Integrand size = 32, antiderivative size = 415 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\left (a d (d e-c f)^2-b c \left (d^2 e^2-2 c d e f+2 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{3 c d^3 \left (c+d x^2\right )^{3/2}}+\frac {b f^2 x^5 \sqrt {a+b x^2}}{3 d \left (c+d x^2\right )^{3/2}}+\frac {2 f (3 b d e-3 b c f+2 a d f) x \sqrt {a+b x^2}}{3 d^3 \sqrt {c+d x^2}}+\frac {2 \left (a d \left (d^2 e^2+c d e f-4 c^2 f^2\right )+b c \left (d^2 e^2-8 c d e f+8 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\left (3 a c d f^2-b \left (d^2 e^2-8 c d e f+8 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 \sqrt {c} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/3*(a*d*(-c*f+d*e)^2-b*c*(2*c^2*f^2-2*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2) /c/d^3/(d*x^2+c)^(3/2)+1/3*b*f^2*x^5*(b*x^2+a)^(1/2)/d/(d*x^2+c)^(3/2)+2/3 *f*(2*a*d*f-3*b*c*f+3*b*d*e)*x*(b*x^2+a)^(1/2)/d^3/(d*x^2+c)^(1/2)+2/3*(a* d*(-4*c^2*f^2+c*d*e*f+d^2*e^2)+b*c*(8*c^2*f^2-8*c*d*e*f+d^2*e^2))*(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^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/3*(3*a*c* d*f^2-b*(8*c^2*f^2-8*c*d*e*f+d^2*e^2))*(b*x^2+a)^(1/2)*InverseJacobiAM(arc tan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/c^(1/2)/d^(7/2)/(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 = 7.55 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (a d (d e-c f) \left (3 c^2 f+2 d^2 e x^2+c d \left (3 e+4 f x^2\right )\right )+b c \left (8 c^3 f^2+2 d^3 e^2 x^2+2 c^2 d f \left (-4 e+5 f x^2\right )+c d^2 \left (e^2-10 e f x^2+f^2 x^4\right )\right )\right )-2 i b c \left (-a d \left (d^2 e^2+c d e f-4 c^2 f^2\right )-b c \left (d^2 e^2-8 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (3 a^2 c d^2 f^2+a b d \left (d^2 e^2+10 c d e f-16 c^2 f^2\right )+2 b^2 c \left (d^2 e^2-8 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} c^2 d^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \] Input:
Integrate[((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(5/2),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(a*d*(d*e - c*f)*(3*c^2*f + 2*d^2*e*x^2 + c*d*( 3*e + 4*f*x^2)) + b*c*(8*c^3*f^2 + 2*d^3*e^2*x^2 + 2*c^2*d*f*(-4*e + 5*f*x ^2) + c*d^2*(e^2 - 10*e*f*x^2 + f^2*x^4))) - (2*I)*b*c*(-(a*d*(d^2*e^2 + c *d*e*f - 4*c^2*f^2)) - b*c*(d^2*e^2 - 8*c*d*e*f + 8*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*(3*a^2*c*d^2*f^2 + a*b*d*(d^2*e^2 + 10*c*d*e*f - 16*c^2 *f^2) + 2*b^2*c*(d^2*e^2 - 8*c*d*e*f + 8*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*Sqrt[b/a]*c^2*d^4*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))
Leaf count is larger than twice the leaf count of optimal. \(906\) vs. \(2(415)=830\).
Time = 1.23 (sec) , antiderivative size = 906, normalized size of antiderivative = 2.18, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 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 {(2 b c-a d) f^2 \sqrt {b x^2+a} x^3}{c d^2 \sqrt {d x^2+c}}-\frac {f^2 \left (b x^2+a\right )^{3/2} x^3}{3 d \left (d x^2+c\right )^{3/2}}+\frac {(8 b c-3 a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 c d^3}-\frac {8 (2 b c-a d) f^2 \sqrt {b x^2+a} x}{3 d^3 \sqrt {d x^2+c}}-\frac {2 (4 b c-a d) e f \sqrt {b x^2+a} x}{3 c d^2 \sqrt {d x^2+c}}+\frac {2 (8 b c-a d) e f \sqrt {b x^2+a} x}{3 c d^2 \sqrt {d x^2+c}}-\frac {2 e f \left (b x^2+a\right )^{3/2} x}{3 d \left (d x^2+c\right )^{3/2}}-\frac {(b c-a d) e^2 \sqrt {b x^2+a} x}{3 c d \left (d x^2+c\right )^{3/2}}+\frac {2 (b c+a d) 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 )}{3 c^{3/2} 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 {8 \sqrt {c} (2 b c-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 d^{7/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 (8 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 \sqrt {c} 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 {b 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 \sqrt {c} 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} (8 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 d^{7/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {8 b \sqrt {c} 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^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(5/2),x]
Output:
-1/3*((b*c - a*d)*e^2*x*Sqrt[a + b*x^2])/(c*d*(c + d*x^2)^(3/2)) - (2*e*f* x*(a + b*x^2)^(3/2))/(3*d*(c + d*x^2)^(3/2)) - (f^2*x^3*(a + b*x^2)^(3/2)) /(3*d*(c + d*x^2)^(3/2)) - (2*(4*b*c - a*d)*e*f*x*Sqrt[a + b*x^2])/(3*c*d^ 2*Sqrt[c + d*x^2]) + (2*(8*b*c - a*d)*e*f*x*Sqrt[a + b*x^2])/(3*c*d^2*Sqrt [c + d*x^2]) - (8*(2*b*c - a*d)*f^2*x*Sqrt[a + b*x^2])/(3*d^3*Sqrt[c + d*x ^2]) - ((2*b*c - a*d)*f^2*x^3*Sqrt[a + b*x^2])/(c*d^2*Sqrt[c + d*x^2]) + ( (8*b*c - 3*a*d)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*c*d^3) + (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)*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*(8*b*c - a*d)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d ]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]*d^(5/2)*Sqrt[(c*(a + b*x^2))/( a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (8*Sqrt[c]*(2*b*c - a*d)*f^2*Sqrt[a + b *x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*d^(7/2)* Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*e^2*Sqrt[a + b *x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[c]* d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (8*b*Sqrt [c]*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/( a*d)])/(3*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(8*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^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*...
