Integrand size = 32, antiderivative size = 631 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=-\frac {2 \left (24 a^3 d^3 f^2-4 a^2 b d^2 f (14 d e+9 c f)-b^3 c \left (70 d^2 e^2+21 c d e f-3 c^2 f^2\right )+a b^2 d \left (35 d^2 e^2+91 c d e f+6 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{105 b^3 d^2 \sqrt {a+b x^2}}+\frac {\left (24 a^2 d^2 f^2-a b d f (56 d e+33 c f)+b^2 \left (35 d^2 e^2+84 c d e f+3 c^2 f^2\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 b^3 d}+\frac {2 f (7 b d e+4 b c f-3 a d f) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 b^2}+\frac {d f^2 x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b}+\frac {2 \sqrt {a} \left (24 a^3 d^3 f^2-4 a^2 b d^2 f (14 d e+9 c f)-b^3 c \left (70 d^2 e^2+21 c d e f-3 c^2 f^2\right )+a b^2 d \left (35 d^2 e^2+91 c d e f+6 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 b^{7/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} \left (105 b^3 c d e^2-24 a^3 d^2 f^2+a^2 b d f (56 d e+33 c f)-a b^2 \left (35 d^2 e^2+84 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{7/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-2/105*(24*a^3*d^3*f^2-4*a^2*b*d^2*f*(9*c*f+14*d*e)-b^3*c*(-3*c^2*f^2+21*c *d*e*f+70*d^2*e^2)+a*b^2*d*(6*c^2*f^2+91*c*d*e*f+35*d^2*e^2))*x*(d*x^2+c)^ (1/2)/b^3/d^2/(b*x^2+a)^(1/2)+1/105*(24*a^2*d^2*f^2-a*b*d*f*(33*c*f+56*d*e )+b^2*(3*c^2*f^2+84*c*d*e*f+35*d^2*e^2))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2) /b^3/d+2/35*f*(-3*a*d*f+4*b*c*f+7*b*d*e)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/ 2)/b^2+1/7*d*f^2*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b+2/105*a^(1/2)*(24*a ^3*d^3*f^2-4*a^2*b*d^2*f*(9*c*f+14*d*e)-b^3*c*(-3*c^2*f^2+21*c*d*e*f+70*d^ 2*e^2)+a*b^2*d*(6*c^2*f^2+91*c*d*e*f+35*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^(7/2)/d^2/(b*x^ 2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/105*a^(1/2)*(105*b^3*c*d*e^2- 24*a^3*d^2*f^2+a^2*b*d*f*(33*c*f+56*d*e)-a*b^2*(3*c^2*f^2+84*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^(7/2)/d/(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 = 5.17 (sec) , antiderivative size = 423, normalized size of antiderivative = 0.67 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (24 a^2 d^2 f^2-a b d f \left (56 d e+33 c f+18 d f x^2\right )+b^2 \left (3 c^2 f^2+12 c d f \left (7 e+2 f x^2\right )+d^2 \left (35 e^2+42 e f x^2+15 f^2 x^4\right )\right )\right )+2 i c \left (24 a^3 d^3 f^2-4 a^2 b d^2 f (14 d e+9 c f)+b^3 c \left (-70 d^2 e^2-21 c d e f+3 c^2 f^2\right )+a b^2 d \left (35 d^2 e^2+91 c d e f+6 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (24 a^2 d^2 f^2-a b d f (56 d e+15 c f)+b^2 \left (35 d^2 e^2+42 c d e f-6 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 b^3 \sqrt {\frac {b}{a}} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((c + d*x^2)^(3/2)*(e + f*x^2)^2)/Sqrt[a + b*x^2],x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(24*a^2*d^2*f^2 - a*b*d*f*(56*d*e + 33*c*f + 18*d*f*x^2) + b^2*(3*c^2*f^2 + 12*c*d*f*(7*e + 2*f*x^2) + d^2*(3 5*e^2 + 42*e*f*x^2 + 15*f^2*x^4))) + (2*I)*c*(24*a^3*d^3*f^2 - 4*a^2*b*d^2 *f*(14*d*e + 9*c*f) + b^3*c*(-70*d^2*e^2 - 21*c*d*e*f + 3*c^2*f^2) + a*b^2 *d*(35*d^2*e^2 + 91*c*d*e*f + 6*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*c*(-(b*c) + a*d )*(24*a^2*d^2*f^2 - a*b*d*f*(56*d*e + 15*c*f) + b^2*(35*d^2*e^2 + 42*c*d*e *f - 6*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSi nh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*b^3*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt [c + d*x^2])
