Integrand size = 32, antiderivative size = 416 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\left (b^3 c e^2-2 a^3 d f^2+a^2 b f (2 d e+c f)-a b^2 e (d e+2 c f)\right ) x \sqrt {c+d x^2}}{3 a b^3 \left (a+b x^2\right )^{3/2}}+\frac {d f^2 x^5 \sqrt {c+d x^2}}{3 b \left (a+b x^2\right )^{3/2}}+\frac {2 f (3 b d e+2 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b^3 \sqrt {a+b x^2}}+\frac {2 \left (b^3 c e^2+8 a^3 d f^2+a b^2 e (d e+c f)-4 a^2 b f (2 d e+c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} b^{7/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\left (b^2 d e^2+8 a^2 d f^2-a b f (8 d e+3 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{7/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/3*(b^3*c*e^2-2*a^3*d*f^2+a^2*b*f*(c*f+2*d*e)-a*b^2*e*(2*c*f+d*e))*x*(d*x ^2+c)^(1/2)/a/b^3/(b*x^2+a)^(3/2)+1/3*d*f^2*x^5*(d*x^2+c)^(1/2)/b/(b*x^2+a )^(3/2)+2/3*f*(-3*a*d*f+2*b*c*f+3*b*d*e)*x*(d*x^2+c)^(1/2)/b^3/(b*x^2+a)^( 1/2)+2/3*(b^3*c*e^2+8*a^3*d*f^2+a*b^2*e*(c*f+d*e)-4*a^2*b*f*(c*f+2*d*e))*( 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))/a^(3/2)/b^(7/2)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/3* (b^2*d*e^2+8*a^2*d*f^2-a*b*f*(3*c*f+8*d*e))*(d*x^2+c)^(1/2)*InverseJacobiA M(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/a^(1/2)/b^(7/2)/(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 = 7.94 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.94 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\left (\frac {b}{a}\right )^{3/2} \left (\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (8 a^4 d f^2+2 b^4 c e^2 x^2+a b^3 e \left (3 c e+2 d e x^2+2 c f x^2\right )+a^3 b f \left (-8 d e-3 c f+10 d f x^2\right )+a^2 b^2 \left (-4 c f^2 x^2+d \left (e^2-10 e f x^2+f^2 x^4\right )\right )\right )+2 i c \left (b^3 c e^2+8 a^3 d f^2+a b^2 e (d e+c f)-4 a^2 b f (2 d e+c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (2 b^3 c e^2+8 a^3 d f^2+a b^2 e (d e+2 c f)-a^2 b f (8 d e+5 c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \] Input:
Integrate[((c + d*x^2)^(3/2)*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x]
Output:
((b/a)^(3/2)*(Sqrt[b/a]*x*(c + d*x^2)*(8*a^4*d*f^2 + 2*b^4*c*e^2*x^2 + a*b ^3*e*(3*c*e + 2*d*e*x^2 + 2*c*f*x^2) + a^3*b*f*(-8*d*e - 3*c*f + 10*d*f*x^ 2) + a^2*b^2*(-4*c*f^2*x^2 + d*(e^2 - 10*e*f*x^2 + f^2*x^4))) + (2*I)*c*(b ^3*c*e^2 + 8*a^3*d*f^2 + a*b^2*e*(d*e + c*f) - 4*a^2*b*f*(2*d*e + c*f))*(a + b*x^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*(2*b^3*c*e^2 + 8*a^3*d*f^2 + a*b^2*e*(d*e + 2 *c*f) - a^2*b*f*(8*d*e + 5*c*f))*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b^5*(a + b* x^2)^(3/2)*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(926\) vs. \(2(416)=832\).
