Integrand size = 32, antiderivative size = 331 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\frac {(b e-a f)^2 x \sqrt {c+d x^2}}{3 a b (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {2 (b e-a f) \left (b^2 c e-a^2 d f-2 a b (d e-c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} b^{3/2} (b c-a d)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\left (b^2 c d e^2+a^2 c d f^2-a b \left (3 d^2 e^2-4 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{3/2} c (b c-a d)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/3*(-a*f+b*e)^2*x*(d*x^2+c)^(1/2)/a/b/(-a*d+b*c)/(b*x^2+a)^(3/2)+2/3*(-a* f+b*e)*(b^2*c*e-a^2*d*f-2*a*b*(-c*f+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^(3/2)/(-a*d+b*c )^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/3*(b^2*c*d*e^2+a^2*c *d*f^2-a*b*(3*c^2*f^2-4*c*d*e*f+3*d^2*e^2))*(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^(3/2)/c/(-a*d+b*c )^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 = 12.25 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.03 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} (-b e+a f) x \left (c+d x^2\right ) \left (a^3 d f-2 b^3 c e x^2+a b^2 \left (-3 c e+4 d e x^2-4 c f x^2\right )+a^2 b \left (5 d e-3 c f+2 d f x^2\right )\right )+2 i c (-b e+a f) \left (-b^2 c e+a^2 d f+2 a b (d e-c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i (-b c+a d) \left (-2 b^2 c e^2+a^2 c f^2+a b e (3 d e-2 c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^3 \left (\frac {b}{a}\right )^{3/2} (b c-a d)^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \] Input:
Integrate[(e + f*x^2)^2/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]
Output:
(Sqrt[b/a]*(-(b*e) + a*f)*x*(c + d*x^2)*(a^3*d*f - 2*b^3*c*e*x^2 + a*b^2*( -3*c*e + 4*d*e*x^2 - 4*c*f*x^2) + a^2*b*(5*d*e - 3*c*f + 2*d*f*x^2)) + (2* I)*c*(-(b*e) + a*f)*(-(b^2*c*e) + a^2*d*f + 2*a*b*(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*(-(b*c) + a*d)*(-2*b^2*c*e^2 + a^2*c*f^2 + a*b*e*(3*d*e - 2*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*a^3*(b/a)^(3/2)*(b*c - a*d)^2*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(779\) vs. \(2(331)=662\).
Time = 0.94 (sec) , antiderivative size = 779, normalized size of antiderivative = 2.35, 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.
| \(\displaystyle \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}+\frac {2 e f x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}+\frac {f^2 x^4}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {b} e^2 \sqrt {c+d x^2} (b c-2 a d) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {a+b x^2} (b c-a d)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} e^2 \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 \sqrt {a} f^2 \sqrt {c+d x^2} (2 b c-a d) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 b^{3/2} \sqrt {a+b x^2} (b c-a d)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {4 c^{3/2} \sqrt {d} e f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {c^{3/2} f^2 \sqrt {a+b x^2} (3 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a b \sqrt {d} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 e f \sqrt {c+d x^2} (a d+b c) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a+b x^2} (b c-a d)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b e^2 x \sqrt {c+d x^2}}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}-\frac {2 e f x \sqrt {c+d x^2}}{3 \left (a+b x^2\right )^{3/2} (b c-a d)}+\frac {a f^2 x \sqrt {c+d x^2}}{3 b \left (a+b x^2\right )^{3/2} (b c-a d)}\) |
Input:
Int[(e + f*x^2)^2/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]
Output:
(b*e^2*x*Sqrt[c + d*x^2])/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)) - (2*e*f*x*S qrt[c + d*x^2])/(3*(b*c - a*d)*(a + b*x^2)^(3/2)) + (a*f^2*x*Sqrt[c + d*x^ 2])/(3*b*(b*c - a*d)*(a + b*x^2)^(3/2)) + (2*Sqrt[b]*(b*c - 2*a*d)*e^2*Sqr t[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(3*a ^(3/2)*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))] ) + (2*(b*c + a*d)*e*f*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a ]], 1 - (a*d)/(b*c)])/(3*Sqrt[a]*Sqrt[b]*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqr t[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (2*Sqrt[a]*(2*b*c - a*d)*f^2*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(3*b^(3/ 2)*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (Sqrt[c]*Sqrt[d]*(b*c - 3*a*d)*e^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[ d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2) )/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (4*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*c - a* d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*(3* b*c - a*d)*f^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*b*Sqrt[d]*(b*c - a*d)^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]
Leaf count of result is larger than twice the leaf count of optimal. \(682\) vs. \(2(308)=616\).
