Integrand size = 30, antiderivative size = 526 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\frac {\left (8 a^3 d^3 f+3 b^3 c^2 (7 d e-2 c f)+a b^2 c d (49 d e+9 c f)-a^2 b d^2 (14 d e+19 c f)\right ) x \sqrt {c+d x^2}}{105 b^2 d^2 \sqrt {a+b x^2}}-\frac {\left (4 a^2 d^2 f-3 b^2 c (7 d e-2 c f)-a b d (7 d e+6 c f)\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 b^2 d}+\frac {(7 b d e-2 b c f+a d f) x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{35 b d}+\frac {f x \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{7 d}-\frac {\sqrt {a} \left (8 a^3 d^3 f+3 b^3 c^2 (7 d e-2 c f)+a b^2 c d (49 d e+9 c f)-a^2 b d^2 (14 d e+19 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 b^{5/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (4 a^2 d^2 f+3 b^2 c (21 d e-c f)-a b d (7 d e+9 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{5/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/105*(8*a^3*d^3*f+3*b^3*c^2*(-2*c*f+7*d*e)+a*b^2*c*d*(9*c*f+49*d*e)-a^2*b *d^2*(19*c*f+14*d*e))*x*(d*x^2+c)^(1/2)/b^2/d^2/(b*x^2+a)^(1/2)-1/105*(4*a ^2*d^2*f-3*b^2*c*(-2*c*f+7*d*e)-a*b*d*(6*c*f+7*d*e))*x*(b*x^2+a)^(1/2)*(d* x^2+c)^(1/2)/b^2/d+1/35*(a*d*f-2*b*c*f+7*b*d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c )^(3/2)/b/d+1/7*f*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(5/2)/d-1/105*a^(1/2)*(8*a^3 *d^3*f+3*b^3*c^2*(-2*c*f+7*d*e)+a*b^2*c*d*(9*c*f+49*d*e)-a^2*b*d^2*(19*c*f +14*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))/b^(5/2)/d^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/ 2)+1/105*a^(3/2)*(4*a^2*d^2*f+3*b^2*c*(-c*f+21*d*e)-a*b*d*(9*c*f+7*d*e))*( d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2) )/b^(5/2)/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 = 3.03 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.70 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a^2 d^2 f-a b d \left (7 d e+9 c f+3 d f x^2\right )-3 b^2 \left (c^2 f+2 c d \left (7 e+4 f x^2\right )+d^2 x^2 \left (7 e+5 f x^2\right )\right )\right )-i c \left (8 a^3 d^3 f+3 b^3 c^2 (7 d e-2 c f)+a b^2 c d (49 d e+9 c f)-a^2 b d^2 (14 d e+19 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) \left (4 a^2 d^2 f+3 b^2 c (-7 d e+2 c f)-a b d (7 d e+6 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 a^2 \left (\frac {b}{a}\right )^{5/2} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2),x]
Output:
(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a^2*d^2*f - a*b*d*(7*d*e + 9*c *f + 3*d*f*x^2) - 3*b^2*(c^2*f + 2*c*d*(7*e + 4*f*x^2) + d^2*x^2*(7*e + 5* f*x^2)))) - I*c*(8*a^3*d^3*f + 3*b^3*c^2*(7*d*e - 2*c*f) + a*b^2*c*d*(49*d *e + 9*c*f) - a^2*b*d^2*(14*d*e + 19*c*f))*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)*(4*a^2*d^2*f + 3*b^2*c*(-7*d*e + 2*c*f) - a*b*d*(7*d*e + 6*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/ (b*c)])/(105*a^2*(b/a)^(5/2)*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.66 (sec) , antiderivative size = 471, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {403, 403, 403, 406, 320, 388, 313}
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.
