Integrand size = 34, antiderivative size = 260 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {(B c-A d) x \sqrt {a-b x^2}}{d (b c+a d) \sqrt {c+d x^2}}+\frac {\sqrt {a} (2 b B c-A b d+a B d) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d^2 (b c+a d) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {\sqrt {a} (2 B c-A d) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d^2 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] 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)
Result contains complex when optimal does not.
Time = 8.00 (sec) , antiderivative size = 231, 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)*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])
Time = 0.78 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, 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 |
| \(\displaystyle \frac {\int -\frac {a (B c-A d)-(2 b B c-A b d+a B d) x^2}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d (a d+b c)}+\frac {x \sqrt {a-b x^2} (B c-A d)}{d \sqrt {c+d x^2} (a d+b c)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {a-b x^2} (B c-A d)}{d \sqrt {c+d x^2} (a d+b c)}-\frac {\int \frac {a (B c-A d)-(2 b B c-A b d+a B d) x^2}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d (a d+b c)}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {x \sqrt {a-b x^2} (B c-A d)}{d \sqrt {c+d x^2} (a d+b c)}-\frac {\frac {(a d+b c) (2 B c-A d) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d}-\frac {(a B d-A b d+2 b B c) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}}{d (a d+b c)}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {x \sqrt {a-b x^2} (B c-A d)}{d \sqrt {c+d x^2} (a d+b c)}-\frac {\frac {\sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 B c-A d) \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {c+d x^2}}-\frac {(a B d-A b d+2 b B c) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}}{d (a d+b c)}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {x \sqrt {a-b x^2} (B c-A d)}{d \sqrt {c+d x^2} (a d+b c)}-\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 B c-A d) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {(a B d-A b d+2 b B c) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}}{d (a d+b c)}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {x \sqrt {a-b x^2} (B c-A d)}{d \sqrt {c+d x^2} (a d+b c)}-\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {(a B d-A b d+2 b B c) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}}{d (a d+b c)}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle \frac {x \sqrt {a-b x^2} (B c-A d)}{d \sqrt {c+d x^2} (a d+b c)}-\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {1-\frac {b x^2}{a}} (a B d-A b d+2 b B c) \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}}{d (a d+b c)}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {x \sqrt {a-b x^2} (B c-A d)}{d \sqrt {c+d x^2} (a d+b c)}-\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a B d-A b d+2 b B c) \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{d (a d+b c)}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {x \sqrt {a-b x^2} (B c-A d)}{d \sqrt {c+d x^2} (a d+b c)}-\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) (2 B c-A d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} (a B d-A b d+2 b B c) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}}{d (a d+b c)}\) |
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]) - (-((Sqrt [a]*(2*b*B*c - A*b*d + a*B*d)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*Elliptic E[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2] *Sqrt[1 + (d*x^2)/c])) + (Sqrt[a]*(b*c + a*d)*(2*B*c - A*d)*Sqrt[1 - (b*x^ 2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/( b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]))/(d*(b*c + a*d))
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])
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]
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]
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]
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]
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])))))
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]
Time = 8.10 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.40
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {\left (-b d \,x^{2}+a d \right ) x \left (A d -B c \right )}{d^{2} \left (a d +b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}+\frac {a \left (A d -B c \right ) \sqrt {1-\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{d \left (a d +b c \right ) \sqrt {\frac {b}{a}}\, \sqrt {-d b \,x^{4}+a d \,x^{2}-b c \,x^{2}+a c}}-\frac {\left (\frac {B}{d}-\frac {b \left (A d -B c \right )}{d \left (a d +b c \right )}\right ) c \sqrt {1-\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-d b \,x^{4}+a d \,x^{2}-b c \,x^{2}+a c}\, d}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}\) | \(363\) |
| default | \(\frac {\left (A \sqrt {\frac {b}{a}}\, b \,d^{2} x^{3}-B \sqrt {\frac {b}{a}}\, b c d \,x^{3}+A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a \,d^{2}+A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c d -A \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c d -2 B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a c d -2 B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,c^{2}+B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a c d +2 B \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,c^{2}-A \sqrt {\frac {b}{a}}\, a \,d^{2} x +B \sqrt {\frac {b}{a}}\, a c d x \right ) \sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{\sqrt {\frac {b}{a}}\, d^{2} \left (a d +b c \right ) \left (-d b \,x^{4}+a d \,x^{2}-b c \,x^{2}+a c \right )}\) | \(512\) |
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))))
Time = 0.10 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.23 \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (2 \, B a b c d + {\left (B a^{2} - A a b\right )} d^{2}\right )} x^{3} + {\left (2 \, B a b c^{2} + {\left (B a^{2} - A a b\right )} c d\right )} x\right )} \sqrt {-b d} \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) - {\left ({\left ({\left (2 \, B a b - B b^{2}\right )} c d + {\left (B a^{2} - A a b + A b^{2}\right )} d^{2}\right )} x^{3} + {\left ({\left (2 \, B a b - B b^{2}\right )} c^{2} + {\left (B a^{2} - A a b + A b^{2}\right )} c d\right )} x\right )} \sqrt {-b d} \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) + {\left (2 \, B b^{2} c^{2} + {\left (B a b - A b^{2}\right )} c d + {\left (B b^{2} c d + B a b d^{2}\right )} x^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (b^{3} c d^{3} + a b^{2} d^{4}\right )} x^{3} + {\left (b^{3} c^{2} d^{2} + a b^{2} c d^{3}\right )} x} \] Input:
integrate(x^2*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),x, algorithm="fri cas")
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)*ellipt ic_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)
\[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (A + B x^{2}\right )}{\sqrt {a - b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] 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)
\[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{2}}{\sqrt {-b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),x, algorithm="max ima")
Output:
integrate((B*x^2 + A)*x^2/(sqrt(-b*x^2 + a)*(d*x^2 + c)^(3/2)), x)
\[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} x^{2}}{\sqrt {-b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x^2*(B*x^2+A)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),x, algorithm="gia c")
Output:
integrate((B*x^2 + A)*x^2/(sqrt(-b*x^2 + a)*(d*x^2 + c)^(3/2)), x)
Timed out. \[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (B\,x^2+A\right )}{\sqrt {a-b\,x^2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] 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)
\[ \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, a x -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}-2 b c d \,x^{4}+2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) a b c d -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}-2 b c d \,x^{4}+2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) a b \,d^{2} x^{2}+2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}-2 b c d \,x^{4}+2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b^{2} c^{2}+2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}\, x^{4}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}-2 b c d \,x^{4}+2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) b^{2} c d \,x^{2}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}-2 b c d \,x^{4}+2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) a^{2} c^{2}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {-b \,x^{2}+a}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}-2 b c d \,x^{4}+2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) a^{2} c d \,x^{2}}{2 b c \left (d \,x^{2}+c \right )} \] 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))