Time = 1.23 (sec) , antiderivative size = 926, normalized size of antiderivative = 2.23, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2}}+\frac {2 e f x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2}}+\frac {f^2 x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {f^2 \left (d x^2+c\right )^{3/2} x^3}{3 b \left (b x^2+a\right )^{3/2}}+\frac {(b c-2 a d) f^2 \sqrt {d x^2+c} x^3}{a b^2 \sqrt {b x^2+a}}-\frac {2 e f \left (d x^2+c\right )^{3/2} x}{3 b \left (b x^2+a\right )^{3/2}}-\frac {(3 b c-8 a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 a b^3}+\frac {2 (b c-4 a d) e f \sqrt {d x^2+c} x}{3 a b^2 \sqrt {b x^2+a}}+\frac {(b c-a d) e^2 \sqrt {d x^2+c} x}{3 a b \left (b x^2+a\right )^{3/2}}+\frac {8 d (b c-2 a d) f^2 \sqrt {b x^2+a} x}{3 b^4 \sqrt {d x^2+c}}-\frac {2 d (b c-8 a d) e f \sqrt {b x^2+a} x}{3 a b^3 \sqrt {d x^2+c}}+\frac {2 (b c+a d) e^2 \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} b^{3/2} \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {8 \sqrt {c} \sqrt {d} (b c-2 a d) 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 )}{3 b^4 \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} \sqrt {d} (b c-8 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 a b^3 \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} \sqrt {d} 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 )}{3 a^2 b \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} (3 b c-8 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 )}{3 a b^3 \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {8 c^{3/2} \sqrt {d} 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 a b^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[((c + d*x^2)^(3/2)*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x]
Output:
(-2*d*(b*c - 8*a*d)*e*f*x*Sqrt[a + b*x^2])/(3*a*b^3*Sqrt[c + d*x^2]) + (8* d*(b*c - 2*a*d)*f^2*x*Sqrt[a + b*x^2])/(3*b^4*Sqrt[c + d*x^2]) + ((b*c - a *d)*e^2*x*Sqrt[c + d*x^2])/(3*a*b*(a + b*x^2)^(3/2)) + (2*(b*c - 4*a*d)*e* f*x*Sqrt[c + d*x^2])/(3*a*b^2*Sqrt[a + b*x^2]) + ((b*c - 2*a*d)*f^2*x^3*Sq rt[c + d*x^2])/(a*b^2*Sqrt[a + b*x^2]) - ((3*b*c - 8*a*d)*f^2*x*Sqrt[a + b *x^2]*Sqrt[c + d*x^2])/(3*a*b^3) - (2*e*f*x*(c + d*x^2)^(3/2))/(3*b*(a + b *x^2)^(3/2)) - (f^2*x^3*(c + d*x^2)^(3/2))/(3*b*(a + b*x^2)^(3/2)) + (2*(b *c + a*d)*e^2*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - ( a*d)/(b*c)])/(3*a^(3/2)*b^(3/2)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) + (2*Sqrt[c]*Sqrt[d]*(b*c - 8*a*d)*e*f*Sqrt[a + b*x^2]*Ellipt icE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*b^3*Sqrt[(c*(a + b *x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (8*Sqrt[c]*Sqrt[d]*(b*c - 2*a*d )*f^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a* d)])/(3*b^4*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3 /2)*Sqrt[d]*e^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*b*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x ^2]) + (8*c^(3/2)*Sqrt[d]*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x) /Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*b^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2)) ]*Sqrt[c + d*x^2]) + (c^(3/2)*(3*b*c - 8*a*d)*f^2*Sqrt[a + b*x^2]*Elliptic F[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*b^3*Sqrt[d]*Sqrt[...
