Integrand size = 32, antiderivative size = 463 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {(b e-a f)^2 x}{a b (b c-a d) \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {\left (3 b^2 c d e^2+3 a^2 c d f^2+a b \left (d^2 e^2-8 c d e f+c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{3 a b c (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {\left (3 b^2 c^2 d e^2-a^2 d \left (2 d^2 e^2+2 c d e f-7 c^2 f^2\right )+a b c \left (7 d^2 e^2-14 c d e f+c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {d} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (3 a^2 c d f^2+3 b^2 c e (3 d e-2 c f)-a b \left (d^2 e^2+10 c d e f-5 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
(-a*f+b*e)^2*x/a/b/(-a*d+b*c)/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)+1/3*(3*b^2*c *d*e^2+3*a^2*c*d*f^2+a*b*(c^2*f^2-8*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)/a/ b/c/(-a*d+b*c)^2/(d*x^2+c)^(3/2)+1/3*(3*b^2*c^2*d*e^2-a^2*d*(-7*c^2*f^2+2* c*d*e*f+2*d^2*e^2)+a*b*c*(c^2*f^2-14*c*d*e*f+7*d^2*e^2))*(b*x^2+a)^(1/2)*E llipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/a/c^(3/2)/ d^(1/2)/(-a*d+b*c)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*( 3*a^2*c*d*f^2+3*b^2*c*e*(-2*c*f+3*d*e)-a*b*(-5*c^2*f^2+10*c*d*e*f+d^2*e^2) )*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1 /2))/a/c^(1/2)/d^(1/2)/(-a*d+b*c)^3/(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 = 12.25 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.17 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} d x \left (-3 b^3 c^2 e^2 \left (c+d x^2\right )^2+a^3 d (d e-c f) \left (3 c^2 f+2 d^2 e x^2+c d \left (3 e+4 f x^2\right )\right )+a b^2 c \left (-7 d^3 e^2 x^4+c^2 d f x^2 \left (22 e-f x^2\right )-2 c^3 f \left (-3 e+f x^2\right )+2 c d^2 e x^2 \left (-4 e+7 f x^2\right )\right )+a^2 b \left (-5 c^4 f^2+2 d^4 e^2 x^4+10 c^3 d f \left (e-f x^2\right )+2 c d^3 e x^2 \left (-2 e+f x^2\right )+c^2 d^2 \left (-8 e^2+8 e f x^2-7 f^2 x^4\right )\right )\right )-i b c \left (3 b^2 c^2 d e^2+a b c \left (7 d^2 e^2-14 c d e f+c^2 f^2\right )+a^2 d \left (-2 d^2 e^2-2 c d e f+7 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (3 b^2 c d e^2+3 a^2 c d f^2+a b \left (d^2 e^2-8 c d e f+c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b c^2 d (-b c+a d)^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \] Input:
Integrate[(e + f*x^2)^2/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)),x]
Output:
(Sqrt[b/a]*(Sqrt[b/a]*d*x*(-3*b^3*c^2*e^2*(c + d*x^2)^2 + a^3*d*(d*e - c*f )*(3*c^2*f + 2*d^2*e*x^2 + c*d*(3*e + 4*f*x^2)) + a*b^2*c*(-7*d^3*e^2*x^4 + c^2*d*f*x^2*(22*e - f*x^2) - 2*c^3*f*(-3*e + f*x^2) + 2*c*d^2*e*x^2*(-4* e + 7*f*x^2)) + a^2*b*(-5*c^4*f^2 + 2*d^4*e^2*x^4 + 10*c^3*d*f*(e - f*x^2) + 2*c*d^3*e*x^2*(-2*e + f*x^2) + c^2*d^2*(-8*e^2 + 8*e*f*x^2 - 7*f^2*x^4) )) - I*b*c*(3*b^2*c^2*d*e^2 + a*b*c*(7*d^2*e^2 - 14*c*d*e*f + c^2*f^2) + a ^2*d*(-2*d^2*e^2 - 2*c*d*e*f + 7*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*(c + d*x^2) *Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c* (-(b*c) + a*d)*(3*b^2*c*d*e^2 + 3*a^2*c*d*f^2 + a*b*(d^2*e^2 - 8*c*d*e*f + c^2*f^2))*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*EllipticF[I *ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b*c^2*d*(-(b*c) + a*d)^3*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))
Leaf count is larger than twice the leaf count of optimal. \(941\) vs. \(2(463)=926\).
