Integrand size = 32, antiderivative size = 445 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {\left (2 a^2 d^2 f^2-a b d f (10 d e-3 c f)-b^2 \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 b d^3 \sqrt {a+b x^2}}+\frac {f (10 b d e-4 b c f+a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b d^2}+\frac {f^2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}+\frac {\sqrt {a} \left (2 a^2 d^2 f^2-a b d f (10 d e-3 c f)-b^2 \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{3/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (a c d f^2-b \left (15 d^2 e^2-10 c d e f+4 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{3/2} c d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/15*(2*a^2*d^2*f^2-a*b*d*f*(-3*c*f+10*d*e)-b^2*(8*c^2*f^2-20*c*d*e*f+15* d^2*e^2))*x*(d*x^2+c)^(1/2)/b/d^3/(b*x^2+a)^(1/2)+1/15*f*(a*d*f-4*b*c*f+10 *b*d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d^2+1/5*f^2*x^3*(b*x^2+a)^(1/2 )*(d*x^2+c)^(1/2)/d+1/15*a^(1/2)*(2*a^2*d^2*f^2-a*b*d*f*(-3*c*f+10*d*e)-b^ 2*(8*c^2*f^2-20*c*d*e*f+15*d^2*e^2))*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a ^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(3/2)/d^3/(b*x^2+a)^(1/2)/(a *(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*a^(3/2)*(a*c*d*f^2-b*(4*c^2*f^2-10*c*d* e*f+15*d^2*e^2))*(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)/c/d^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a) )^(1/2)
Result contains complex when optimal does not.
Time = 2.31 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d f x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a d f+b \left (10 d e-4 c f+3 d f x^2\right )\right )+i c \left (2 a^2 d^2 f^2+a b d f (-10 d e+3 c f)+b^2 \left (-15 d^2 e^2+20 c d e f-8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i (-b c+a d) \left (a c d f^2+b \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 b \sqrt {\frac {b}{a}} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*(e + f*x^2)^2)/Sqrt[c + d*x^2],x]
Output:
(Sqrt[b/a]*d*f*x*(a + b*x^2)*(c + d*x^2)*(a*d*f + b*(10*d*e - 4*c*f + 3*d* f*x^2)) + I*c*(2*a^2*d^2*f^2 + a*b*d*f*(-10*d*e + 3*c*f) + b^2*(-15*d^2*e^ 2 + 20*c*d*e*f - 8*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellip ticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(-(b*c) + a*d)*(a*c*d*f^2 + b*(15*d^2*e^2 - 20*c*d*e*f + 8*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x ^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*b*Sqrt[b/a]*d^3 *Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 1.04 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.86, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2 \sqrt {a+b x^2}}{\sqrt {c+d x^2}}+\frac {2 e f x^2 \sqrt {a+b x^2}}{\sqrt {c+d x^2}}+\frac {f^2 x^4 \sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d}-\frac {(4 b c-a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b d^2}+\frac {2 e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 d}+\frac {e^2 \sqrt {b x^2+a} x}{\sqrt {d x^2+c}}+\frac {\left (8 b^2 c^2-3 a b d c-2 a^2 d^2\right ) f^2 \sqrt {b x^2+a} x}{15 b^2 d^2 \sqrt {d x^2+c}}-\frac {2 (2 b c-a d) e f \sqrt {b x^2+a} x}{3 b d \sqrt {d x^2+c}}-\frac {\sqrt {c} e^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} \left (8 b^2 c^2-3 a b d c-2 a^2 d^2\right ) f^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 \sqrt {c} (2 b c-a d) e f \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {c} e^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {c^{3/2} (4 b c-a d) f^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 c^{3/2} e f \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[(Sqrt[a + b*x^2]*(e + f*x^2)^2)/Sqrt[c + d*x^2],x]
Output:
(e^2*x*Sqrt[a + b*x^2])/Sqrt[c + d*x^2] - (2*(2*b*c - a*d)*e*f*x*Sqrt[a + b*x^2])/(3*b*d*Sqrt[c + d*x^2]) + ((8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*f^2 *x*Sqrt[a + b*x^2])/(15*b^2*d^2*Sqrt[c + d*x^2]) + (2*e*f*x*Sqrt[a + b*x^2 ]*Sqrt[c + d*x^2])/(3*d) - ((4*b*c - a*d)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d *x^2])/(15*b*d^2) + (f^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d) - (Sqr t[c]*e^2*Sqrt[a + b*x^2]*EllipticE[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]) + (2*Sqrt[c]*(2*b*c - a*d)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/ Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^ 2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*f^2*S qrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(1 5*b^2*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sq rt[c]*e^2*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]) - (2*c^(3/2)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d *x^2]) + (c^(3/2)*(4*b*c - a*d)*f^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt [d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b*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 = 7.34 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.02
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f^{2} x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 d}+\frac {\left (a \,f^{2}+2 b e f -\frac {f^{2} \left (4 a d +4 b c \right )}{5 d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (a \,e^{2}-\frac {\left (a \,f^{2}+2 b e f -\frac {f^{2} \left (4 a d +4 b c \right )}{5 d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (2 a e f +b \,e^{2}-\frac {3 a c \,f^{2}}{5 d}-\frac {\left (a \,f^{2}+2 b e f -\frac {f^{2} \left (4 a d +4 b c \right )}{5 d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(453\) |