Time = 1.51 (sec) , antiderivative size = 1074, normalized size of antiderivative = 1.70, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2 \left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}}+\frac {2 e f x^2 \left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}}+\frac {f^2 x^4 \left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^5}{7 b}+\frac {2 (4 b c-3 a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 b^2}+\frac {2 d e f \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 b}+\frac {d e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 b}+\frac {\left (b^2 c^2-11 a b d c+8 a^2 d^2\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{35 b^3 d}+\frac {4 (3 b c-2 a d) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b^2}+\frac {2 d (2 b c-a d) e^2 \sqrt {b x^2+a} x}{3 b^2 \sqrt {d x^2+c}}-\frac {2 (b c-2 a d) \left (b^2 c^2+4 a b d c-4 a^2 d^2\right ) f^2 \sqrt {b x^2+a} x}{35 b^4 d \sqrt {d x^2+c}}+\frac {2 \left (3 b^2 c^2-13 a b d c+8 a^2 d^2\right ) e f \sqrt {b x^2+a} x}{15 b^3 \sqrt {d x^2+c}}-\frac {2 \sqrt {c} \sqrt {d} (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 b^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} (b c-2 a d) \left (b^2 c^2+4 a b d c-4 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^4 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} \left (3 b^2 c^2-13 a b d c+8 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^3 \sqrt {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-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 a b \sqrt {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} \left (b^2 c^2-11 a b d c+8 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^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 {4 c^{3/2} (3 b c-2 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 b^2 \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[((c + d*x^2)^(3/2)*(e + f*x^2)^2)/Sqrt[a + b*x^2],x]
Output:
(2*d*(2*b*c - a*d)*e^2*x*Sqrt[a + b*x^2])/(3*b^2*Sqrt[c + d*x^2]) + (2*(3* b^2*c^2 - 13*a*b*c*d + 8*a^2*d^2)*e*f*x*Sqrt[a + b*x^2])/(15*b^3*Sqrt[c + d*x^2]) - (2*(b*c - 2*a*d)*(b^2*c^2 + 4*a*b*c*d - 4*a^2*d^2)*f^2*x*Sqrt[a + b*x^2])/(35*b^4*d*Sqrt[c + d*x^2]) + (d*e^2*x*Sqrt[a + b*x^2]*Sqrt[c + d *x^2])/(3*b) + (4*(3*b*c - 2*a*d)*e*f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/( 15*b^2) + ((b^2*c^2 - 11*a*b*c*d + 8*a^2*d^2)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*b^3*d) + (2*d*e*f*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*b ) + (2*(4*b*c - 3*a*d)*f^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*b^2) + (d*f^2*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7*b) - (2*Sqrt[c]*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*b^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2 ]) - (2*Sqrt[c]*(3*b^2*c^2 - 13*a*b*c*d + 8*a^2*d^2)*e*f*Sqrt[a + b*x^2]*E llipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^3*Sqrt[d]*Sq rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c]*(b*c - 2 *a*d)*(b^2*c^2 + 4*a*b*c*d - 4*a^2*d^2)*f^2*Sqrt[a + b*x^2]*EllipticE[ArcT an[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(35*b^4*d^(3/2)*Sqrt[(c*(a + b* x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*(3*b*c - a*d)*e^2*Sqrt[ a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*b *Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (4*c^(3/ 2)*(3*b*c - 2*a*d)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqr...