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]
Leaf count of result is larger than twice the leaf count of optimal. \(850\) vs. \(2(381)=762\).
Time = 20.51 (sec) , antiderivative size = 851, normalized size of antiderivative = 2.05
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (a^{3} d \,f^{2}-a^{2} c \,f^{2} b -2 a^{2} b d e f +2 a c e f \,b^{2}+a \,b^{2} d \,e^{2}-b^{3} c \,e^{2}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a \,b^{5} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {2 \left (b d \,x^{2}+b c \right ) \left (4 a^{3} d \,f^{2}-2 a^{2} c \,f^{2} b -5 a^{2} b d e f +a c e f \,b^{2}+a \,b^{2} d \,e^{2}+b^{3} c \,e^{2}\right ) x}{3 a^{2} b^{4} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {d \,f^{2} x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b^{3}}+\frac {\left (\frac {3 a^{2} d^{2} f^{2}-4 a b c d \,f^{2}-4 a b \,d^{2} e f +b^{2} c^{2} f^{2}+4 b^{2} c d e f +b^{2} d^{2} e^{2}}{b^{4}}-\frac {\left (a^{3} d \,f^{2}-a^{2} c \,f^{2} b -2 a^{2} b d e f +2 a c e f \,b^{2}+a \,b^{2} d \,e^{2}-b^{3} c \,e^{2}\right ) d}{3 b^{4} a}-\frac {2 \left (4 a^{3} d \,f^{2}-2 a^{2} c \,f^{2} b -5 a^{2} b d e f +a c e f \,b^{2}+a \,b^{2} d \,e^{2}+b^{3} c \,e^{2}\right ) \left (a d -b c \right )}{3 b^{4} a^{2}}-\frac {2 c \left (4 a^{3} d \,f^{2}-2 a^{2} c \,f^{2} b -5 a^{2} b d e f +a c e f \,b^{2}+a \,b^{2} d \,e^{2}+b^{3} c \,e^{2}\right )}{3 b^{3} a^{2}}-\frac {d \,f^{2} a c}{3 b^{3}}\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 {2 d f \left (a d f -b c f -b d e \right )}{b^{3}}-\frac {2 \left (4 a^{3} d \,f^{2}-2 a^{2} c \,f^{2} b -5 a^{2} b d e f +a c e f \,b^{2}+a \,b^{2} d \,e^{2}+b^{3} c \,e^{2}\right ) d}{3 b^{3} a^{2}}-\frac {d \,f^{2} \left (2 a d +2 b c \right )}{3 b^{3}}\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}}\) | \(851\) |
| risch | \(\text {Expression too large to display}\) | \(1255\) |
| default | \(\text {Expression too large to display}\) | \(1967\) |
Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(5/2),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)*(-1/3*(a^3*d*f ^2-a^2*b*c*f^2-2*a^2*b*d*e*f+2*a*b^2*c*e*f+a*b^2*d*e^2-b^3*c*e^2)/a/b^5*x* (b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+a/b)^2+2/3*(b*d*x^2+b*c)*(4*a^3*d *f^2-2*a^2*b*c*f^2-5*a^2*b*d*e*f+a*b^2*c*e*f+a*b^2*d*e^2+b^3*c*e^2)/a^2/b^ 4*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+1/3/b^3*d*f^2*x*(b*d*x^4+a*d*x^2+b*c*x ^2+a*c)^(1/2)+((3*a^2*d^2*f^2-4*a*b*c*d*f^2-4*a*b*d^2*e*f+b^2*c^2*f^2+4*b^ 2*c*d*e*f+b^2*d^2*e^2)/b^4-1/3*(a^3*d*f^2-a^2*b*c*f^2-2*a^2*b*d*e*f+2*a*b^ 2*c*e*f+a*b^2*d*e^2-b^3*c*e^2)/b^4*d/a-2/3*(4*a^3*d*f^2-2*a^2*b*c*f^2-5*a^ 2*b*d*e*f+a*b^2*c*e*f+a*b^2*d*e^2+b^3*c*e^2)/b^4*(a*d-b*c)/a^2-2/3/b^3*c*( 4*a^3*d*f^2-2*a^2*b*c*f^2-5*a^2*b*d*e*f+a*b^2*c*e*f+a*b^2*d*e^2+b^3*c*e^2) /a^2-1/3/b^3*d*f^2*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/b^3*d*f*(a*d*f-b*c*f-b*d*e)-2/3*(4*a^3*d*f^2-2*a^2*b*c*f^2 -5*a^2*b*d*e*f+a*b^2*c*e*f+a*b^2*d*e^2+b^3*c*e^2)/b^3*d/a^2-1/3/b^3*d*f^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))))
Leaf count of result is larger than twice the leaf count of optimal. 911 vs. \(2 (381) = 762\).
