Integrand size = 32, antiderivative size = 636 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {2 \left (3 a^3 d^3 f^2-3 a^2 b d^2 f (7 d e-2 c f)+b^3 c \left (35 d^2 e^2-56 c d e f+24 c^2 f^2\right )-a b^2 d \left (70 d^2 e^2-91 c d e f+36 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{105 b d^4 \sqrt {a+b x^2}}+\frac {\left (3 a^2 d^2 f^2+3 a b d f (28 d e-11 c f)+b^2 \left (35 d^2 e^2-56 c d e f+24 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 b d^3}+\frac {2 f (7 b d e-3 b c f+4 a d f) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 d^2}+\frac {b f^2 x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 d}+\frac {2 \sqrt {a} \left (3 a^3 d^3 f^2-3 a^2 b d^2 f (7 d e-2 c f)+b^3 c \left (35 d^2 e^2-56 c d e f+24 c^2 f^2\right )-a b^2 d \left (70 d^2 e^2-91 c d e f+36 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 b^{3/2} d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (3 a^2 c d^2 f^2-3 a b d \left (35 d^2 e^2-28 c d e f+11 c^2 f^2\right )+b^2 c \left (35 d^2 e^2-56 c d e f+24 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{3/2} c d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-2/105*(3*a^3*d^3*f^2-3*a^2*b*d^2*f*(-2*c*f+7*d*e)+b^3*c*(24*c^2*f^2-56*c* d*e*f+35*d^2*e^2)-a*b^2*d*(36*c^2*f^2-91*c*d*e*f+70*d^2*e^2))*x*(d*x^2+c)^ (1/2)/b/d^4/(b*x^2+a)^(1/2)+1/105*(3*a^2*d^2*f^2+3*a*b*d*f*(-11*c*f+28*d*e )+b^2*(24*c^2*f^2-56*c*d*e*f+35*d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2 )/b/d^3+2/35*f*(4*a*d*f-3*b*c*f+7*b*d*e)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/ 2)/d^2+1/7*b*f^2*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d+2/105*a^(1/2)*(3*a^ 3*d^3*f^2-3*a^2*b*d^2*f*(-2*c*f+7*d*e)+b^3*c*(24*c^2*f^2-56*c*d*e*f+35*d^2 *e^2)-a*b^2*d*(36*c^2*f^2-91*c*d*e*f+70*d^2*e^2))*(d*x^2+c)^(1/2)*Elliptic E(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(3/2)/d^4/(b*x^ 2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/105*a^(3/2)*(3*a^2*c*d^2*f^2- 3*a*b*d*(11*c^2*f^2-28*c*d*e*f+35*d^2*e^2)+b^2*c*(24*c^2*f^2-56*c*d*e*f+35 *d^2*e^2))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a* d/b/c)^(1/2))/b^(3/2)/c/d^3/(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 = 5.23 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (3 a^2 d^2 f^2+3 a b d f \left (28 d e-11 c f+8 d f x^2\right )+b^2 \left (24 c^2 f^2-2 c d f \left (28 e+9 f x^2\right )+d^2 \left (35 e^2+42 e f x^2+15 f^2 x^4\right )\right )\right )+2 i c \left (3 a^3 d^3 f^2+3 a^2 b d^2 f (-7 d e+2 c f)+a b^2 d \left (-70 d^2 e^2+91 c d e f-36 c^2 f^2\right )+b^3 c \left (35 d^2 e^2-56 c d e f+24 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i (-b c+a d) \left (3 a^2 c d^2 f^2+3 a b d \left (35 d^2 e^2-42 c d e f+16 c^2 f^2\right )-2 b^2 c \left (35 d^2 e^2-56 c d e f+24 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 b \sqrt {\frac {b}{a}} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((a + b*x^2)^(3/2)*(e + f*x^2)^2)/Sqrt[c + d*x^2],x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(3*a^2*d^2*f^2 + 3*a*b*d*f*(28*d*e - 11*c*f + 8*d*f*x^2) + b^2*(24*c^2*f^2 - 2*c*d*f*(28*e + 9*f*x^2) + d^2*( 35*e^2 + 42*e*f*x^2 + 15*f^2*x^4))) + (2*I)*c*(3*a^3*d^3*f^2 + 3*a^2*b*d^2 *f*(-7*d*e + 2*c*f) + a*b^2*d*(-70*d^2*e^2 + 91*c*d*e*f - 36*c^2*f^2) + b^ 3*c*(35*d^2*e^2 - 56*c*d*e*f + 24*c^2*f^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 )*(3*a^2*c*d^2*f^2 + 3*a*b*d*(35*d^2*e^2 - 42*c*d*e*f + 16*c^2*f^2) - 2*b^ 2*c*(35*d^2*e^2 - 56*c*d*e*f + 24*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + ( d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*b*Sqrt[b/a] *d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 1.48 (sec) , antiderivative size = 1062, normalized size of antiderivative = 1.67, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx\) |
