}} \] 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*(a*B*d - b*(2*B*c + A*d))* 
Sqrt[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 = 0.80 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {440, 25, 399, 323, 323, 321, 331, 330, 327}

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

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

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])) + (-((S 
qrt[c]*(a*B*d - b*(2*B*c + A*d))*Sqrt[a - b*x^2]*Sqrt[1 - (d*x^2)/c]*Ellip 
ticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[1 - (b*x^2 
)/a]*Sqrt[c - d*x^2])) - (a*B*Sqrt[c]*(b*c - a*d)*Sqrt[1 - (b*x^2)/a]*Sqrt 
[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], (b*c)/(a*d)])/(b*Sq 
rt[d]*Sqrt[a - b*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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 331

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

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 = 8.28 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.41

method	result	size
elliptic	\(\frac {\sqrt {\left (-d \,x^{2}+c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {\left (b d \,x^{2}-a d \right ) x \left (A d +B c \right )}{d^{2} \left (a d -b c \right ) \sqrt {\left (x^{2}-\frac {c}{d}\right ) \left (b d \,x^{2}-a d \right )}}-\frac {a \left (A d +B c \right ) \sqrt {1-\frac {x^{2} d}{c}}\, \sqrt {1-\frac {x^{2} b}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )}{d \left (a d -b c \right ) \sqrt {\frac {d}{c}}\, \sqrt {d b \,x^{4}-a d \,x^{2}-b c \,x^{2}+a c}}+\frac {\left (-\frac {B}{d}+\frac {b \left (A d +B c \right )}{d \left (a d -b c \right )}\right ) a \sqrt {1-\frac {x^{2} d}{c}}\, \sqrt {1-\frac {x^{2} b}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d -b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {d b \,x^{4}-a d \,x^{2}-b c \,x^{2}+a c}\, b}\right )}{\sqrt {-d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}\)	\(371\)
default	\(\frac {\left (-A \sqrt {\frac {d}{c}}\, b^{2} d \,x^{3}-B \sqrt {\frac {d}{c}}\, b^{2} c \,x^{3}-A \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) a b d -B \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) a^{2} d +B \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) a b c +B \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) a^{2} d -2 B \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {\frac {b c}{a d}}\right ) a b c +A \sqrt {\frac {d}{c}}\, a b d x +B \sqrt {\frac {d}{c}}\, a b c x \right ) \sqrt {-d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{d b \sqrt {\frac {d}{c}}\, \left (a d -b c \right ) \left (d b \,x^{4}-a d \,x^{2}-b c \,x^{2}+a c \right )}\)	\(407\)

Input:

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

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)/(d/c)^(1/2)*(1-1/c*x^2*d)^(1/2)*(1-1/a*x^2*b)^(1/2)/(b*d*x^4 
-a*d*x^2-b*c*x^2+a*c)^(1/2)*EllipticF(x*(d/c)^(1/2),(-1-(-a*d-b*c)/a/d)^(1 
/2))+(-B/d+b/d*(A*d+B*c)/(a*d-b*c))*a/(d/c)^(1/2)*(1-1/c*x^2*d)^(1/2)*(1-1 
/a*x^2*b)^(1/2)/(b*d*x^4-a*d*x^2-b*c*x^2+a*c)^(1/2)/b*(EllipticF(x*(d/c)^( 
1/2),(-1-(-a*d-b*c)/a/d)^(1/2))-EllipticE(x*(d/c)^(1/2),(-1-(-a*d-b*c)/a/d 
)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.25 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \left (c-d x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (2 \, B a b c d - {\left (B a^{2} - A a b\right )} d^{2}\right )} x^{3} - {\left (2 \, B a b c^{2} - {\left (B a^{2} - A a b\right )} c d\right )} x\right )} \sqrt {b d} \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,\frac {b c}{a d}) - {\left ({\left ({\left (2 \, B a b - B b^{2}\right )} c d - {\left (B a^{2} - A a b + A b^{2}\right )} d^{2}\right )} x^{3} - {\left ({\left (2 \, B a b - B b^{2}\right )} c^{2} - {\left (B a^{2} - A a b + A b^{2}\right )} c d\right )} x\right )} \sqrt {b d} \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,\frac {b c}{a d}) - {\left (2 \, B b^{2} c^{2} - {\left (B a b - A b^{2}\right )} c d - {\left (B b^{2} c d - B a b d^{2}\right )} x^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {-d x^{2} + c}}{{\left (b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{3} - {\left (b^{3} c^{2} d^{2} - a b^{2} c d^{3}\right )} x} \] Input:

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

Output:

-(((2*B*a*b*c*d - (B*a^2 - A*a*b)*d^2)*x^3 - (2*B*a*b*c^2 - (B*a^2 - A*a*b 
)*c*d)*x)*sqrt(b*d)*sqrt(a/b)*elliptic_e(arcsin(sqrt(a/b)/x), b*c/(a*d)) - 
 (((2*B*a*b - B*b^2)*c*d - (B*a^2 - A*a*b + A*b^2)*d^2)*x^3 - ((2*B*a*b - 
B*b^2)*c^2 - (B*a^2 - A*a*b + A*b^2)*c*d)*x)*sqrt(b*d)*sqrt(a/b)*elliptic_ 
f(arcsin(sqrt(a/b)/x), b*c/(a*d)) - (2*B*b^2*c^2 - (B*a*b - A*b^2)*c*d - ( 
B*b^2*c*d - B*a*b*d^2)*x^2)*sqrt(-b*x^2 + a)*sqrt(-d*x^2 + c))/((b^3*c*d^3 
 - a*b^2*d^4)*x^3 - (b^3*c^2*d^2 - a*b^2*c*d^3)*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="ma 
xima")
 

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="gi 
ac")
 

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