| \(\displaystyle \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int \sqrt {b x^2+a} \sqrt {d x^2+c} \left ((7 b d e+3 b c f-4 a d f) x^2+c (7 b e-a f)\right )dx}{7 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {b x^2+a} \left (\left (3 c (14 d e+c f) b^2-a d (14 d e+15 c f) b+8 a^2 d^2 f\right ) x^2+c \left (4 d f a^2-7 b d e a-8 b c f a+35 b^2 c e\right )\right )}{\sqrt {d x^2+c}}dx}{5 b}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+7 b d e)}{5 b}}{7 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (3 c^2 (7 d e-2 c f) b^3+a c d (49 d e+9 c f) b^2-a^2 d^2 (14 d e+19 c f) b+8 a^3 d^3 f\right ) x^2+a c \left (3 c (21 d e-c f) b^2-a d (7 d e+9 c f) b+4 a^2 d^2 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (15 c f+14 d e)+3 b^2 c (c f+14 d e)\right )}{3 d}}{5 b}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+7 b d e)}{5 b}}{7 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {\frac {a c \left (4 a^2 d^2 f-a b d (9 c f+7 d e)+3 b^2 c (21 d e-c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (8 a^3 d^3 f-a^2 b d^2 (19 c f+14 d e)+a b^2 c d (9 c f+49 d e)+3 b^3 c^2 (7 d e-2 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (15 c f+14 d e)+3 b^2 c (c f+14 d e)\right )}{3 d}}{5 b}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+7 b d e)}{5 b}}{7 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {\left (8 a^3 d^3 f-a^2 b d^2 (19 c f+14 d e)+a b^2 c d (9 c f+49 d e)+3 b^3 c^2 (7 d e-2 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^2 d^2 f-a b d (9 c f+7 d e)+3 b^2 c (21 d e-c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (15 c f+14 d e)+3 b^2 c (c f+14 d e)\right )}{3 d}}{5 b}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+7 b d e)}{5 b}}{7 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {\frac {\left (8 a^3 d^3 f-a^2 b d^2 (19 c f+14 d e)+a b^2 c d (9 c f+49 d e)+3 b^3 c^2 (7 d e-2 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^2 d^2 f-a b d (9 c f+7 d e)+3 b^2 c (21 d e-c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (15 c f+14 d e)+3 b^2 c (c f+14 d e)\right )}{3 d}}{5 b}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+7 b d e)}{5 b}}{7 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (8 a^2 d^2 f-a b d (15 c f+14 d e)+3 b^2 c (c f+14 d e)\right )}{3 d}+\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (4 a^2 d^2 f-a b d (9 c f+7 d e)+3 b^2 c (21 d e-c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (8 a^3 d^3 f-a^2 b d^2 (19 c f+14 d e)+a b^2 c d (9 c f+49 d e)+3 b^3 c^2 (7 d e-2 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{5 b}+\frac {x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-4 a d f+3 b c f+7 b d e)}{5 b}}{7 b}+\frac {f x \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{7 b}\) |
Input:
Int[Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)*(e + f*x^2),x]
Output:
(f*x*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2))/(7*b) + (((7*b*d*e + 3*b*c*f - 4 *a*d*f)*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*b) + (((8*a^2*d^2*f + 3*b^ 2*c*(14*d*e + c*f) - a*b*d*(14*d*e + 15*c*f))*x*Sqrt[a + b*x^2]*Sqrt[c + d *x^2])/(3*d) + ((8*a^3*d^3*f + 3*b^3*c^2*(7*d*e - 2*c*f) + a*b^2*c*d*(49*d *e + 9*c*f) - a^2*b*d^2*(14*d*e + 19*c*f))*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]] , 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[ c + d*x^2])) + (c^(3/2)*(4*a^2*d^2*f + 3*b^2*c*(21*d*e - c*f) - a*b*d*(7*d *e + 9*c*f))*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b *c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2] ))/(3*d))/(5*b))/(7*b)
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Time = 5.76 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.32
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {d f \,x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7}+\frac {\left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {d f \left (6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (\frac {9 a c d f}{7}+a \,d^{2} e +b \,c^{2} f +2 b c d e -\frac {\left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {d f \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (a \,c^{2} e -\frac {\left (\frac {9 a c d f}{7}+a \,d^{2} e +b \,c^{2} f +2 b c d e -\frac {\left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {d f \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (a \,c^{2} f +2 a c d e +b \,c^{2} e -\frac {3 \left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {d f \left (6 a d +6 b c \right )}{7}\right ) a c}{5 b d}-\frac {\left (\frac {9 a c d f}{7}+a \,d^{2} e +b \,c^{2} f +2 b c d e -\frac {\left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {d f \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(695\) |
| risch | \(-\frac {x \left (-15 f \,x^{4} b^{2} d^{2}-3 a b \,d^{2} f \,x^{2}-24 b^{2} c f \,x^{2} d -21 b^{2} d^{2} e \,x^{2}+4 f \,d^{2} a^{2}-9 f d c b a -7 a b \,d^{2} e -3 f \,c^{2} b^{2}-42 d \,b^{2} c e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{105 d \,b^{2}}+\frac {\left (-\frac {\left (8 f \,d^{3} a^{3}-19 a^{2} b c \,d^{2} f -14 a^{2} b \,d^{3} e +9 a \,b^{2} c^{2} d f +49 a \,b^{2} c \,d^{2} e -6 b^{3} c^{3} f +21 b^{3} c^{2} d e \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}-\frac {3 a \,b^{2} c^{3} f \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 {4 a^{3} c \,d^{2} f \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 {63 a \,c^{2} e d \,b^{2} \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 {7 a^{2} b c \,d^{2} e \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 {9 a^{2} b \,c^{2} d f \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}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{105 d \,b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(834\) |
