Integrand size = 32, antiderivative size = 364 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\left (b^2 e^2-2 a b e f+4 a^2 f^2\right ) x \sqrt {c+d x^2}}{3 a b^2 \left (a+b x^2\right )^{3/2}}+\frac {f^2 x^3 \sqrt {c+d x^2}}{b \left (a+b x^2\right )^{3/2}}+\frac {\left (2 b^3 c e^2+8 a^3 d f^2-a b^2 e (d e-2 c f)-a^2 b f (4 d e+7 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} b^{5/2} (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\left (b^2 d e^2+4 a^2 d f^2-a b f (2 d e+3 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{5/2} (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/3*(4*a^2*f^2-2*a*b*e*f+b^2*e^2)*x*(d*x^2+c)^(1/2)/a/b^2/(b*x^2+a)^(3/2)+ f^2*x^3*(d*x^2+c)^(1/2)/b/(b*x^2+a)^(3/2)+1/3*(2*b^3*c*e^2+8*a^3*d*f^2-a*b ^2*e*(-2*c*f+d*e)-a^2*b*f*(7*c*f+4*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^(5/2)/(-a*d+b*c) /(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+4*a^2*d*f^ 2-a*b*f*(3*c*f+2*d*e))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^ (1/2)),(1-a*d/b/c)^(1/2))/a^(1/2)/b^(5/2)/(-a*d+b*c)/(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 = 5.52 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\left (\frac {b}{a}\right )^{3/2} \left (-i c \left (2 b^3 c e^2+8 a^3 d f^2+a b^2 e (-d e+2 c f)-a^2 b f (4 d e+7 c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(-b e+a f) \left (\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (4 a^3 d f-2 b^3 c e x^2+a b^2 \left (-3 c e+d e x^2-4 c f x^2\right )+a^2 b \left (2 d e-3 c f+5 d f x^2\right )\right )-2 i c (-b c+a d) (b e+2 a f) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )\right )}{3 b^4 (-b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[c + d*x^2]*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x]
Output:
((b/a)^(3/2)*((-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*(4*d*e + 7*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)] - (-(b*e) + a*f)*(Sqr t[b/a]*x*(c + d*x^2)*(4*a^3*d*f - 2*b^3*c*e*x^2 + a*b^2*(-3*c*e + d*e*x^2 - 4*c*f*x^2) + a^2*b*(2*d*e - 3*c*f + 5*d*f*x^2)) - (2*I)*c*(-(b*c) + a*d) *(b*e + 2*a*f)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipti cF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])))/(3*b^4*(-(b*c) + a*d)*(a + b*x^ 2)^(3/2)*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(857\) vs. \(2(364)=728\).
Time = 1.04 (sec) , antiderivative size = 857, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}}+\frac {2 e f x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}}+\frac {f^2 x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {f^2 \sqrt {d x^2+c} x^3}{3 b \left (b x^2+a\right )^{3/2}}-\frac {(3 b c-4 a d) f^2 \sqrt {d x^2+c} x}{3 b^2 (b c-a d) \sqrt {b x^2+a}}+\frac {e^2 \sqrt {d x^2+c} x}{3 a \left (b x^2+a\right )^{3/2}}-\frac {2 e f \sqrt {d x^2+c} x}{3 b \left (b x^2+a\right )^{3/2}}+\frac {d (7 b c-8 a d) f^2 \sqrt {b x^2+a} x}{3 b^3 (b c-a d) \sqrt {d x^2+c}}+\frac {(2 b c-a d) e^2 \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}+\frac {2 (b c-2 a d) e f \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{3/2} (b c-a d) \sqrt {b x^2+a} \sqrt {\frac {a \left (d x^2+c\right )}{c \left (b x^2+a\right )}}}-\frac {\sqrt {c} \sqrt {d} (7 b c-8 a d) f^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^3 (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {c^{3/2} \sqrt {d} e^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {c^{3/2} (3 b c-4 a d) f^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a b^2 \sqrt {d} (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 c^{3/2} \sqrt {d} e f \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a b (b c-a d) \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[(Sqrt[c + d*x^2]*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x]