Time = 8.53 (sec) , antiderivative size = 683, normalized size of antiderivative = 2.06
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {x \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b^{3} a \left (a d -b c \right ) \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {2 \left (b d \,x^{2}+b c \right ) x \left (a^{3} d \,f^{2}-2 a^{2} c \,f^{2} b +a^{2} b d e f +a c e f \,b^{2}-2 a \,b^{2} d \,e^{2}+b^{3} c \,e^{2}\right )}{3 b^{2} a^{2} \left (a d -b c \right )^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {f^{2}}{b^{2}}-\frac {d \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{3 b^{2} \left (a d -b c \right ) a}-\frac {2 \left (a^{3} d \,f^{2}-2 a^{2} c \,f^{2} b +a^{2} b d e f +a c e f \,b^{2}-2 a \,b^{2} d \,e^{2}+b^{3} c \,e^{2}\right )}{3 \left (a d -b c \right ) b^{2} a^{2}}-\frac {2 c \left (a^{3} d \,f^{2}-2 a^{2} c \,f^{2} b +a^{2} b d e f +a c e f \,b^{2}-2 a \,b^{2} d \,e^{2}+b^{3} c \,e^{2}\right )}{3 b \,a^{2} \left (a d -b c \right )^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {2 \left (a^{3} d \,f^{2}-2 a^{2} c \,f^{2} b +a^{2} b d e f +a c e f \,b^{2}-2 a \,b^{2} d \,e^{2}+b^{3} c \,e^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 b \left (a d -b c \right )^{2} a^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(683\) |
| default | \(\text {Expression too large to display}\) | \(2078\) |
Input:
int((f*x^2+e)^2/(b*x^2+a)^(5/2)/(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)*(-1/3/b^3/a/(a *d-b*c)*x*(a^2*f^2-2*a*b*e*f+b^2*e^2)*(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)/b^2/a^2/(a*d-b*c)^2*x*(a^3*d*f^2-2*a^2*b*c*f ^2+a^2*b*d*e*f+a*b^2*c*e*f-2*a*b^2*d*e^2+b^3*c*e^2)/((x^2+a/b)*(b*d*x^2+b* c))^(1/2)+(f^2/b^2-1/3/b^2*d*(a^2*f^2-2*a*b*e*f+b^2*e^2)/(a*d-b*c)/a-2/3/( a*d-b*c)/b^2*(a^3*d*f^2-2*a^2*b*c*f^2+a^2*b*d*e*f+a*b^2*c*e*f-2*a*b^2*d*e^ 2+b^3*c*e^2)/a^2-2/3/b*c/a^2/(a*d-b*c)^2*(a^3*d*f^2-2*a^2*b*c*f^2+a^2*b*d* e*f+a*b^2*c*e*f-2*a*b^2*d*e^2+b^3*c*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))+2/3/b*(a^3*d*f^2-2*a^2*b*c*f^2+a^2*b*d*e*f+a *b^2*c*e*f-2*a*b^2*d*e^2+b^3*c*e^2)/(a*d-b*c)^2/a^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)*(Elliptic F(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. 950 vs. \(2 (308) = 616\).
