Integrand size = 32, antiderivative size = 329 \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \, dx=-\frac {(d e-c f)^2 x \sqrt {a+b x^2}}{3 c d (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {2 (d e-c f) (b c (2 d e+c f)-a d (d e+2 c f)) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} d^{3/2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (3 b^2 c d e^2+3 a^2 c d f^2-a b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} d^{3/2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
-1/3*(-c*f+d*e)^2*x*(b*x^2+a)^(1/2)/c/d/(-a*d+b*c)/(d*x^2+c)^(3/2)-2/3*(-c *f+d*e)*(b*c*(c*f+2*d*e)-a*d*(2*c*f+d*e))*(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))/c^(3/2)/d^(3/2)/(-a*d+b*c )^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/3*(3*b^2*c*d*e^2+3*a ^2*c*d*f^2-a*b*(c^2*f^2+4*c*d*e*f+d^2*e^2))*(b*x^2+a)^(1/2)*InverseJacobiA M(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a/c^(1/2)/d^(3/2)/(-a*d+b*c )^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 = 11.73 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.06 \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d (d e-c f) x \left (a+b x^2\right ) \left (-b c \left (c^2 f+4 d^2 e x^2+c d \left (5 e+2 f x^2\right )\right )+a d \left (3 c^2 f+2 d^2 e x^2+c d \left (3 e+4 f x^2\right )\right )\right )-2 i b c (-d e+c f) (-b c (2 d e+c f)+a d (d e+2 c f)) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (3 a c d f^2+b \left (d^2 e^2-2 c d e f-2 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} c^2 d^2 (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \] Input:
Integrate[(e + f*x^2)^2/(Sqrt[a + b*x^2]*(c + d*x^2)^(5/2)),x]
Output:
(Sqrt[b/a]*d*(d*e - c*f)*x*(a + b*x^2)*(-(b*c*(c^2*f + 4*d^2*e*x^2 + c*d*( 5*e + 2*f*x^2))) + a*d*(3*c^2*f + 2*d^2*e*x^2 + c*d*(3*e + 4*f*x^2))) - (2 *I)*b*c*(-(d*e) + c*f)*(-(b*c*(2*d*e + c*f)) + a*d*(d*e + 2*c*f))*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*a*c*d*f^2 + b*(d^2*e^2 - 2*c*d*e* f - 2*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Ellipt icF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*c^2*d^2*(b*c - a*d) ^2*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))
Leaf count is larger than twice the leaf count of optimal. \(772\) vs. \(2(329)=658\).
Time = 0.94 (sec) , antiderivative size = 772, 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.
| \(\displaystyle \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}+\frac {2 e f x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}+\frac {f^2 x^4}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {d} e^2 \sqrt {a+b x^2} (2 b c-a d) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {c} f^2 \sqrt {a+b x^2} (b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 d^{3/2} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 \sqrt {c} f^2 \sqrt {a+b x^2} (b c-2 a d) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 d^{3/2} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b e^2 \sqrt {a+b x^2} (3 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {4 b \sqrt {c} e f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 \sqrt {d} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 e f \sqrt {a+b x^2} (a d+b c) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {d e^2 x \sqrt {a+b x^2}}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {2 e f x \sqrt {a+b x^2}}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {c f^2 x \sqrt {a+b x^2}}{3 d \left (c+d x^2\right )^{3/2} (b c-a d)}\) |
Input:
Int[(e + f*x^2)^2/(Sqrt[a + b*x^2]*(c + d*x^2)^(5/2)),x]
Output:
-1/3*(d*e^2*x*Sqrt[a + b*x^2])/(c*(b*c - a*d)*(c + d*x^2)^(3/2)) + (2*e*f* x*Sqrt[a + b*x^2])/(3*(b*c - a*d)*(c + d*x^2)^(3/2)) - (c*f^2*x*Sqrt[a + b *x^2])/(3*d*(b*c - a*d)*(c + d*x^2)^(3/2)) - (2*Sqrt[d]*(2*b*c - a*d)*e^2* Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/( 3*c^(3/2)*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x ^2]) + (2*(b*c + a*d)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqr t[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]*Sqrt[d]*(b*c - a*d)^2*Sqrt[(c*(a + b*x ^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c]*(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*d^ (3/2)*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*(3*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]*Sqrt[d]*(b*c - a*d)^2*Sqrt[(c*(a + b*x ^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (4*b*Sqrt[c]*e*f*Sqrt[a + b*x^2]* EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[d]*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]* (b*c - 3*a*d)*f^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*d^(3/2)*(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. \(698\) vs. \(2(306)=612\).