Time = 1.21 (sec) , antiderivative size = 941, normalized size of antiderivative = 2.03, 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.
| \(\displaystyle \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}+\frac {2 e f x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}+\frac {f^2 x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d} \left (3 b^2 c^2+7 a b d c-2 a^2 d^2\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right ) e^2}{3 a c^{3/2} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {d} (9 b c-a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) e^2}{3 a \sqrt {c} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d (3 b c+a d) x \sqrt {b x^2+a} e^2}{3 a c (b c-a d)^2 \left (d x^2+c\right )^{3/2}}+\frac {b x e^2}{a (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}-\frac {2 \sqrt {d} (7 b c+a d) f \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right ) e}{3 \sqrt {c} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 b \sqrt {c} (3 b c+5 a d) 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 ) e}{3 a \sqrt {d} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {8 d f x \sqrt {b x^2+a} e}{3 (b c-a d)^2 \left (d x^2+c\right )^{3/2}}-\frac {2 f x e}{(b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {\sqrt {c} (b c+7 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 \sqrt {d} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} (5 b c+3 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 \sqrt {d} (b c-a d)^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {(b c+3 a d) f^2 x \sqrt {b x^2+a}}{3 b (b c-a d)^2 \left (d x^2+c\right )^{3/2}}+\frac {a f^2 x}{b (b c-a d) \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}\) |
Input:
Int[(e + f*x^2)^2/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)),x]
Output:
(b*e^2*x)/(a*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) - (2*e*f*x)/(( b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) + (a*f^2*x)/(b*(b*c - a*d)*S qrt[a + b*x^2]*(c + d*x^2)^(3/2)) + (d*(3*b*c + a*d)*e^2*x*Sqrt[a + b*x^2] )/(3*a*c*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - (8*d*e*f*x*Sqrt[a + b*x^2])/(3 *(b*c - a*d)^2*(c + d*x^2)^(3/2)) + ((b*c + 3*a*d)*f^2*x*Sqrt[a + b*x^2])/ (3*b*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (Sqrt[d]*(3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*e^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*c^(3/2)*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^ 2))]*Sqrt[c + d*x^2]) - (2*Sqrt[d]*(7*b*c + a*d)*e*f*Sqrt[a + b*x^2]*Ellip ticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]*(b*c - a*d) ^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*(b*c + 7*a*d)*f^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b *c)/(a*d)])/(3*Sqrt[d]*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))] *Sqrt[c + d*x^2]) - (b*Sqrt[d]*(9*b*c - a*d)*e^2*Sqrt[a + b*x^2]*EllipticF [ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*Sqrt[c]*(b*c - a*d)^3 *Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*b*Sqrt[c]*(3* b*c + 5*a*d)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*Sqrt[d]*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d* x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(5*b*c + 3*a*d)*f^2*Sqrt[a + b*x^2]*Ell ipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[d]*(b*c -...