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 = 13.62 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.15
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {d \,f^{2} x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 b}+\frac {\left (2 c d \,f^{2}+2 d^{2} e f -\frac {d \,f^{2} \left (6 a d +6 b c \right )}{7 b}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (c^{2} f^{2}+4 c d e f +d^{2} e^{2}-\frac {5 a c d \,f^{2}}{7 b}-\frac {\left (2 c d \,f^{2}+2 d^{2} e f -\frac {d \,f^{2} \left (6 a d +6 b c \right )}{7 b}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (c^{2} e^{2}-\frac {\left (c^{2} f^{2}+4 c d e f +d^{2} e^{2}-\frac {5 a c d \,f^{2}}{7 b}-\frac {\left (2 c d \,f^{2}+2 d^{2} e f -\frac {d \,f^{2} \left (6 a d +6 b c \right )}{7 b}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (2 c^{2} e f +2 c d \,e^{2}-\frac {3 \left (2 c d \,f^{2}+2 d^{2} e f -\frac {d \,f^{2} \left (6 a d +6 b c \right )}{7 b}\right ) a c}{5 b d}-\frac {\left (c^{2} f^{2}+4 c d e f +d^{2} e^{2}-\frac {5 a c d \,f^{2}}{7 b}-\frac {\left (2 c d \,f^{2}+2 d^{2} e f -\frac {d \,f^{2} \left (6 a d +6 b c \right )}{7 b}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(723\) |
| risch | \(\text {Expression too large to display}\) | \(1098\) |
| default | \(\text {Expression too large to display}\) | \(1724\) |
Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(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*c*d*f^2+2*d^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*(c^2*f^2+ 4*c*d*e*f+d^2*e^2-5/7*a/b*c*d*f^2-1/5*(2*c*d*f^2+2*d^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)+( c^2*e^2-1/3*(c^2*f^2+4*c*d*e*f+d^2*e^2-5/7*a/b*c*d*f^2-1/5*(2*c*d*f^2+2*d^ 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*c^2*e*f+2*c*d*e^2-3/5* (2*c*d*f^2+2*d^2*e*f-1/7/b*d*f^2*(6*a*d+6*b*c))/b/d*a*c-1/3*(c^2*f^2+4*c*d *e*f+d^2*e^2-5/7*a/b*c*d*f^2-1/5*(2*c*d*f^2+2*d^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 = 0.11 (sec) , antiderivative size = 618, normalized size of antiderivative = 0.98 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=-\frac {2 \, {\left (35 \, {\left (2 \, b^{3} c^{2} d^{2} - a b^{2} c d^{3}\right )} e^{2} + 7 \, {\left (3 \, b^{3} c^{3} d - 13 \, a b^{2} c^{2} d^{2} + 8 \, a^{2} b c d^{3}\right )} e f - 3 \, {\left (b^{3} c^{4} + 2 \, a b^{2} c^{3} d - 12 \, a^{2} b c^{2} d^{2} + 8 \, a^{3} c d^{3}\right )} f^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (35 \, {\left (4 \, b^{3} c^{2} d^{2} - a b^{2} d^{4} - {\left (2 \, a b^{2} - 3 \, b^{3}\right )} c d^{3}\right )} e^{2} + 14 \, {\left (3 \, b^{3} c^{3} d - 13 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b d^{4} + 2 \, {\left (4 \, a^{2} b - 3 \, a b^{2}\right )} c d^{3}\right )} e f - 3 \, {\left (2 \, b^{3} c^{4} + 4 \, a b^{2} c^{3} d + 8 \, a^{3} d^{4} - {\left (24 \, a^{2} b - a b^{2}\right )} c^{2} d^{2} + {\left (16 \, a^{3} - 11 \, a^{2} b\right )} c d^{3}\right )} f^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, b^{3} d^{4} f^{2} x^{6} + 6 \, {\left (7 \, b^{3} d^{4} e f + {\left (4 \, b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} f^{2}\right )} x^{4} + 70 \, {\left (2 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} e^{2} + 