\(\Big \downarrow \) 433 |
\(\displaystyle \int \left (\frac {e^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}}+\frac {2 e f x^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}}+\frac {f^2 x^4 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 d}-\frac {2 (3 b c-4 a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 d^2}+\frac {2 b e f \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d}+\frac {b e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 d}+\frac {\left (8 b^2 c^2-11 a b d c+a^2 d^2\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{35 b d^3}-\frac {4 (2 b c-3 a d) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 d^2}-\frac {2 (b c-2 a d) e^2 \sqrt {b x^2+a} x}{3 d \sqrt {d x^2+c}}-\frac {2 (2 b c-a d) \left (4 b^2 c^2-4 a b d c-a^2 d^2\right ) f^2 \sqrt {b x^2+a} x}{35 b^2 d^3 \sqrt {d x^2+c}}-\frac {2 \left (-\frac {3 d a^2}{b}+13 c a-\frac {8 b c^2}{d}\right ) e f \sqrt {b x^2+a} x}{15 d \sqrt {d x^2+c}}+\frac {2 \sqrt {c} (b c-2 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 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 {2 \sqrt {c} (2 b c-a d) \left (4 b^2 c^2-4 a b d c-a^2 d^2\right ) 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 )}{35 b^2 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 \sqrt {c} \left (8 b^2 c^2-13 a b d c+3 a^2 d^2\right ) 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 )}{15 b 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 {\sqrt {c} (b c-3 a 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 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 {c^{3/2} \left (8 b^2 c^2-11 a b d c+a^2 d^2\right ) 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 )}{35 b 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 {4 c^{3/2} (2 b c-3 a 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 )}{15 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)/Sqrt[c + d*x^2],x]
Output:
(-2*(b*c - 2*a*d)*e^2*x*Sqrt[a + b*x^2])/(3*d*Sqrt[c + d*x^2]) - (2*(13*a* c - (8*b*c^2)/d - (3*a^2*d)/b)*e*f*x*Sqrt[a + b*x^2])/(15*d*Sqrt[c + d*x^2 ]) - (2*(2*b*c - a*d)*(4*b^2*c^2 - 4*a*b*c*d - a^2*d^2)*f^2*x*Sqrt[a + b*x ^2])/(35*b^2*d^3*Sqrt[c + d*x^2]) + (b*e^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^ 2])/(3*d) - (4*(2*b*c - 3*a*d)*e*f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15* d^2) + ((8*b^2*c^2 - 11*a*b*c*d + a^2*d^2)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*b*d^3) + (2*b*e*f*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d) - (2*(3*b*c - 4*a*d)*f^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*d^2) + (b *f^2*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7*d) + (2*Sqrt[c]*(b*c - 2*a*d) *e^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d )])/(3*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2 *Sqrt[c]*(8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*e*f*Sqrt[a + b*x^2]*Elliptic E[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c]*(2*b*c - a*d)*(4* b^2*c^2 - 4*a*b*c*d - a^2*d^2)*f^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[ d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b^2*d^(7/2)*Sqrt[(c*(a + b*x^2))/(a* (c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(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*d^(3/2)*Sqrt [(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (4*c^(3/2)*(2*b*c - 3 *a*d)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b...