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. \(861\) vs. \(2(380)=760\).
Time = 16.20 (sec) , antiderivative size = 862, normalized size of antiderivative = 2.08
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}+a \,d^{3} e^{2}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 c \,d^{5} \left (x^{2}+\frac {c}{d}\right )^{2}}-\frac {2 \left (b d \,x^{2}+a d \right ) \left (2 a \,c^{2} d \,f^{2}-a c e f \,d^{2}-a \,d^{3} e^{2}-4 b \,c^{3} f^{2}+5 b \,c^{2} d e f -b c \,d^{2} e^{2}\right ) x}{3 c^{2} d^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {b \,f^{2} x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 d^{3}}+\frac {\left (\frac {a^{2} d^{2} f^{2}-4 a b c d \,f^{2}+4 a b \,d^{2} e f +3 b^{2} c^{2} f^{2}-4 b^{2} c d e f +b^{2} d^{2} e^{2}}{d^{4}}+\frac {\left (a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}+a \,d^{3} e^{2}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}\right ) b}{3 d^{4} c}-\frac {2 \left (2 a \,c^{2} d \,f^{2}-a c e f \,d^{2}-a \,d^{3} e^{2}-4 b \,c^{3} f^{2}+5 b \,c^{2} d e f -b c \,d^{2} e^{2}\right ) \left (a d -b c \right )}{3 d^{4} c^{2}}+\frac {2 a \left (2 a \,c^{2} d \,f^{2}-a c e f \,d^{2}-a \,d^{3} e^{2}-4 b \,c^{3} f^{2}+5 b \,c^{2} d e f -b c \,d^{2} e^{2}\right )}{3 d^{3} c^{2}}-\frac {a b c \,f^{2}}{3 d^{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 b f \left (a d f -b c f +b d e \right )}{d^{3}}+\frac {2 \left (2 a \,c^{2} d \,f^{2}-a c e f \,d^{2}-a \,d^{3} e^{2}-4 b \,c^{3} f^{2}+5 b \,c^{2} d e f -b c \,d^{2} e^{2}\right ) b}{3 d^{3} c^{2}}-\frac {b \,f^{2} \left (2 a d +2 b c \right )}{3 d^{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}}\) | \(862\) |
| risch | \(\text {Expression too large to display}\) | \(1248\) |
| default | \(\text {Expression too large to display}\) | \(2120\) |
Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)^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*(a*c^2*d* f^2-2*a*c*d^2*e*f+a*d^3*e^2-b*c^3*f^2+2*b*c^2*d*e*f-b*c*d^2*e^2)/c/d^5*x*( 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)*(2*a*c^2* d*f^2-a*c*d^2*e*f-a*d^3*e^2-4*b*c^3*f^2+5*b*c^2*d*e*f-b*c*d^2*e^2)/c^2/d^4 *x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+1/3*b*f^2/d^3*x*(b*d*x^4+a*d*x^2+b*c*x^ 2+a*c)^(1/2)+((a^2*d^2*f^2-4*a*b*c*d*f^2+4*a*b*d^2*e*f+3*b^2*c^2*f^2-4*b^2 *c*d*e*f+b^2*d^2*e^2)/d^4+1/3*(a*c^2*d*f^2-2*a*c*d^2*e*f+a*d^3*e^2-b*c^3*f ^2+2*b*c^2*d*e*f-b*c*d^2*e^2)/d^4*b/c-2/3*(2*a*c^2*d*f^2-a*c*d^2*e*f-a*d^3 *e^2-4*b*c^3*f^2+5*b*c^2*d*e*f-b*c*d^2*e^2)/d^4*(a*d-b*c)/c^2+2/3*a/d^3*(2 *a*c^2*d*f^2-a*c*d^2*e*f-a*d^3*e^2-4*b*c^3*f^2+5*b*c^2*d*e*f-b*c*d^2*e^2)/ c^2-1/3*a*b*c/d^3*f^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*b/d^3*f*(a*d*f-b*c*f+b*d*e)+2/3*(2*a*c^2*d*f^2-a*c*d^2*e*f-a *d^3*e^2-4*b*c^3*f^2+5*b*c^2*d*e*f-b*c*d^2*e^2)/d^3*b/c^2-1/3*b*f^2/d^3*(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. 937 vs. \(2 (380) = 760\).