Time = 0.12 (sec) , antiderivative size = 911, normalized size of antiderivative = 2.19 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="fricas ")
Output:
1/3*(2*(((b^5*c^2 + a*b^4*c*d)*e^2 + (a*b^4*c^2 - 8*a^2*b^3*c*d)*e*f - 4*( a^2*b^3*c^2 - 2*a^3*b^2*c*d)*f^2)*x^5 + 2*((a*b^4*c^2 + a^2*b^3*c*d)*e^2 + (a^2*b^3*c^2 - 8*a^3*b^2*c*d)*e*f - 4*(a^3*b^2*c^2 - 2*a^4*b*c*d)*f^2)*x^ 3 + ((a^2*b^3*c^2 + a^3*b^2*c*d)*e^2 + (a^3*b^2*c^2 - 8*a^4*b*c*d)*e*f - 4 *(a^4*b*c^2 - 2*a^5*c*d)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sq rt(-c/d)/x), a*d/(b*c)) - (((2*b^5*c^2 + 2*a*b^4*c*d + a*b^4*d^2)*e^2 + 2* (a*b^4*c^2 - 8*a^2*b^3*c*d - 4*a^2*b^3*d^2)*e*f - (8*a^2*b^3*c^2 - 8*a^3*b ^2*d^2 - (16*a^3*b^2 - 3*a^2*b^3)*c*d)*f^2)*x^5 + 2*((2*a*b^4*c^2 + 2*a^2* b^3*c*d + a^2*b^3*d^2)*e^2 + 2*(a^2*b^3*c^2 - 8*a^3*b^2*c*d - 4*a^3*b^2*d^ 2)*e*f - (8*a^3*b^2*c^2 - 8*a^4*b*d^2 - (16*a^4*b - 3*a^3*b^2)*c*d)*f^2)*x ^3 + ((2*a^2*b^3*c^2 + 2*a^3*b^2*c*d + a^3*b^2*d^2)*e^2 + 2*(a^3*b^2*c^2 - 8*a^4*b*c*d - 4*a^4*b*d^2)*e*f - (8*a^4*b*c^2 - 8*a^5*d^2 - (16*a^5 - 3*a ^4*b)*c*d)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a *d/(b*c)) + (a^2*b^3*d^2*f^2*x^6 + 2*(3*a^2*b^3*d^2*e*f + (2*a^2*b^3*c*d - 3*a^3*b^2*d^2)*f^2)*x^4 - 2*(a^2*b^3*c*d + a^3*b^2*d^2)*e^2 - 2*(a^3*b^2* c*d - 8*a^4*b*d^2)*e*f + 8*(a^4*b*c*d - 2*a^5*d^2)*f^2 - ((a*b^4*c*d + 3*a ^2*b^3*d^2)*e^2 + 4*(a^2*b^3*c*d - 6*a^3*b^2*d^2)*e*f - (13*a^3*b^2*c*d - 24*a^4*b*d^2)*f^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a^2*b^6*d*x^5 + 2*a^3*b^5*d*x^3 + a^4*b^4*d*x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((d*x**2+c)**(3/2)*(f*x**2+e)**2/(b*x**2+a)**(5/2),x)
Output:
Integral((c + d*x**2)**(3/2)*(e + f*x**2)**2/(a + b*x**2)**(5/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="maxima ")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)^2/(b*x^2 + a)^(5/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)^2/(b*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(((c + d*x^2)^(3/2)*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x)
Output:
int(((c + d*x^2)^(3/2)*(e + f*x^2)^2)/(a + b*x^2)^(5/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(5/2),x)
Output:
(9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d*f**2*x - 6*sqrt(c + d*x**2)* sqrt(a + b*x**2)*a**2*d**2*f**2*x**3 - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2) *a*b*c**2*f**2*x - 9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*e*f*x + 10* sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*f**2*x**3 + 6*sqrt(c + d*x**2)*s qrt(a + b*x**2)*a*b*d**2*e*f*x**3 + sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b* d**2*f**2*x**5 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*e*f*x - 4*s qrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f**2*x**3 + 3*sqrt(c + d*x**2)* sqrt(a + b*x**2)*b**2*c*d*e**2*x - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b** 2*c*d*e*f*x**3 - sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*f**2*x**5 + 24 *int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**4*c*d + a**4*d**2*x**2 - a**3*b*c**2 + 2*a**3*b*c*d*x**2 + 3*a**3*b*d**2*x**4 - 3*a**2*b**2*c**2*x **2 + 3*a**2*b**2*d**2*x**6 - 3*a*b**3*c**2*x**4 - 2*a*b**3*c*d*x**6 + a*b **3*d**2*x**8 - b**4*c**2*x**6 - b**4*c*d*x**8),x)*a**6*d**4*f**2 - 60*int ((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**4*c*d + a**4*d**2*x**2 - a** 3*b*c**2 + 2*a**3*b*c*d*x**2 + 3*a**3*b*d**2*x**4 - 3*a**2*b**2*c**2*x**2 + 3*a**2*b**2*d**2*x**6 - 3*a*b**3*c**2*x**4 - 2*a*b**3*c*d*x**6 + a*b**3* d**2*x**8 - b**4*c**2*x**6 - b**4*c*d*x**8),x)*a**5*b*c*d**3*f**2 - 24*int ((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**4*c*d + a**4*d**2*x**2 - a** 3*b*c**2 + 2*a**3*b*c*d*x**2 + 3*a**3*b*d**2*x**4 - 3*a**2*b**2*c**2*x**2 + 3*a**2*b**2*d**2*x**6 - 3*a*b**3*c**2*x**4 - 2*a*b**3*c*d*x**6 + a*b*...