Output:
(d*(7*b*c - 8*a*d)*f^2*x*Sqrt[a + b*x^2])/(3*b^3*(b*c - a*d)*Sqrt[c + d*x^ 2]) + (e^2*x*Sqrt[c + d*x^2])/(3*a*(a + b*x^2)^(3/2)) - (2*e*f*x*Sqrt[c + d*x^2])/(3*b*(a + b*x^2)^(3/2)) - (f^2*x^3*Sqrt[c + d*x^2])/(3*b*(a + b*x^ 2)^(3/2)) - ((3*b*c - 4*a*d)*f^2*x*Sqrt[c + d*x^2])/(3*b^2*(b*c - a*d)*Sqr t[a + b*x^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)*Sqrt[b]*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) + (2*(b*c - 2*a*d)*e*f*Sqrt[ c + d*x^2]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(3*Sqr t[a]*b^(3/2)*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^ 2))]) - (Sqrt[c]*Sqrt[d]*(7*b*c - 8*a*d)*f^2*Sqrt[a + b*x^2]*EllipticE[Arc Tan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^3*(b*c - a*d)*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*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*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*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*S qrt[c + d*x^2]) + (c^(3/2)*(3*b*c - 4*a*d)*f^2*Sqrt[a + b*x^2]*EllipticF[A rcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*b^2*Sqrt[d]*(b*c - a*d) *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. \(2(337)=674\).
Time = 7.02 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.92
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a \,b^{4} \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {\left (b d \,x^{2}+b c \right ) x \left (5 a^{3} d \,f^{2}-4 a^{2} c \,f^{2} b -4 a^{2} b d e f +2 a c e f \,b^{2}-a \,b^{2} d \,e^{2}+2 b^{3} c \,e^{2}\right )}{3 b^{3} a^{2} \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (-\frac {f \left (2 a d f -b c f -2 b d e \right )}{b^{3}}+\frac {\left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) d}{3 b^{3} a}+\frac {5 a^{3} d \,f^{2}-4 a^{2} c \,f^{2} b -4 a^{2} b d e f +2 a c e f \,b^{2}-a \,b^{2} d \,e^{2}+2 b^{3} c \,e^{2}}{3 b^{3} a^{2}}+\frac {c \left (5 a^{3} d \,f^{2}-4 a^{2} c \,f^{2} b -4 a^{2} b d e f +2 a c e f \,b^{2}-a \,b^{2} d \,e^{2}+2 b^{3} c \,e^{2}\right )}{3 b^{2} a^{2} \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {f^{2} d}{b^{2}}+\frac {d \left (5 a^{3} d \,f^{2}-4 a^{2} c \,f^{2} b -4 a^{2} b d e f +2 a c e f \,b^{2}-a \,b^{2} d \,e^{2}+2 b^{3} c \,e^{2}\right )}{3 b^{2} \left (a d -b c \right ) a^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(699\) |
| default | \(\text {Expression too large to display}\) | \(1955\) |
Input:
int((d*x^2+c)^(1/2)*(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^2*f^2- 2*a*b*e*f+b^2*e^2)/a/b^4*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+a/b)^2 -1/3*(b*d*x^2+b*c)/b^3/a^2/(a*d-b*c)*x*(5*a^3*d*f^2-4*a^2*b*c*f^2-4*a^2*b* d*e*f+2*a*b^2*c*e*f-a*b^2*d*e^2+2*b^3*c*e^2)/((x^2+a/b)*(b*d*x^2+b*c))^(1/ 2)+(-f*(2*a*d*f-b*c*f-2*b*d*e)/b^3+1/3*(a^2*f^2-2*a*b*e*f+b^2*e^2)/b^3*d/a +1/3/b^3*(5*a^3*d*f^2-4*a^2*b*c*f^2-4*a^2*b*d*e*f+2*a*b^2*c*e*f-a*b^2*d*e^ 2+2*b^3*c*e^2)/a^2+1/3/b^2*c/a^2/(a*d-b*c)*(5*a^3*d*f^2-4*a^2*b*c*f^2-4*a^ 2*b*d*e*f+2*a*b^2*c*e*f-a*b^2*d*e^2+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))-(f^2*d/b^2+1/3/b^2*d*(5*a^3*d*f^2-4* a^2*b*c*f^2-4*a^2*b*d*e*f+2*a*b^2*c*e*f-a*b^2*d*e^2+2*b^3*c*e^2)/(a*d-b*c) /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)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-E llipticE(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. 922 vs. \(2 (337) = 674\).