| default | \(\text {Expression too large to display}\) | \(1331\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e),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*d*f*x^5*( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(a*d^2*f+2*b*c*d*f+b*d^2*e-1/7*d*f* (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*(9/7*a*c*d* f+a*d^2*e+b*c^2*f+2*b*c*d*e-1/5*(a*d^2*f+2*b*c*d*f+b*d^2*e-1/7*d*f*(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*c^ 2*e-1/3*(9/7*a*c*d*f+a*d^2*e+b*c^2*f+2*b*c*d*e-1/5*(a*d^2*f+2*b*c*d*f+b*d^ 2*e-1/7*d*f*(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)*Elliptic F(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(a*c^2*f+2*a*c*d*e+b*c^2*e-3/5* (a*d^2*f+2*b*c*d*f+b*d^2*e-1/7*d*f*(6*a*d+6*b*c))/b/d*a*c-1/3*(9/7*a*c*d*f +a*d^2*e+b*c^2*f+2*b*c*d*e-1/5*(a*d^2*f+2*b*c*d*f+b*d^2*e-1/7*d*f*(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 = 494, normalized size of antiderivative = 0.94 \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=-\frac {\sqrt {b d} {\left (7 \, {\left (3 \, b^{3} c^{3} d + 7 \, a b^{2} c^{2} d^{2} - 2 \, a^{2} b c d^{3}\right )} e - {\left (6 \, b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 19 \, a^{2} b c^{2} d^{2} - 8 \, a^{3} c d^{3}\right )} f\right )} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (7 \, {\left (3 \, b^{3} c^{3} d + 7 \, a b^{2} c^{2} d^{2} - a^{2} b d^{4} - {\left (2 \, a^{2} b - 9 \, a b^{2}\right )} c d^{3}\right )} e - {\left (6 \, b^{3} c^{4} - 9 \, a b^{2} c^{3} d - 4 \, a^{3} d^{4} + {\left (19 \, a^{2} b + 3 \, a b^{2}\right )} c^{2} d^{2} - {\left (8 \, a^{3} - 9 \, a^{2} b\right )} c d^{3}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, b^{3} d^{4} f x^{6} + 3 \, {\left (7 \, b^{3} d^{4} e + {\left (8 \, b^{3} c d^{3} + a b^{2} d^{4}\right )} f\right )} x^{4} + {\left (7 \, {\left (6 \, b^{3} c d^{3} + a b^{2} d^{4}\right )} e + {\left (3 \, b^{3} c^{2} d^{2} + 9 \, a b^{2} c d^{3} - 4 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 7 \, {\left (3 \, b^{3} c^{2} d^{2} + 7 \, a b^{2} c d^{3} - 2 \, a^{2} b d^{4}\right )} e - {\left (6 \, b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 19 \, a^{2} b c d^{3} - 8 \, a^{3} d^{4}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b^{3} d^{3} x} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x, algorithm="fricas")
Output:
-1/105*(sqrt(b*d)*(7*(3*b^3*c^3*d + 7*a*b^2*c^2*d^2 - 2*a^2*b*c*d^3)*e - ( 6*b^3*c^4 - 9*a*b^2*c^3*d + 19*a^2*b*c^2*d^2 - 8*a^3*c*d^3)*f)*x*sqrt(-c/d )*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - sqrt(b*d)*(7*(3*b^3*c^3*d + 7*a*b^2*c^2*d^2 - a^2*b*d^4 - (2*a^2*b - 9*a*b^2)*c*d^3)*e - (6*b^3*c^4 - 9*a*b^2*c^3*d - 4*a^3*d^4 + (19*a^2*b + 3*a*b^2)*c^2*d^2 - (8*a^3 - 9*a^ 2*b)*c*d^3)*f)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (15*b^3*d^4*f*x^6 + 3*(7*b^3*d^4*e + (8*b^3*c*d^3 + a*b^2*d^4)*f)*x^4 + (7 *(6*b^3*c*d^3 + a*b^2*d^4)*e + (3*b^3*c^2*d^2 + 9*a*b^2*c*d^3 - 4*a^2*b*d^ 4)*f)*x^2 + 7*(3*b^3*c^2*d^2 + 7*a*b^2*c*d^3 - 2*a^2*b*d^4)*e - (6*b^3*c^3 *d - 9*a*b^2*c^2*d^2 + 19*a^2*b*c*d^3 - 8*a^3*d^4)*f)*sqrt(b*x^2 + a)*sqrt (d*x^2 + c))/(b^3*d^3*x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int \sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)*(f*x**2+e),x)
Output:
Integral(sqrt(a + b*x**2)*(c + d*x**2)**(3/2)*(e + f*x**2), x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int { \sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e), x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int { \sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*(f*x^2 + e), x)
Timed out. \[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx=\int \sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}\,\left (f\,x^2+e\right ) \,d x \] Input:
int((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2),x)
Output:
int((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)*(e + f*x^2), x)
\[ \int \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right ) \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e),x)
Output:
( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*f*x + 9*sqrt(c + d*x**2) *sqrt(a + b*x**2)*a*b*c*d*f*x + 7*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d* *2*e*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*f*x**3 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f*x + 42*sqrt(c + d*x**2)*sqrt(a + b*x* *2)*b**2*c*d*e*x + 24*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*f*x**3 + 21*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d**2*e*x**3 + 15*sqrt(c + d*x**2 )*sqrt(a + b*x**2)*b**2*d**2*f*x**5 + 8*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 - 19*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 - 14*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 + 9*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 + 49*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 - 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 + 21*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**2*d*e + 4*int((sqrt(c + d*x**2)*sqrt(a + b* x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*c*d**2*f - 9*int((sq rt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x) *a**2*b*c**2*d*f - 7*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d...