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 = 18.11 (sec) , antiderivative size = 867, normalized size of antiderivative = 1.87
| 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}+b c \right ) x \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{a \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {x \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 c \,d^{2} \left (a d -b c \right )^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}-\frac {\left (b d \,x^{2}+a d \right ) x \left (4 a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}-2 a \,d^{3} e^{2}+b \,c^{3} f^{2}-8 b \,c^{2} d e f +7 b c \,d^{2} e^{2}\right )}{3 c^{2} d \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {a^{2} f^{2}-2 a b f e +b^{2} e^{2}}{\left (a d -b c \right )^{2} a}+\frac {b c \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{a \left (a d -b c \right )^{3}}+\frac {b \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}{3 d \left (a d -b c \right )^{2} c}-\frac {4 a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}-2 a \,d^{3} e^{2}+b \,c^{3} f^{2}-8 b \,c^{2} d e f +7 b c \,d^{2} e^{2}}{3 \left (a d -b c \right )^{2} d \,c^{2}}+\frac {a \left (4 a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}-2 a \,d^{3} e^{2}+b \,c^{3} f^{2}-8 b \,c^{2} d e f +7 b c \,d^{2} e^{2}\right )}{3 c^{2} \left (a d -b c \right )^{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 {b d \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{\left (a d -b c \right )^{3} a}+\frac {b \left (4 a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}-2 a \,d^{3} e^{2}+b \,c^{3} f^{2}-8 b \,c^{2} d e f +7 b c \,d^{2} e^{2}\right )}{3 \left (a d -b c \right )^{3} c^{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}}\) | \(867\) |
| default | \(\text {Expression too large to display}\) | \(2511\) |
Input:
int((f*x^2+e)^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(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)*(-(b*d*x^2+b*c )/a/(a*d-b*c)^3*x*(a^2*f^2-2*a*b*e*f+b^2*e^2)/((x^2+a/b)*(b*d*x^2+b*c))^(1 /2)+1/3/c/d^2/(a*d-b*c)^2*x*(c^2*f^2-2*c*d*e*f+d^2*e^2)*(b*d*x^4+a*d*x^2+b *c*x^2+a*c)^(1/2)/(x^2+c/d)^2-1/3*(b*d*x^2+a*d)/c^2/d/(a*d-b*c)^3*x*(4*a*c ^2*d*f^2-2*a*c*d^2*e*f-2*a*d^3*e^2+b*c^3*f^2-8*b*c^2*d*e*f+7*b*c*d^2*e^2)/ ((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+((a^2*f^2-2*a*b*e*f+b^2*e^2)/(a*d-b*c)^2/a +b*c/a/(a*d-b*c)^3*(a^2*f^2-2*a*b*e*f+b^2*e^2)+1/3*b/d*(c^2*f^2-2*c*d*e*f+ d^2*e^2)/(a*d-b*c)^2/c-1/3/(a*d-b*c)^2/d*(4*a*c^2*d*f^2-2*a*c*d^2*e*f-2*a* d^3*e^2+b*c^3*f^2-8*b*c^2*d*e*f+7*b*c*d^2*e^2)/c^2+1/3*a/c^2/(a*d-b*c)^3*( 4*a*c^2*d*f^2-2*a*c*d^2*e*f-2*a*d^3*e^2+b*c^3*f^2-8*b*c^2*d*e*f+7*b*c*d^2* e^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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(b*d*( a^2*f^2-2*a*b*e*f+b^2*e^2)/(a*d-b*c)^3/a+1/3*b*(4*a*c^2*d*f^2-2*a*c*d^2*e* f-2*a*d^3*e^2+b*c^3*f^2-8*b*c^2*d*e*f+7*b*c*d^2*e^2)/(a*d-b*c)^3/c^2)*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. 1949 vs. \(2 (436) = 872\).
Time = 0.16 (sec) , antiderivative size = 1949, normalized size of antiderivative = 4.21 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x, algorithm="fricas ")
Output:
-1/3*((((3*b^5*c^2*d^3 + 7*a*b^4*c*d^4 - 2*a^2*b^3*d^5)*e^2 - 2*(7*a*b^4*c ^2*d^3 + a^2*b^3*c*d^4)*e*f + (a*b^4*c^3*d^2 + 7*a^2*b^3*c^2*d^3)*f^2)*x^6 + ((6*b^5*c^3*d^2 + 17*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 - 2*a^3*b^2*d^5)*e ^2 - 2*(14*a*b^4*c^3*d^2 + 9*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4)*e*f + (2*a*b ^4*c^4*d + 15*a^2*b^3*c^3*d^2 + 7*a^3*b^2*c^2*d^3)*f^2)*x^4 + (3*a*b^4*c^4 *d + 7*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2*d^3)*e^2 - 2*(7*a^2*b^3*c^4*d + a^3 *b^2*c^3*d^2)*e*f + (a^2*b^3*c^5 + 7*a^3*b^2*c^4*d)*f^2 + ((3*b^5*c^4*d + 13*a*b^4*c^3*d^2 + 12*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4)*e^2 - 2*(7*a*b^4* c^4*d + 15*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3)*e*f + (a*b^4*c^5 + 9*a^2*b ^3*c^4*d + 14*a^3*b^2*c^3*d^2)*f^2)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(a rcsin(x*sqrt(-b/a)), a*d/(b*c)) - (((3*b^5*c^2*d^3 + (9*a^2*b^3 + 7*a*b^4) *c*d^4 - (a^3*b^2 + 2*a^2*b^3)*d^5)*e^2 - 2*((3*a^2*b^3 + 7*a*b^4)*c^2*d^3 + (5*a^3*b^2 + a^2*b^3)*c*d^4)*e*f + (a*b^4*c^3*d^2 + 3*a^4*b*c*d^4 + (5* a^3*b^2 + 7*a^2*b^3)*c^2*d^3)*f^2)*x^6 + ((6*b^5*c^3*d^2 + (18*a^2*b^3 + 1 7*a*b^4)*c^2*d^3 + (7*a^3*b^2 + 3*a^2*b^3)*c*d^4 - (a^4*b + 2*a^3*b^2)*d^5 )*e^2 - 2*(2*(3*a^2*b^3 + 7*a*b^4)*c^3*d^2 + (13*a^3*b^2 + 9*a^2*b^3)*c^2* d^3 + (5*a^4*b + a^3*b^2)*c*d^4)*e*f + (2*a*b^4*c^4*d + 3*a^5*c*d^4 + 5*(2 *a^3*b^2 + 3*a^2*b^3)*c^3*d^2 + (11*a^4*b + 7*a^3*b^2)*c^2*d^3)*f^2)*x^4 + (3*a*b^4*c^4*d + (9*a^3*b^2 + 7*a^2*b^3)*c^3*d^2 - (a^4*b + 2*a^3*b^2)*c^ 2*d^3)*e^2 - 2*((3*a^3*b^2 + 7*a^2*b^3)*c^4*d + (5*a^4*b + a^3*b^2)*c^3...
Timed out. \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((f*x**2+e)**2/(b*x**2+a)**(3/2)/(d*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x, algorithm="maxima ")
Output:
integrate((f*x^2 + e)^2/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(5/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x, algorithm="giac")
Output:
integrate((f*x^2 + e)^2/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(5/2)), x)
Timed out. \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^2}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:
int((e + f*x^2)^2/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)),x)
Output:
int((e + f*x^2)^2/((a + b*x^2)^(3/2)*(c + d*x^2)^(5/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((f*x^2+e)^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(5/2),x)
Output:
( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*e*f*x + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3 + 3*a**2*c**2*d*x**2 + 3*a**2*c*d**2*x**4 + a* *2*d**3*x**6 + 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 + 6*a*b*c*d**2*x**6 + 2 *a*b*d**3*x**8 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**8 + b**2*d**3*x**10),x)*a**2*c**2*d*f**2 + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3 + 3*a**2*c**2*d*x**2 + 3*a**2*c*d**2*x**4 + a**2* d**3*x**6 + 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 + 6*a*b*c*d**2*x**6 + 2*a* b*d**3*x**8 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**8 + b **2*d**3*x**10),x)*a**2*c*d**2*f**2*x**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3 + 3*a**2*c**2*d*x**2 + 3*a**2*c*d**2*x**4 + a**2* d**3*x**6 + 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 + 6*a*b*c*d**2*x**6 + 2*a* b*d**3*x**8 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**8 + b **2*d**3*x**10),x)*a**2*d**3*f**2*x**4 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3 + 3*a**2*c**2*d*x**2 + 3*a**2*c*d**2*x**4 + a**2* d**3*x**6 + 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 + 6*a*b*c*d**2*x**6 + 2*a* b*d**3*x**8 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**8 + b **2*d**3*x**10),x)*a*b*c**2*d*e*f + int((sqrt(c + d*x**2)*sqrt(a + b*x**2) *x**4)/(a**2*c**3 + 3*a**2*c**2*d*x**2 + 3*a**2*c*d**2*x**4 + a**2*d**3*x* *6 + 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 + 6*a*b*c*d**2*x**6 + 2*a*b*d**3* x**8 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**8 + b**2*...