Time = 0.12 (sec) , antiderivative size = 922, normalized size of antiderivative = 2.53 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((d*x^2+c)^(1/2)*(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 + 2*(a*b^4*c^2 - 2*a^2*b^3*c*d)*e*f - ( 7*a^2*b^3*c^2 - 8*a^3*b^2*c*d)*f^2)*x^5 + 2*((2*a*b^4*c^2 - a^2*b^3*c*d)*e ^2 + 2*(a^2*b^3*c^2 - 2*a^3*b^2*c*d)*e*f - (7*a^3*b^2*c^2 - 8*a^4*b*c*d)*f ^2)*x^3 + ((2*a^2*b^3*c^2 - a^3*b^2*c*d)*e^2 + 2*(a^3*b^2*c^2 - 2*a^4*b*c* d)*e*f - (7*a^4*b*c^2 - 8*a^5*c*d)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e (arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (((2*b^5*c^2 - a*b^4*c*d + a*b^4*d^2)* e^2 + 2*(a*b^4*c^2 - 2*a^2*b^3*c*d - a^2*b^3*d^2)*e*f - (7*a^2*b^3*c^2 - 4 *a^3*b^2*d^2 - (8*a^3*b^2 - 3*a^2*b^3)*c*d)*f^2)*x^5 + 2*((2*a*b^4*c^2 - a ^2*b^3*c*d + a^2*b^3*d^2)*e^2 + 2*(a^2*b^3*c^2 - 2*a^3*b^2*c*d - a^3*b^2*d ^2)*e*f - (7*a^3*b^2*c^2 - 4*a^4*b*d^2 - (8*a^4*b - 3*a^3*b^2)*c*d)*f^2)*x ^3 + ((2*a^2*b^3*c^2 - a^3*b^2*c*d + a^3*b^2*d^2)*e^2 + 2*(a^3*b^2*c^2 - 2 *a^4*b*c*d - a^4*b*d^2)*e*f - (7*a^4*b*c^2 - 4*a^5*d^2 - (8*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)) + (3*(a^2*b^3*c*d - 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 - 2*a^4*b*d^2)*e*f + (7*a^4*b*c*d - 8*a^5*d^2)*f^ 2 - (a*b^4*c*d*e^2 + 2*(2*a^2*b^3*c*d - 3*a^3*b^2*d^2)*e*f - (11*a^3*b^2*c *d - 12*a^4*b*d^2)*f^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/((a^2*b^6*c* d - a^3*b^5*d^2)*x^5 + 2*(a^3*b^5*c*d - a^4*b^4*d^2)*x^3 + (a^4*b^4*c*d - a^5*b^3*d^2)*x)
\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((d*x**2+c)**(1/2)*(f*x**2+e)**2/(b*x**2+a)**(5/2),x)
Output:
Integral(sqrt(c + d*x**2)*(e + f*x**2)**2/(a + b*x**2)**(5/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="maxima ")
Output:
integrate(sqrt(d*x^2 + c)*(f*x^2 + e)^2/(b*x^2 + a)^(5/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)*(f*x^2+e)^2/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x^2 + c)*(f*x^2 + e)^2/(b*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(((c + d*x^2)^(1/2)*(e + f*x^2)^2)/(a + b*x^2)^(5/2),x)
Output:
int(((c + d*x^2)^(1/2)*(e + f*x^2)^2)/(a + b*x^2)^(5/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((d*x^2+c)^(1/2)*(f*x^2+e)^2/(b*x^2+a)^(5/2),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*f**2*x + 2*sqrt(c + d*x**2)*sq rt(a + b*x**2)*a*d*f**2*x**3 + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*e*f *x - 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*f**2*x**3 + sqrt(c + d*x**2)* sqrt(a + b*x**2)*b*d*e**2*x - 8*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*d**3*f**2 + 15*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*b*c*d**2*f**2 + 4*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*b*d**3*e*f - 16*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*b*d**3*f**2*x**2 - 9*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/...