Time = 0.13 (sec) , antiderivative size = 950, normalized size of antiderivative = 2.87 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="fricas ")
Output:
-1/3*(2*(((b^6*c^2 - 2*a*b^5*c*d)*e^2 + (a*b^5*c^2 + a^2*b^4*c*d)*e*f - (2 *a^2*b^4*c^2 - a^3*b^3*c*d)*f^2)*x^4 + (a^2*b^4*c^2 - 2*a^3*b^3*c*d)*e^2 + (a^3*b^3*c^2 + a^4*b^2*c*d)*e*f - (2*a^4*b^2*c^2 - a^5*b*c*d)*f^2 + 2*((a *b^5*c^2 - 2*a^2*b^4*c*d)*e^2 + (a^2*b^4*c^2 + a^3*b^3*c*d)*e*f - (2*a^3*b ^3*c^2 - a^4*b^2*c*d)*f^2)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*s qrt(-b/a)), a*d/(b*c)) - (((2*b^6*c^2 - 3*a^3*b^3*d^2 + (a^2*b^4 - 4*a*b^5 )*c*d)*e^2 + 2*(a*b^5*c^2 + (2*a^3*b^3 + a^2*b^4)*c*d)*e*f - ((3*a^3*b^3 + 4*a^2*b^4)*c^2 - (a^4*b^2 + 2*a^3*b^3)*c*d)*f^2)*x^4 + (2*a^2*b^4*c^2 - 3 *a^5*b*d^2 + (a^4*b^2 - 4*a^3*b^3)*c*d)*e^2 + 2*(a^3*b^3*c^2 + (2*a^5*b + a^4*b^2)*c*d)*e*f - ((3*a^5*b + 4*a^4*b^2)*c^2 - (a^6 + 2*a^5*b)*c*d)*f^2 + 2*((2*a*b^5*c^2 - 3*a^4*b^2*d^2 + (a^3*b^3 - 4*a^2*b^4)*c*d)*e^2 + 2*(a^ 2*b^4*c^2 + (2*a^4*b^2 + a^3*b^3)*c*d)*e*f - ((3*a^4*b^2 + 4*a^3*b^3)*c^2 - (a^5*b + 2*a^4*b^2)*c*d)*f^2)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsi n(x*sqrt(-b/a)), a*d/(b*c)) - (2*((a*b^5*c^2 - 2*a^2*b^4*c*d)*e^2 + (a^2*b ^4*c^2 + a^3*b^3*c*d)*e*f - (2*a^3*b^3*c^2 - a^4*b^2*c*d)*f^2)*x^3 + (4*a^ 4*b^2*c*d*e*f + (3*a^2*b^4*c^2 - 5*a^3*b^3*c*d)*e^2 - (3*a^4*b^2*c^2 - a^5 *b*c*d)*f^2)*x)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a^5*b^4*c^3 - 2*a^6*b^3* c^2*d + a^7*b^2*c*d^2 + (a^3*b^6*c^3 - 2*a^4*b^5*c^2*d + a^5*b^4*c*d^2)*x^ 4 + 2*(a^4*b^5*c^3 - 2*a^5*b^4*c^2*d + a^6*b^3*c*d^2)*x^2)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int \frac {\left (e + f x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate((f*x**2+e)**2/(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)
Output:
Integral((e + f*x**2)**2/((a + b*x**2)**(5/2)*sqrt(c + d*x**2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="maxima ")
Output:
integrate((f*x^2 + e)^2/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate((f*x^2 + e)^2/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)), x)
Timed out. \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^2}{{\left (b\,x^2+a\right )}^{5/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int((e + f*x^2)^2/((a + b*x^2)^(5/2)*(c + d*x^2)^(1/2)),x)
Output:
int((e + f*x^2)^2/((a + b*x^2)^(5/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\text {too large to display} \] Input:
int((f*x^2+e)^2/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),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**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**4*d**2*f**2 - 2*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**3*b*c*d*f**2 + 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**3*b*d**2*e*f + 2*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**3*b*d**2*f**2*x**2 + 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*...