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 = 13.68 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.14
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {b \,f^{2} x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 d}+\frac {\left (2 b a \,f^{2}+2 b^{2} e f -\frac {b \,f^{2} \left (6 a d +6 b c \right )}{7 d}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (a^{2} f^{2}+4 a b f e +b^{2} e^{2}-\frac {5 a b c \,f^{2}}{7 d}-\frac {\left (2 b a \,f^{2}+2 b^{2} e f -\frac {b \,f^{2} \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (a^{2} e^{2}-\frac {\left (a^{2} f^{2}+4 a b f e +b^{2} e^{2}-\frac {5 a b c \,f^{2}}{7 d}-\frac {\left (2 b a \,f^{2}+2 b^{2} e f -\frac {b \,f^{2} \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (2 a^{2} e f +2 a b \,e^{2}-\frac {3 \left (2 b a \,f^{2}+2 b^{2} e f -\frac {b \,f^{2} \left (6 a d +6 b c \right )}{7 d}\right ) a c}{5 b d}-\frac {\left (a^{2} f^{2}+4 a b f e +b^{2} e^{2}-\frac {5 a b c \,f^{2}}{7 d}-\frac {\left (2 b a \,f^{2}+2 b^{2} e f -\frac {b \,f^{2} \left (6 a d +6 b c \right )}{7 d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(723\) |
risch | \(\text {Expression too large to display}\) | \(1098\) |
default | \(\text {Expression too large to display}\) | \(1784\) |
Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)^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)*(1/7*b/d*f^2*x ^5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(2*b*a*f^2+2*b^2*e*f-1/7*b/d*f^ 2*(6*a*d+6*b*c))/b/d*x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(a^2*f^2+ 4*a*b*f*e+b^2*e^2-5/7*a*b*c/d*f^2-1/5*(2*b*a*f^2+2*b^2*e*f-1/7*b/d*f^2*(6* a*d+6*b*c))/b/d*(4*a*d+4*b*c))/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+( a^2*e^2-1/3*(a^2*f^2+4*a*b*f*e+b^2*e^2-5/7*a*b*c/d*f^2-1/5*(2*b*a*f^2+2*b^ 2*e*f-1/7*b/d*f^2*(6*a*d+6*b*c))/b/d*(4*a*d+4*b*c))/b/d*a*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)*El lipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(2*a^2*e*f+2*a*b*e^2-3/5* (2*b*a*f^2+2*b^2*e*f-1/7*b/d*f^2*(6*a*d+6*b*c))/b/d*a*c-1/3*(a^2*f^2+4*a*b *f*e+b^2*e^2-5/7*a*b*c/d*f^2-1/5*(2*b*a*f^2+2*b^2*e*f-1/7*b/d*f^2*(6*a*d+6 *b*c))/b/d*(4*a*d+4*b*c))/b/d*(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))))
Time = 0.10 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {2 \, {\left (35 \, {\left (b^{3} c^{3} d^{2} - 2 \, a b^{2} c^{2} d^{3}\right )} e^{2} - 7 \, {\left (8 \, b^{3} c^{4} d - 13 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3}\right )} e f + 3 \, {\left (8 \, b^{3} c^{5} - 12 \, a b^{2} c^{4} d + 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )} f^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (35 \, {\left (2 \, b^{3} c^{3} d^{2} - 4 \, a b^{2} c^{2} d^{3} + a b^{2} c d^{4} - 3 \, a^{2} b d^{5}\right )} e^{2} - 14 \, {\left (8 \, b^{3} c^{4} d - 13 \, a b^{2} c^{3} d^{2} - 6 \, a^{2} b c d^{4} + {\left (3 \, a^{2} b + 4 \, a b^{2}\right )} c^{2} d^{3}\right )} e f + 3 \, {\left (16 \, b^{3} c^{5} - 24 \, a b^{2} c^{4} d + a^{3} c d^{4} + 4 \, {\left (a^{2} b + 2 \, a b^{2}\right )} c^{3} d^{2} + {\left (2 \, a^{3} - 11 \, a^{2} b\right )} c^{2} d^{3}\right )} f^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (15 \, b^{3} c d^{4} f^{2} x^{6} + 6 \, {\left (7 \, b^{3} c d^{4} e f - {\left (3 \, b^{3} c^{2} d^{3} - 4 \, a b^{2} c d^{4}\right )} f^{2}\right )} x^{4} - 70 \, {\left (b^{3} c^{2} d^{3} - 2 \, a b^{2} c d^{4}\right )} e^{2} + 14 \, {\left (8 \, b^{3} c^{3} d^{2} - 13 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} e f - 6 \, {\left (8 \, b^{3} c^{4} d - 12 \, a b^{2} c^{3} d^{2} + 2 \, a^{2} b c^{2} d^{3} + a^{3} c d^{4}\right )} f^{2} + {\left (35 \, b^{3} c d^{4} e^{2} - 28 \, {\left (2 \, b^{3} c^{2} d^{3} - 3 \, a b^{2} c d^{4}\right )} e f + 3 \, {\left (8 \, b^{3} c^{3} d^{2} - 11 \, a b^{2} c^{2} d^{3} + a^{2} b c d^{4}\right )} f^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b^{2} c d^{5} x} \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x, algorithm="fricas ")
