Integrand size = 32, antiderivative size = 306 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {f (6 b d e-4 b c f+a d f) x \sqrt {a+b x^2}}{3 b d^2 \sqrt {c+d x^2}}+\frac {f^2 x^3 \sqrt {a+b x^2}}{3 d \sqrt {c+d x^2}}-\frac {\left (a c d f^2-b \left (3 d^2 e^2-12 c d e f+8 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b \sqrt {c} d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {2 \sqrt {c} f (3 d e-2 c f) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/3*f*(a*d*f-4*b*c*f+6*b*d*e)*x*(b*x^2+a)^(1/2)/b/d^2/(d*x^2+c)^(1/2)+1/3* f^2*x^3*(b*x^2+a)^(1/2)/d/(d*x^2+c)^(1/2)-1/3*(a*c*d*f^2-b*(8*c^2*f^2-12*c *d*e*f+3*d^2*e^2))*(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/c^(1/2)/d^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2 )/(d*x^2+c)^(1/2)+2/3*c^(1/2)*f*(-2*c*f+3*d*e)*(b*x^2+a)^(1/2)*InverseJaco biAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/d^(5/2)/(c*(b*x^2+a)/a/( d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 4.12 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (3 d^2 e^2+4 c^2 f^2+c d f \left (-6 e+f x^2\right )\right )-i c \left (a c d f^2+b \left (-3 d^2 e^2+12 c d e f-8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c \left (a d f (-6 d e+5 c f)+b \left (-3 d^2 e^2+12 c d e f-8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} c d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(3*d^2*e^2 + 4*c^2*f^2 + c*d*f*(-6*e + f*x^2)) - I*c*(a*c*d*f^2 + b*(-3*d^2*e^2 + 12*c*d*e*f - 8*c^2*f^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*(a*d*f*(-6*d*e + 5*c*f) + b*(-3*d^2*e^2 + 12*c*d*e*f - 8*c^2*f^2))*Sq rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a *d)/(b*c)])/(3*Sqrt[b/a]*c*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.77 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {433, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}}+\frac {2 e f x^2 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}}+\frac {f^2 x^4 \sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 d^{5/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 \sqrt {c} e f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {4 \sqrt {c} e f \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c} f^2 \sqrt {a+b x^2} (8 b c-a d) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b d^{5/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {e^2 \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {4 f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d^2}-\frac {f^2 x \sqrt {a+b x^2} (8 b c-a d)}{3 b d^2 \sqrt {c+d x^2}}+\frac {2 e f x \sqrt {a+b x^2}}{d \sqrt {c+d x^2}}-\frac {f^2 x^3 \sqrt {a+b x^2}}{d \sqrt {c+d x^2}}\) |
Input:
Int[(Sqrt[a + b*x^2]*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x]
Output:
(2*e*f*x*Sqrt[a + b*x^2])/(d*Sqrt[c + d*x^2]) - ((8*b*c - a*d)*f^2*x*Sqrt[ a + b*x^2])/(3*b*d^2*Sqrt[c + d*x^2]) - (f^2*x^3*Sqrt[a + b*x^2])/(d*Sqrt[ c + d*x^2]) + (4*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d^2) + (e^2*Sqr t[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqr t[c]*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (4*S qrt[c]*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c )/(a*d)])/(d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*(8*b*c - a*d)*f^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/ Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^ 2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c]*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(S qrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (4*c^(3/2)*f^2*Sqrt[a + b*x^2]*EllipticF[Arc Tan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*d^(5/2)*Sqrt[(c*(a + b*x^2) )/(a*(c + d*x^2))]*Sqrt[c + d*x^2])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2) ^q*(e + f*x^2)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x]
Time = 11.67 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.74
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+a d \right ) \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) x}{c \,d^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {f^{2} x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 d^{2}}+\frac {\left (-\frac {a c d \,f^{2}-2 a \,d^{2} e f -b \,c^{2} f^{2}+2 b c d e f -b \,d^{2} e^{2}}{d^{3}}+\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \left (a d -b c \right )}{d^{3} c}-\frac {a \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}{d^{2} c}-\frac {a c \,f^{2}}{3 d^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {f \left (a d f -b c f +2 b d e \right )}{d^{2}}-\frac {\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) b}{d^{2} c}-\frac {f^{2} \left (2 a d +2 b c \right )}{3 d^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(532\) |
| risch | \(\frac {f^{2} x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 d^{2}}+\frac {\left (-\frac {f \left (a d f -5 b c f +6 b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {3 \left (a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}+a \,d^{3} e^{2}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}\right ) \left (\frac {\left (b d \,x^{2}+a d \right ) x}{c \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{\left (a d -b c \right ) c}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{d}-\frac {\left (4 a c d \,f^{2}-6 a \,d^{2} e f -3 b \,c^{2} f^{2}+6 b c d e f -3 b \,d^{2} e^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(685\) |