14 \, {\left (3 \, b^{3} c^{2} d^{2} - 13 \, a b^{2} c d^{3} + 8 \, a^{2} b d^{4}\right )} e f - 6 \, {\left (b^{3} c^{3} d + 2 \, a b^{2} c^{2} d^{2} - 12 \, a^{2} b c d^{3} + 8 \, a^{3} d^{4}\right )} f^{2} + {\left (35 \, b^{3} d^{4} e^{2} + 28 \, {\left (3 \, b^{3} c d^{3} - 2 \, a b^{2} d^{4}\right )} e f + 3 \, {\left (b^{3} c^{2} d^{2} - 11 \, a b^{2} c d^{3} + 8 \, a^{2} b d^{4}\right )} f^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b^{4} d^{3} x} \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(1/2),x, algorithm="fricas ")
Output:
-1/105*(2*(35*(2*b^3*c^2*d^2 - a*b^2*c*d^3)*e^2 + 7*(3*b^3*c^3*d - 13*a*b^ 2*c^2*d^2 + 8*a^2*b*c*d^3)*e*f - 3*(b^3*c^4 + 2*a*b^2*c^3*d - 12*a^2*b*c^2 *d^2 + 8*a^3*c*d^3)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/ d)/x), a*d/(b*c)) - (35*(4*b^3*c^2*d^2 - a*b^2*d^4 - (2*a*b^2 - 3*b^3)*c*d ^3)*e^2 + 14*(3*b^3*c^3*d - 13*a*b^2*c^2*d^2 + 4*a^2*b*d^4 + 2*(4*a^2*b - 3*a*b^2)*c*d^3)*e*f - 3*(2*b^3*c^4 + 4*a*b^2*c^3*d + 8*a^3*d^4 - (24*a^2*b - a*b^2)*c^2*d^2 + (16*a^3 - 11*a^2*b)*c*d^3)*f^2)*sqrt(b*d)*x*sqrt(-c/d) *elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (15*b^3*d^4*f^2*x^6 + 6*(7* b^3*d^4*e*f + (4*b^3*c*d^3 - 3*a*b^2*d^4)*f^2)*x^4 + 70*(2*b^3*c*d^3 - a*b ^2*d^4)*e^2 + 14*(3*b^3*c^2*d^2 - 13*a*b^2*c*d^3 + 8*a^2*b*d^4)*e*f - 6*(b ^3*c^3*d + 2*a*b^2*c^2*d^2 - 12*a^2*b*c*d^3 + 8*a^3*d^4)*f^2 + (35*b^3*d^4 *e^2 + 28*(3*b^3*c*d^3 - 2*a*b^2*d^4)*e*f + 3*(b^3*c^2*d^2 - 11*a*b^2*c*d^ 3 + 8*a^2*b*d^4)*f^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^4*d^3*x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{2}}{\sqrt {a + b x^{2}}}\, dx \] Input:
integrate((d*x**2+c)**(3/2)*(f*x**2+e)**2/(b*x**2+a)**(1/2),x)
Output:
Integral((c + d*x**2)**(3/2)*(e + f*x**2)**2/sqrt(a + b*x**2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(1/2),x, algorithm="maxima ")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)^2/sqrt(b*x^2 + a), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)^2/sqrt(b*x^2 + a), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2}{\sqrt {b\,x^2+a}} \,d x \] Input:
int(((c + d*x^2)^(3/2)*(e + f*x^2)^2)/(a + b*x^2)^(1/2),x)
Output:
int(((c + d*x^2)^(3/2)*(e + f*x^2)^2)/(a + b*x^2)^(1/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(1/2),x)
Output:
(24*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 - 56*sqrt(c + d*x**2)*sqrt(a + b*x**2)* a*b*d**2*e*f*x - 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*f**2*x**3 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f**2*x + 84*sqrt(c + d*x**2 )*sqrt(a + b*x**2)*b**2*c*d*e*f*x + 24*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 - 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)*a**3*d**3*f** 2 + 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**2*b*c*d**2*f**2 + 112*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 - 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*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 - 70*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 - 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)*b**3*c**3*f**2 + 42*in t((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 + 140*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)...