Time = 0.12 (sec) , antiderivative size = 937, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(5/2),x, algorithm="fricas ")
Output:
1/3*(2*(((b^2*c^2*d^4 + a*b*c*d^5)*e^2 - (8*b^2*c^3*d^3 - a*b*c^2*d^4)*e*f + 4*(2*b^2*c^4*d^2 - a*b*c^3*d^3)*f^2)*x^5 + 2*((b^2*c^3*d^3 + a*b*c^2*d^ 4)*e^2 - (8*b^2*c^4*d^2 - a*b*c^3*d^3)*e*f + 4*(2*b^2*c^5*d - a*b*c^4*d^2) *f^2)*x^3 + ((b^2*c^4*d^2 + a*b*c^3*d^3)*e^2 - (8*b^2*c^5*d - a*b*c^4*d^2) *e*f + 4*(2*b^2*c^6 - a*b*c^5*d)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(a rcsin(sqrt(-c/d)/x), a*d/(b*c)) - (((2*b^2*c^2*d^4 + 2*a*b*c*d^5 + a*b*d^6 )*e^2 - 2*(8*b^2*c^3*d^3 - a*b*c^2*d^4 + 4*a*b*c*d^5)*e*f + (16*b^2*c^4*d^ 2 - 8*a*b*c^3*d^3 + 8*a*b*c^2*d^4 - 3*a^2*c*d^5)*f^2)*x^5 + 2*((2*b^2*c^3* d^3 + 2*a*b*c^2*d^4 + a*b*c*d^5)*e^2 - 2*(8*b^2*c^4*d^2 - a*b*c^3*d^3 + 4* a*b*c^2*d^4)*e*f + (16*b^2*c^5*d - 8*a*b*c^4*d^2 + 8*a*b*c^3*d^3 - 3*a^2*c ^2*d^4)*f^2)*x^3 + ((2*b^2*c^4*d^2 + 2*a*b*c^3*d^3 + a*b*c^2*d^4)*e^2 - 2* (8*b^2*c^5*d - a*b*c^4*d^2 + 4*a*b*c^3*d^3)*e*f + (16*b^2*c^6 - 8*a*b*c^5* d + 8*a*b*c^4*d^2 - 3*a^2*c^3*d^3)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f (arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (b^2*c^2*d^4*f^2*x^6 + 2*(3*b^2*c^2*d^ 4*e*f - (3*b^2*c^3*d^3 - 2*a*b*c^2*d^4)*f^2)*x^4 - 2*(b^2*c^3*d^3 + a*b*c^ 2*d^4)*e^2 + 2*(8*b^2*c^4*d^2 - a*b*c^3*d^3)*e*f - 8*(2*b^2*c^5*d - a*b*c^ 4*d^2)*f^2 - ((3*b^2*c^2*d^4 + a*b*c*d^5)*e^2 - 4*(6*b^2*c^3*d^3 - a*b*c^2 *d^4)*e*f + (24*b^2*c^4*d^2 - 13*a*b*c^3*d^3)*f^2)*x^2)*sqrt(b*x^2 + a)*sq rt(d*x^2 + c))/(b*c^2*d^7*x^5 + 2*b*c^3*d^6*x^3 + b*c^4*d^5*x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(f*x**2+e)**2/(d*x**2+c)**(5/2),x)
Output:
Integral((a + b*x**2)**(3/2)*(e + f*x**2)**2/(c + d*x**2)**(5/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(5/2),x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^2/(d*x^2 + c)^(5/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(5/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^2/(d*x^2 + c)^(5/2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(5/2),x)
Output:
int(((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(5/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(5/2),x)
Output:
(6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d*f**2*x - 3*sqrt(c + d*x**2)* sqrt(a + b*x**2)*a**2*d**2*e*f*x + 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a** 2*d**2*f**2*x**3 - 9*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)*sqr t(a + b*x**2)*a*b*c*d*f**2*x**3 - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b* d**2*e**2*x + 6*sqrt(c + d*x**2)*sqrt(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 + 6*sqrt(c + d*x**2)*sqrt (a + b*x**2)*b**2*c**2*f**2*x**3 - 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 + 3* 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**4*c**2*d**4*f**2 + 6* 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**4*c*d**5*f**2*x**2 + 3*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...