Integrand size = 33, antiderivative size = 381 \[ \int \frac {x^4 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {(9 a b c d f+2 (b c+a d) (5 b d e-4 b c f-4 a d f)) x \sqrt {c+d x^2}}{15 b^2 d^3 \sqrt {a+b x^2}}+\frac {(5 b d e-4 b c f-4 a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d^2}+\frac {f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}+\frac {\sqrt {a} (9 a b c d f+2 (b c+a d) (5 b d e-4 b c f-4 a d f)) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{5/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (5 b d e-4 b c f-4 a d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{5/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/15*(9*a*b*c*d*f+2*(a*d+b*c)*(-4*a*d*f-4*b*c*f+5*b*d*e))*x*(d*x^2+c)^(1/ 2)/b^2/d^3/(b*x^2+a)^(1/2)+1/15*(-4*a*d*f-4*b*c*f+5*b*d*e)*x*(b*x^2+a)^(1/ 2)*(d*x^2+c)^(1/2)/b^2/d^2+1/5*f*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d+1 /15*a^(1/2)*(9*a*b*c*d*f+2*(a*d+b*c)*(-4*a*d*f-4*b*c*f+5*b*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^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*a^(3/2)*(-4 *a*d*f-4*b*c*f+5*b*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^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 = 5.61 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.75 \[ \int \frac {x^4 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d f+b \left (-5 d e+4 c f-3 d f x^2\right )\right )-i c \left (8 a^2 d^2 f+2 b^2 c (-5 d e+4 c f)+a b d (-10 d e+7 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c \left (4 a^2 d^2 f+a b d (-5 d e+3 c f)+2 b^2 c (-5 d e+4 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(x^4*(e + f*x^2))/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
Output:
(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a*d*f + b*(-5*d*e + 4*c*f - 3* d*f*x^2))) - I*c*(8*a^2*d^2*f + 2*b^2*c*(-5*d*e + 4*c*f) + a*b*d*(-10*d*e + 7*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*(4*a^2*d^2*f + a*b*d*(-5*d*e + 3*c*f) + 2*b^2 *c*(-5*d*e + 4*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*A rcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*a^2*(b/a)^(5/2)*d^3*Sqrt[a + b*x^2] *Sqrt[c + d*x^2])
Time = 0.57 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {444, 444, 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.
| \(\displaystyle \int \frac {x^4 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\int \frac {x^2 \left (3 a c f-(5 b d e-4 b c f-4 a d f) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {-\frac {\int -\frac {a c (5 b d e-4 b c f-4 a d f)-\left (-2 c (5 d e-4 c f) b^2-a d (10 d e-7 c f) b+8 a^2 d^2 f\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {\int \frac {a c (5 b d e-4 b c f-4 a d f)-\left (-2 c (5 d e-4 c f) b^2-a d (10 d e-7 c f) b+8 a^2 d^2 f\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {a c (-4 a d f-4 b c f+5 b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\left (8 a^2 d^2 f-a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (-4 a d f-4 b c f+5 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (8 a^2 d^2 f-a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (-4 a d f-4 b c f+5 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (8 a^2 d^2 f-a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (-4 a d f-4 b c f+5 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (8 a^2 d^2 f-a b d (10 d e-7 c f)-2 b^2 c (5 d e-4 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-4 a d f-4 b c f+5 b d e)}{3 b d}}{5 b d}\) |
Input:
Int[(x^4*(e + f*x^2))/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
Output:
(f*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*b*d) - (-1/3*((5*b*d*e - 4*b*c* f - 4*a*d*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(b*d) + (-((8*a^2*d^2*f - a*b*d*(10*d*e - 7*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)/S qrt[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)*(5*b*d*e - 4*b*c*f - 4*a*d*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*b*d))/(5*b*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[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]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
Time = 8.38 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.09
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (e -\frac {f \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}-\frac {\left (e -\frac {f \left (4 a d +4 b c \right )}{5 b d}\right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {3 a c f}{5 b d}-\frac {\left (e -\frac {f \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(414\) |
| risch | \(-\frac {x \left (-3 b d f \,x^{2}+4 a d f +4 b c f -5 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b^{2} d^{2}}+\frac {\left (-\frac {\left (8 f \,d^{2} a^{2}+7 f d c b a -10 a b \,d^{2} e +8 f \,c^{2} b^{2}-10 d \,b^{2} c e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {4 a b \,c^{2} f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {4 a^{2} c d f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {5 a c d e b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b^{2} d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(540\) |