| risch | \(\frac {f x \left (3 b d f \,x^{2}+a d f -4 b c f +10 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b \,d^{2}}-\frac {\left (-\frac {\left (2 a^{2} d^{2} f^{2}+3 a b c d \,f^{2}-10 a b \,d^{2} e f -8 b^{2} c^{2} f^{2}+20 b^{2} c d e f -15 b^{2} d^{2} e^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a^{2} c d \,f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {4 a b \,c^{2} f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {15 a b \,d^{2} e^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {10 a b c d e f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b \,d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(658\) |
| default | \(\text {Expression too large to display}\) | \(1059\) |
Input:
int((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),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)*(1/5*f^2/d*x^3 *(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(a*f^2+2*b*e*f-1/5*f^2/d*(4*a*d+4 *b*c))/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(a*e^2-1/3*(a*f^2+2*b*e*f -1/5*f^2/d*(4*a*d+4*b*c))/b/d*a*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)*EllipticF(x*(-b/a)^(1/2),(-1 +(a*d+b*c)/c/b)^(1/2))-(2*a*e*f+b*e^2-3/5*a*c/d*f^2-1/3*(a*f^2+2*b*e*f-1/5 *f^2/d*(4*a*d+4*b*c))/b/d*(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 = 0.11 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {{\left (15 \, b^{2} c^{2} d^{2} e^{2} - 10 \, {\left (2 \, b^{2} c^{3} d - a b c^{2} d^{2}\right )} e f + {\left (8 \, b^{2} c^{4} - 3 \, a b c^{3} d - 2 \, a^{2} c^{2} d^{2}\right )} f^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, {\left (b^{2} c^{2} d^{2} + a b d^{4}\right )} e^{2} - 10 \, {\left (2 \, b^{2} c^{3} d - a b c^{2} d^{2} + a b c d^{3}\right )} e f + {\left (8 \, b^{2} c^{4} - 3 \, a b c^{3} d - a^{2} c d^{3} - 2 \, {\left (a^{2} - 2 \, a b\right )} c^{2} d^{2}\right )} f^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{2} c d^{3} f^{2} x^{4} + 15 \, b^{2} c d^{3} e^{2} - 10 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3}\right )} e f + {\left (8 \, b^{2} c^{3} d - 3 \, a b c^{2} d^{2} - 2 \, a^{2} c d^{3}\right )} f^{2} + {\left (10 \, b^{2} c d^{3} e f - {\left (4 \, b^{2} c^{2} d^{2} - a b c d^{3}\right )} f^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b^{2} c d^{4} x} \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x, algorithm="fricas ")
Output:
-1/15*((15*b^2*c^2*d^2*e^2 - 10*(2*b^2*c^3*d - a*b*c^2*d^2)*e*f + (8*b^2*c ^4 - 3*a*b*c^3*d - 2*a^2*c^2*d^2)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(a rcsin(sqrt(-c/d)/x), a*d/(b*c)) - (15*(b^2*c^2*d^2 + a*b*d^4)*e^2 - 10*(2* b^2*c^3*d - a*b*c^2*d^2 + a*b*c*d^3)*e*f + (8*b^2*c^4 - 3*a*b*c^3*d - a^2* c*d^3 - 2*(a^2 - 2*a*b)*c^2*d^2)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(ar csin(sqrt(-c/d)/x), a*d/(b*c)) - (3*b^2*c*d^3*f^2*x^4 + 15*b^2*c*d^3*e^2 - 10*(2*b^2*c^2*d^2 - a*b*c*d^3)*e*f + (8*b^2*c^3*d - 3*a*b*c^2*d^2 - 2*a^2 *c*d^3)*f^2 + (10*b^2*c*d^3*e*f - (4*b^2*c^2*d^2 - a*b*c*d^3)*f^2)*x^2)*sq rt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*c*d^4*x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (e + f x^{2}\right )^{2}}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(f*x**2+e)**2/(d*x**2+c)**(1/2),x)
Output:
Integral(sqrt(a + b*x**2)*(e + f*x**2)**2/sqrt(c + d*x**2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x, algorithm="maxima ")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^2/sqrt(d*x^2 + c), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}^{2}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)^2/sqrt(d*x^2 + c), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (f\,x^2+e\right )}^2}{\sqrt {d\,x^2+c}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(e + f*x^2)^2)/(c + d*x^2)^(1/2),x)
Output:
int(((a + b*x^2)^(1/2)*(e + f*x^2)^2)/(c + d*x^2)^(1/2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d \,f^{2} x -4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c \,f^{2} x +10 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d e f x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d \,f^{2} x^{3}-2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2} f^{2}-3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d \,f^{2}+10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,d^{2} e f +8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2} f^{2}-20 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c d e f +15 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} d^{2} e^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c d \,f^{2}+4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2} f^{2}-10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d e f +15 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,d^{2} e^{2}}{15 b \,d^{2}} \] Input:
int((b*x^2+a)^(1/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x)
Output:
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f**2*x - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*f**2*x + 10*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*f*x + 3* sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*f**2*x**3 - 2*int((sqrt(c + d*x**2)* sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*d**2 *f**2 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b *c*x**2 + b*d*x**4),x)*a*b*c*d*f**2 + 10*int((sqrt(c + d*x**2)*sqrt(a + b* x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b*d**2*e*f + 8*int ((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d *x**4),x)*b**2*c**2*f**2 - 20*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2) /(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b**2*c*d*e*f + 15*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)* b**2*d**2*e**2 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*c*d*f**2 + 4*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b*c**2*f**2 - 10*int( (sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4) ,x)*a*b*c*d*e*f + 15*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x* *2 + b*c*x**2 + b*d*x**4),x)*a*b*d**2*e**2)/(15*b*d**2)