Integrand size = 30, antiderivative size = 393 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {\left (3 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) x \sqrt {c+d x^2}}{15 d^3 \sqrt {a+b x^2}}+\frac {(5 b d e-4 b c f+3 a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d^2}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 d}-\frac {\sqrt {a} \left (3 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 \sqrt {b} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (b c (5 d e-4 c f)-3 a d (5 d e-2 c f)) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 \sqrt {b} c d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/15*(3*a^2*d^2*f+a*b*d*(-13*c*f+20*d*e)-2*b^2*c*(-4*c*f+5*d*e))*x*(d*x^2+ c)^(1/2)/d^3/(b*x^2+a)^(1/2)+1/15*(3*a*d*f-4*b*c*f+5*b*d*e)*x*(b*x^2+a)^(1 /2)*(d*x^2+c)^(1/2)/d^2+1/5*f*x*(b*x^2+a)^(3/2)*(d*x^2+c)^(1/2)/d-1/15*a^( 1/2)*(3*a^2*d^2*f+a*b*d*(-13*c*f+20*d*e)-2*b^2*c*(-4*c*f+5*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^ (1/2)/d^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*a^(3/2)*(b* c*(-4*c*f+5*d*e)-3*a*d*(-2*c*f+5*d*e))*(d*x^2+c)^(1/2)*InverseJacobiAM(arc tan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/d^2/(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 = 3.72 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a d f+b \left (5 d e-4 c f+3 d f x^2\right )\right )-i c \left (3 a^2 d^2 f+a b d (20 d e-13 c f)+2 b^2 c (-5 d e+4 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-b c+a d) (2 b c (5 d e-4 c f)+3 a d (-5 d e+3 c f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 \sqrt {\frac {b}{a}} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((a + b*x^2)^(3/2)*(e + f*x^2))/Sqrt[c + d*x^2],x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(6*a*d*f + b*(5*d*e - 4*c*f + 3*d*f *x^2)) - I*c*(3*a^2*d^2*f + a*b*d*(20*d*e - 13*c*f) + 2*b^2*c*(-5*d*e + 4* 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*(-(b*c) + a*d)*(2*b*c*(5*d*e - 4*c*f) + 3*a*d*(-5*d *e + 3*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[S qrt[b/a]*x], (a*d)/(b*c)])/(15*Sqrt[b/a]*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^ 2])
Time = 0.52 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {403, 403, 25, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int \frac {\sqrt {b x^2+a} \left ((5 b d e-4 b c f+3 a d f) x^2+a (5 d e-c f)\right )}{\sqrt {d x^2+c}}dx}{5 d}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int -\frac {a (b c (5 d e-4 c f)-3 a d (5 d e-2 c f))-\left (-2 c (5 d e-4 c f) b^2+a d (20 d e-13 c f) b+3 a^2 d^2 f\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d f-4 b c f+5 b d e)}{3 d}}{5 d}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d f-4 b c f+5 b d e)}{3 d}-\frac {\int \frac {a (b c (5 d e-4 c f)-3 a d (5 d e-2 c f))-\left (-2 c (5 d e-4 c f) b^2+a d (20 d e-13 c f) b+3 a^2 d^2 f\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{5 d}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 d}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d f-4 b c f+5 b d e)}{3 d}-\frac {a (b c (5 d e-4 c f)-3 a d (5 d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\left (3 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{5 d}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 d}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d f-4 b c f+5 b d e)}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (b c (5 d e-4 c f)-3 a d (5 d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (3 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{5 d}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 d}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d f-4 b c f+5 b d e)}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (b c (5 d e-4 c f)-3 a d (5 d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (3 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{3 d}}{5 d}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 d}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 a d f-4 b c f+5 b d e)}{3 d}-\frac {\frac {\sqrt {c} \sqrt {a+b x^2} (b c (5 d e-4 c f)-3 a d (5 d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (3 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}}{5 d}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 d}\) |
Input:
Int[((a + b*x^2)^(3/2)*(e + f*x^2))/Sqrt[c + d*x^2],x]
Output:
(f*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*d) + (((5*b*d*e - 4*b*c*f + 3*a *d*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) - (-((3*a^2*d^2*f + a*b*d*( 20*d*e - 13*c*f) - 2*b^2*c*(5*d*e - 4*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]))) + (Sqrt[c]*(b*c*(5*d*e - 4*c*f) - 3*a*d*(5*d*e - 2*c*f))*Sq rt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sq rt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*d))/(5*d)
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 = 11.82 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.14
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f b \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 d}+\frac {\left (2 a f b +b^{2} e -\frac {f b \left (4 a d +4 b c \right )}{5 d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (a^{2} e -\frac {\left (2 a f b +b^{2} e -\frac {f b \left (4 a d +4 b c \right )}{5 d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (a^{2} f +2 a e b -\frac {3 a b c f}{5 d}-\frac {\left (2 a f b +b^{2} e -\frac {f b \left (4 a d +4 b c \right )}{5 d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(448\) |
| risch | \(\frac {x \left (3 b d f \,x^{2}+6 a d f -4 b c f +5 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 d^{2}}-\frac {\left (\frac {\left (3 f \,d^{2} a^{2}-13 f d c b a +20 a b \,d^{2} e +8 f \,c^{2} b^{2}-10 d \,b^{2} c e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}-\frac {15 a^{2} d^{2} e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {4 a b \,c^{2} f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {6 a^{2} c d f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {5 a c d e b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(627\) |