| default | \(-\frac {\left (-3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} f \,x^{7}+\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}+4 \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}+4 \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 +3 \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 -5 \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 -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 ) a^{2} c \,d^{2} f -7 \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 +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 ) 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 +4 \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 ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 d^{3} b^{2} \sqrt {-\frac {b}{a}}\, \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(868\) |
Input:
int(x^4*(f*x^2+e)/(b*x^2+a)^(1/2)/(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)*(1/5/b/d*f*x^3 *(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(e-1/5/b/d*f*(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)-1/3*(e-1/5/b/d*f*(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))-(-3/5*a/b *c/d*f-1/3*(e-1/5/b/d*f*(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 = 0.09 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.84 \[ \int \frac {x^4 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {b d} {\left (10 \, {\left (b^{2} c^{2} d + a b c d^{2}\right )} e - {\left (8 \, b^{2} c^{3} + 7 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (5 \, {\left (2 \, b^{2} c^{2} d + 2 \, a b c d^{2} + a b d^{3}\right )} e - {\left (8 \, b^{2} c^{3} + 7 \, a b c^{2} d + 4 \, a^{2} d^{3} + 4 \, {\left (2 \, a^{2} + a b\right )} c d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (3 \, b^{2} d^{3} f x^{4} + {\left (5 \, b^{2} d^{3} e - 4 \, {\left (b^{2} c d^{2} + a b d^{3}\right )} f\right )} x^{2} - 10 \, {\left (b^{2} c d^{2} + a b d^{3}\right )} e + {\left (8 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b^{3} d^{4} x} \] Input:
integrate(x^4*(f*x^2+e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fric as")
Output:
1/15*(sqrt(b*d)*(10*(b^2*c^2*d + a*b*c*d^2)*e - (8*b^2*c^3 + 7*a*b*c^2*d + 8*a^2*c*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^2*d + 2*a*b*c*d^2 + a*b*d^3)*e - (8*b^2*c^3 + 7*a* b*c^2*d + 4*a^2*d^3 + 4*(2*a^2 + a*b)*c*d^2)*f)*x*sqrt(-c/d)*elliptic_f(ar csin(sqrt(-c/d)/x), a*d/(b*c)) + (3*b^2*d^3*f*x^4 + (5*b^2*d^3*e - 4*(b^2* c*d^2 + a*b*d^3)*f)*x^2 - 10*(b^2*c*d^2 + a*b*d^3)*e + (8*b^2*c^2*d + 7*a* b*c*d^2 + 8*a^2*d^3)*f)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^3*d^4*x)
\[ \int \frac {x^4 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{4} \left (e + f x^{2}\right )}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate(x**4*(f*x**2+e)/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(x**4*(e + f*x**2)/(sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )} x^{4}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(f*x^2+e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="maxi ma")
Output:
integrate((f*x^2 + e)*x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )} x^{4}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^4*(f*x^2+e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac ")
Output:
integrate((f*x^2 + e)*x^4/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^4 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^4\,\left (f\,x^2+e\right )}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int((x^4*(e + f*x^2))/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
Output:
int((x^4*(e + f*x^2))/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {x^4 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {-4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d f x -4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c f x +5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d e x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d f \,x^{3}+8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2} f +7 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d f -10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,d^{2} e +8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2} f -10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c d e +4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c d f +4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2} f -5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d e}{15 b^{2} d^{2}} \] Input:
int(x^4*(f*x^2+e)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
Output:
( - 4*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 + 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**2*d**2*f + 7* 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*c*d*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)*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 + 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**2*c*d*f + 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*b**2*d**2)