Time = 8.72 (sec) , antiderivative size = 699, normalized size of antiderivative = 2.12
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {x \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 c \,d^{3} \left (a d -b c \right ) \left (x^{2}+\frac {c}{d}\right )^{2}}-\frac {2 \left (b d \,x^{2}+a d \right ) x \left (2 a \,c^{2} d \,f^{2}-a c e f \,d^{2}-a \,d^{3} e^{2}-b \,c^{3} f^{2}-b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{3 c^{2} d^{2} \left (a d -b c \right )^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {f^{2}}{d^{2}}+\frac {b \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}{3 d^{2} \left (a d -b c \right ) c}-\frac {2 \left (2 a \,c^{2} d \,f^{2}-a c e f \,d^{2}-a \,d^{3} e^{2}-b \,c^{3} f^{2}-b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{3 d^{2} \left (a d -b c \right ) c^{2}}+\frac {2 a \left (2 a \,c^{2} d \,f^{2}-a c e f \,d^{2}-a \,d^{3} e^{2}-b \,c^{3} f^{2}-b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right )}{3 d \,c^{2} \left (a d -b c \right )^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {2 b \left (2 a \,c^{2} d \,f^{2}-a c e f \,d^{2}-a \,d^{3} e^{2}-b \,c^{3} f^{2}-b \,c^{2} d e f +2 b c \,d^{2} e^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 d^{2} \left (a d -b c \right )^{2} c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(699\) |
| default | \(\text {Expression too large to display}\) | \(2106\) |
Input:
int((f*x^2+e)^2/(b*x^2+a)^(1/2)/(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)*(1/3/c/d^3/(a* d-b*c)*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)^2-2/3*(b*d*x^2+a*d)/c^2/d^2/(a*d-b*c)^2*x*(2*a*c^2*d*f^2-a*c*d^2* e*f-a*d^3*e^2-b*c^3*f^2-b*c^2*d*e*f+2*b*c*d^2*e^2)/((x^2+c/d)*(b*d*x^2+a*d ))^(1/2)+(f^2/d^2+1/3/d^2*b*(c^2*f^2-2*c*d*e*f+d^2*e^2)/(a*d-b*c)/c-2/3/d^ 2/(a*d-b*c)*(2*a*c^2*d*f^2-a*c*d^2*e*f-a*d^3*e^2-b*c^3*f^2-b*c^2*d*e*f+2*b *c*d^2*e^2)/c^2+2/3*a/d/c^2/(a*d-b*c)^2*(2*a*c^2*d*f^2-a*c*d^2*e*f-a*d^3*e ^2-b*c^3*f^2-b*c^2*d*e*f+2*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))-2/3*b/d^2*(2*a*c^2*d*f^2-a*c*d^2*e*f-a*d^3*e^ 2-b*c^3*f^2-b*c^2*d*e*f+2*b*c*d^2*e^2)/(a*d-b*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)*(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. 981 vs. \(2 (306) = 612\).
Time = 0.12 (sec) , antiderivative size = 981, normalized size of antiderivative = 2.98 \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(5/2),x, algorithm="fricas ")
Output:
1/3*(2*(((2*b^3*c*d^4 - a*b^2*d^5)*e^2 - (b^3*c^2*d^3 + a*b^2*c*d^4)*e*f - (b^3*c^3*d^2 - 2*a*b^2*c^2*d^3)*f^2)*x^4 + (2*b^3*c^3*d^2 - a*b^2*c^2*d^3 )*e^2 - (b^3*c^4*d + a*b^2*c^3*d^2)*e*f - (b^3*c^5 - 2*a*b^2*c^4*d)*f^2 + 2*((2*b^3*c^2*d^3 - a*b^2*c*d^4)*e^2 - (b^3*c^3*d^2 + a*b^2*c^2*d^3)*e*f - (b^3*c^4*d - 2*a*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*a*b^2 + 4*b^3)*c*d^4 - (a^2*b + 2* a*b^2)*d^5)*e^2 - 2*(b^3*c^2*d^3 + (2*a^2*b + a*b^2)*c*d^4)*e*f - (2*b^3*c ^3*d^2 - 3*a^3*c*d^4 + (a^2*b - 4*a*b^2)*c^2*d^3)*f^2)*x^4 + ((3*a*b^2 + 4 *b^3)*c^3*d^2 - (a^2*b + 2*a*b^2)*c^2*d^3)*e^2 - 2*(b^3*c^4*d + (2*a^2*b + a*b^2)*c^3*d^2)*e*f - (2*b^3*c^5 - 3*a^3*c^3*d^2 + (a^2*b - 4*a*b^2)*c^4* d)*f^2 + 2*(((3*a*b^2 + 4*b^3)*c^2*d^3 - (a^2*b + 2*a*b^2)*c*d^4)*e^2 - 2* (b^3*c^3*d^2 + (2*a^2*b + a*b^2)*c^2*d^3)*e*f - (2*b^3*c^4*d - 3*a^3*c^2*d ^3 + (a^2*b - 4*a*b^2)*c^3*d^2)*f^2)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_f( arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (2*((2*a*b^2*c*d^4 - a^2*b*d^5)*e^2 - ( a*b^2*c^2*d^3 + a^2*b*c*d^4)*e*f - (a*b^2*c^3*d^2 - 2*a^2*b*c^2*d^3)*f^2)* x^3 - (4*a*b^2*c^3*d^2*e*f - (5*a*b^2*c^2*d^3 - 3*a^2*b*c*d^4)*e^2 + (a*b^ 2*c^4*d - 3*a^2*b*c^3*d^2)*f^2)*x)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a*b^3 *c^6*d^2 - 2*a^2*b^2*c^5*d^3 + a^3*b*c^4*d^4 + (a*b^3*c^4*d^4 - 2*a^2*b^2* c^3*d^5 + a^3*b*c^2*d^6)*x^4 + 2*(a*b^3*c^5*d^3 - 2*a^2*b^2*c^4*d^4 + a^3* b*c^3*d^5)*x^2)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (e + f x^{2}\right )^{2}}{\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((f*x**2+e)**2/(b*x**2+a)**(1/2)/(d*x**2+c)**(5/2),x)
Output:
Integral((e + f*x**2)**2/(sqrt(a + b*x**2)*(c + d*x**2)**(5/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(5/2),x, algorithm="maxima ")
Output:
integrate((f*x^2 + e)^2/(sqrt(b*x^2 + a)*(d*x^2 + c)^(5/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(5/2),x, algorithm="giac")
Output:
integrate((f*x^2 + e)^2/(sqrt(b*x^2 + a)*(d*x^2 + c)^(5/2)), x)
Timed out. \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^2}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:
int((e + f*x^2)^2/((a + b*x^2)^(1/2)*(c + d*x^2)^(5/2)),x)
Output:
int((e + f*x^2)^2/((a + b*x^2)^(1/2)*(c + d*x^2)^(5/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((f*x^2+e)^2/(b*x^2+a)^(1/2)/(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*d + 3*a**2*c**2*d**2*x**2 + 3*a**2*c*d**3*x**4 + a**2*d**4*x**6 - a*b*c**4 - 2*a*b*c**3*d*x**2 + 2*a*b*c*d**3*x**6 + a*b *d**4*x**8 - b**2*c**4*x**2 - 3*b**2*c**3*d*x**4 - 3*b**2*c**2*d**2*x**6 - b**2*c*d**3*x**8),x)*a**2*c**2*d**2*f**2 + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3*d + 3*a**2*c**2*d**2*x**2 + 3*a**2*c*d**3*x**4 + a**2*d**4*x**6 - a*b*c**4 - 2*a*b*c**3*d*x**2 + 2*a*b*c*d**3*x**6 + a*b *d**4*x**8 - b**2*c**4*x**2 - 3*b**2*c**3*d*x**4 - 3*b**2*c**2*d**2*x**6 - b**2*c*d**3*x**8),x)*a**2*c*d**3*f**2*x**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3*d + 3*a**2*c**2*d**2*x**2 + 3*a**2*c*d**3*x**4 + a**2*d**4*x**6 - a*b*c**4 - 2*a*b*c**3*d*x**2 + 2*a*b*c*d**3*x**6 + a*b *d**4*x**8 - b**2*c**4*x**2 - 3*b**2*c**3*d*x**4 - 3*b**2*c**2*d**2*x**6 - b**2*c*d**3*x**8),x)*a**2*d**4*f**2*x**4 - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c**3*d + 3*a**2*c**2*d**2*x**2 + 3*a**2*c*d**3*x**4 + a**2*d**4*x**6 - a*b*c**4 - 2*a*b*c**3*d*x**2 + 2*a*b*c*d**3*x**6 + a*b *d**4*x**8 - b**2*c**4*x**2 - 3*b**2*c**3*d*x**4 - 3*b**2*c**2*d**2*x**6 - b**2*c*d**3*x**8),x)*a*b*c**3*d*f**2 - int((sqrt(c + d*x**2)*sqrt(a + b*x **2)*x**4)/(a**2*c**3*d + 3*a**2*c**2*d**2*x**2 + 3*a**2*c*d**3*x**4 + a** 2*d**4*x**6 - a*b*c**4 - 2*a*b*c**3*d*x**2 + 2*a*b*c*d**3*x**6 + a*b*d**4* x**8 - b**2*c**4*x**2 - 3*b**2*c**3*d*x**4 - 3*b**2*c**2*d**2*x**6 - b*...