Output:
1/105*(2*(35*(b^3*c^3*d^2 - 2*a*b^2*c^2*d^3)*e^2 - 7*(8*b^3*c^4*d - 13*a*b ^2*c^3*d^2 + 3*a^2*b*c^2*d^3)*e*f + 3*(8*b^3*c^5 - 12*a*b^2*c^4*d + 2*a^2* b*c^3*d^2 + a^3*c^2*d^3)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqr t(-c/d)/x), a*d/(b*c)) - (35*(2*b^3*c^3*d^2 - 4*a*b^2*c^2*d^3 + a*b^2*c*d^ 4 - 3*a^2*b*d^5)*e^2 - 14*(8*b^3*c^4*d - 13*a*b^2*c^3*d^2 - 6*a^2*b*c*d^4 + (3*a^2*b + 4*a*b^2)*c^2*d^3)*e*f + 3*(16*b^3*c^5 - 24*a*b^2*c^4*d + a^3* c*d^4 + 4*(a^2*b + 2*a*b^2)*c^3*d^2 + (2*a^3 - 11*a^2*b)*c^2*d^3)*f^2)*sqr t(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (15*b^3* c*d^4*f^2*x^6 + 6*(7*b^3*c*d^4*e*f - (3*b^3*c^2*d^3 - 4*a*b^2*c*d^4)*f^2)* x^4 - 70*(b^3*c^2*d^3 - 2*a*b^2*c*d^4)*e^2 + 14*(8*b^3*c^3*d^2 - 13*a*b^2* c^2*d^3 + 3*a^2*b*c*d^4)*e*f - 6*(8*b^3*c^4*d - 12*a*b^2*c^3*d^2 + 2*a^2*b *c^2*d^3 + a^3*c*d^4)*f^2 + (35*b^3*c*d^4*e^2 - 28*(2*b^3*c^2*d^3 - 3*a*b^ 2*c*d^4)*e*f + 3*(8*b^3*c^3*d^2 - 11*a*b^2*c^2*d^3 + a^2*b*c*d^4)*f^2)*x^2 )*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*c*d^5*x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{2}}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(f*x**2+e)**2/(d*x**2+c)**(1/2),x)
Output:
Integral((a + b*x**2)**(3/2)*(e + f*x**2)**2/sqrt(c + d*x**2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^2/sqrt(d*x^2 + c), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^2/sqrt(d*x^2 + c), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2}{\sqrt {d\,x^2+c}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(1/2),x)
Output:
int(((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(1/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(1/2),x)
Output:
(3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*f**2*x - 33*sqrt(c + d*x**2 )*sqrt(a + b*x**2)*a*b*c*d*f**2*x + 84*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a *b*d**2*e*f*x + 24*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*f**2*x**3 + 24*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f**2*x - 56*sqrt(c + d*x**2 )*sqrt(a + b*x**2)*b**2*c*d*e*f*x - 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b **2*c*d*f**2*x**3 + 35*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d**2*e**2*x + 42*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d**2*e*f*x**3 + 15*sqrt(c + d* x**2)*sqrt(a + b*x**2)*b**2*d**2*f**2*x**5 - 6*int((sqrt(c + d*x**2)*sqrt( a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*d**3*f**2 - 12*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x **2 + b*d*x**4),x)*a**2*b*c*d**2*f**2 + 42*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*b*d**3*e*f + 72*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b**2*c**2*d*f**2 - 182*int((sqrt(c + d*x**2)*sqrt(a + b* x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b**2*c*d**2*e*f + 140*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x** 2 + b*d*x**4),x)*a*b**2*d**3*e**2 - 48*int((sqrt(c + d*x**2)*sqrt(a + b*x* *2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b**3*c**3*f**2 + 112*i nt((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b *d*x**4),x)*b**3*c**2*d*e*f - 70*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)...