| default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} f \,x^{7}-9 \sqrt {-\frac {b}{a}}\, a b \,d^{3} f \,x^{5}+\sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} f \,x^{5}-5 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} e \,x^{5}-6 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} f \,x^{3}-5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} f \,x^{3}-5 \sqrt {-\frac {b}{a}}\, a b \,d^{3} e \,x^{3}+4 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d f \,x^{3}-5 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} e \,x^{3}+9 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2} f -15 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} d^{3} e -17 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d f +25 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c \,d^{2} e +8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3} f -10 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} d e -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2} f +13 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d f -20 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c \,d^{2} e -8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3} f +10 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} d e -6 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} f x +4 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d f x -5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} e x \right )}{15 d^{3} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {-\frac {b}{a}}}\) | \(923\) |
Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)/(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)*(1/5*f*b/d*x^3 *(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(2*a*f*b+b^2*e-1/5*f*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-1/3*(2*a*f*b+b^2*e -1/5*f*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)*EllipticF(x*(-b/a)^(1/2),(-1 +(a*d+b*c)/c/b)^(1/2))-(a^2*f+2*a*e*b-3/5*a*b*c/d*f-1/3*(2*a*f*b+b^2*e-1/5 *f*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 = 355, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {b d} {\left (10 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2}\right )} e - {\left (8 \, b^{2} c^{4} - 13 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (5 \, {\left (2 \, b^{2} c^{3} d - 4 \, a b c^{2} d^{2} + a b c d^{3} - 3 \, a^{2} d^{4}\right )} e - {\left (8 \, b^{2} c^{4} - 13 \, a b c^{3} d - 6 \, a^{2} c d^{3} + {\left (3 \, a^{2} + 4 \, a b\right )} c^{2} d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (3 \, b^{2} c d^{3} f x^{4} + {\left (5 \, b^{2} c d^{3} e - 2 \, {\left (2 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3}\right )} f\right )} x^{2} - 10 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3}\right )} e + {\left (8 \, b^{2} c^{3} d - 13 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b c d^{4} x} \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
1/15*(sqrt(b*d)*(10*(b^2*c^3*d - 2*a*b*c^2*d^2)*e - (8*b^2*c^4 - 13*a*b*c^ 3*d + 3*a^2*c^2*d^2)*f)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/ (b*c)) - sqrt(b*d)*(5*(2*b^2*c^3*d - 4*a*b*c^2*d^2 + a*b*c*d^3 - 3*a^2*d^4 )*e - (8*b^2*c^4 - 13*a*b*c^3*d - 6*a^2*c*d^3 + (3*a^2 + 4*a*b)*c^2*d^2)*f )*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (3*b^2*c*d^3* f*x^4 + (5*b^2*c*d^3*e - 2*(2*b^2*c^2*d^2 - 3*a*b*c*d^3)*f)*x^2 - 10*(b^2* c^2*d^2 - 2*a*b*c*d^3)*e + (8*b^2*c^3*d - 13*a*b*c^2*d^2 + 3*a^2*c*d^3)*f) *sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b*c*d^4*x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}{\sqrt {c + d x^{2}}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(f*x**2+e)/(d*x**2+c)**(1/2),x)
Output:
Integral((a + b*x**2)**(3/2)*(e + f*x**2)/sqrt(c + d*x**2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)/sqrt(d*x^2 + c), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}}{\sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)/sqrt(d*x^2 + c), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\right )}{\sqrt {d\,x^2+c}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(e + f*x^2))/(c + d*x^2)^(1/2),x)
Output:
int(((a + b*x^2)^(3/2)*(e + f*x^2))/(c + d*x^2)^(1/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {6 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d f x -4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c f x +5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d e x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d f \,x^{3}+3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2} f -13 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d f +20 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,d^{2} e +8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2} f -10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c d e -6 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c d f +15 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2} e +4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2} f -5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d e}{15 d^{2}} \] Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x)
Output:
(6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f*x - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*f*x + 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*f*x**3 + 3*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*d**2*f - 13*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)*a*b*c*d*f + 20*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*d**2*e + 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)*b**2*c* *2*f - 10*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**2*c*d*e - 6*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**2*c*d*f + 15*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**2 *d**2*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*b*c**2*f - 5*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*b*c*d*e)/(15*d**2)