| default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-\sqrt {-\frac {b}{a}}\, b c \,d^{2} f^{2} x^{5}-\sqrt {-\frac {b}{a}}\, a c \,d^{2} f^{2} x^{3}-4 \sqrt {-\frac {b}{a}}\, b \,c^{2} d \,f^{2} x^{3}+6 \sqrt {-\frac {b}{a}}\, b c \,d^{2} e f \,x^{3}-3 \sqrt {-\frac {b}{a}}\, b \,d^{3} e^{2} x^{3}+5 \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^{2} d \,f^{2}-6 \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 \,d^{2} e f -8 \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^{3} f^{2}+12 \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^{2} d e f -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 ) b c \,d^{2} 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^{2} d \,f^{2}+8 \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^{3} f^{2}-12 \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^{2} d e f +3 \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 \,d^{2} e^{2}-4 \sqrt {-\frac {b}{a}}\, a \,c^{2} d \,f^{2} x +6 \sqrt {-\frac {b}{a}}\, a c \,d^{2} e f x -3 \sqrt {-\frac {b}{a}}\, a \,d^{3} e^{2} x \right )}{3 \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) d^{3} c \sqrt {-\frac {b}{a}}}\) | \(735\) |
Input:
int((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),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)*((b*d*x^2+a*d) *(c^2*f^2-2*c*d*e*f+d^2*e^2)/c/d^3*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+1/3*f ^2/d^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(-(a*c*d*f^2-2*a*d^2*e*f-b*c^ 2*f^2+2*b*c*d*e*f-b*d^2*e^2)/d^3+(c^2*f^2-2*c*d*e*f+d^2*e^2)/d^3*(a*d-b*c) /c-a/d^2*(c^2*f^2-2*c*d*e*f+d^2*e^2)/c-1/3*a*c/d^2*f^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)*Ellipti cF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(1/d^2*f*(a*d*f-b*c*f+2*b*d*e) -(c^2*f^2-2*c*d*e*f+d^2*e^2)/d^2*b/c-1/3*f^2/d^2*(2*a*d+2*b*c))*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))))
Time = 0.09 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (3 \, b c d^{3} e^{2} - 12 \, b c^{2} d^{2} e f + {\left (8 \, b c^{3} d - a c^{2} d^{2}\right )} f^{2}\right )} x^{3} + {\left (3 \, b c^{2} d^{2} e^{2} - 12 \, b c^{3} d e f + {\left (8 \, b c^{4} - a c^{3} d\right )} f^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (3 \, b c d^{3} e^{2} - 6 \, {\left (2 \, b c^{2} d^{2} + a d^{4}\right )} e f + {\left (8 \, b c^{3} d - a c^{2} d^{2} + 4 \, a c d^{3}\right )} f^{2}\right )} x^{3} + {\left (3 \, b c^{2} d^{2} e^{2} - 6 \, {\left (2 \, b c^{3} d + a c d^{3}\right )} e f + {\left (8 \, b c^{4} - a c^{3} d + 4 \, a c^{2} d^{2}\right )} f^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (b c d^{3} f^{2} x^{4} - 3 \, b c d^{3} e^{2} + 12 \, b c^{2} d^{2} e f - {\left (8 \, b c^{3} d - a c^{2} d^{2}\right )} f^{2} + {\left (6 \, b c d^{3} e f - {\left (4 \, b c^{2} d^{2} - a c d^{3}\right )} f^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (b c d^{5} x^{3} + b c^{2} d^{4} x\right )}} \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="fricas ")
Output:
1/3*(((3*b*c*d^3*e^2 - 12*b*c^2*d^2*e*f + (8*b*c^3*d - a*c^2*d^2)*f^2)*x^3 + (3*b*c^2*d^2*e^2 - 12*b*c^3*d*e*f + (8*b*c^4 - a*c^3*d)*f^2)*x)*sqrt(b* d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((3*b*c*d^3*e^ 2 - 6*(2*b*c^2*d^2 + a*d^4)*e*f + (8*b*c^3*d - a*c^2*d^2 + 4*a*c*d^3)*f^2) *x^3 + (3*b*c^2*d^2*e^2 - 6*(2*b*c^3*d + a*c*d^3)*e*f + (8*b*c^4 - a*c^3*d + 4*a*c^2*d^2)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/ x), a*d/(b*c)) + (b*c*d^3*f^2*x^4 - 3*b*c*d^3*e^2 + 12*b*c^2*d^2*e*f - (8* b*c^3*d - a*c^2*d^2)*f^2 + (6*b*c*d^3*e*f - (4*b*c^2*d^2 - a*c*d^3)*f^2)*x ^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b*c*d^5*x^3 + b*c^2*d^4*x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (e + f x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(f*x**2+e)**2/(d*x**2+c)**(3/2),x)
Output:
Integral(sqrt(a + b*x**2)*(e + f*x**2)**2/(c + d*x**2)**(3/2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="maxima ")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^2/(d*x^2 + c)^(3/2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^2/(d*x^2 + c)^(3/2), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (f\,x^2+e\right )}^2}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x)
Output:
int(((a + b*x^2)^(1/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*f**2*x + 6*sqrt(c + d*x**2)*sq rt(a + b*x**2)*a*d*e*f*x + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*f**2*x* *3 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e**2*x + 5*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**2*d*f**2 - 6*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**2*e*f + 5*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**2*f**2*x**2 - 6*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**3*e*f*x**2 - 8*int((sqr t(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**3*f**2 + 12*int((sq rt(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*d*e*f - 8*int((s qrt(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*d*f**2*x**2 - 3 *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**2*e**2 + 12*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2...