Integrand size = 26, antiderivative size = 332 \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\left (9 a b c d-8 (b c+a d)^2\right ) x \sqrt {c+d x^2}}{15 b^2 d^3 \sqrt {a+b x^2}}-\frac {4 (b c+a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d^2}+\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}+\frac {\sqrt {a} \left (9 a b c d-8 (b c+a d)^2\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{5/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {4 a^{3/2} (b c+a d) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{5/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-1/15*(9*a*b*c*d-8*(a*d+b*c)^2)*x*(d*x^2+c)^(1/2)/b^2/d^3/(b*x^2+a)^(1/2)- 4/15*(a*d+b*c)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^2+1/5*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-8*(a*d+b*c)^2)*(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)+4/15*a^(3/2)*(a *d+b*c)*(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 = 2.02 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.75 \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 b c+4 a d-3 b d x^2\right )-i c \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c \left (8 b^2 c^2+3 a b c d+4 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[x^6/(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*b*c + 4*a*d - 3*b*d*x^2)) - I* c*(8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^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*(8*b^2*c^2 + 3*a* b*c*d + 4*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*Arc Sinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*a^2*(b/a)^(5/2)*d^3*Sqrt[a + b*x^2]*S qrt[c + d*x^2])
Time = 0.48 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {381, 444, 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.
| \(\displaystyle \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 381 |
| \(\displaystyle \frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\int \frac {x^2 \left (4 (b c+a d) x^2+3 a c\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {4 x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d+b c)}{3 b d}-\frac {\int \frac {\left (8 b^2 c^2+7 a b d c+8 a^2 d^2\right ) x^2+4 a c (b c+a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}}{5 b d}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {4 x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d+b c)}{3 b d}-\frac {\left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+4 a c (a d+b c) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}}{5 b d}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {4 x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d+b c)}{3 b d}-\frac {\left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {4 c^{3/2} \sqrt {a+b x^2} (a d+b c) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}}{5 b d}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {4 x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d+b c)}{3 b d}-\frac {\left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {4 c^{3/2} \sqrt {a+b x^2} (a d+b c) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}}{5 b d}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d}-\frac {\frac {4 x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d+b c)}{3 b d}-\frac {\left (8 a^2 d^2+7 a b c d+8 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {4 c^{3/2} \sqrt {a+b x^2} (a d+b c) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}}{5 b d}\) |
Input:
Int[x^6/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
Output:
(x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*b*d) - ((4*(b*c + a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b*d) - ((8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*(( x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*Elliptic E[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])) + (4*c^(3/2)*(b*c + a*d)*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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q) + 1))), x] - Simp[e^4/(b*d*(m + 2*(p + q) + 1)) Int[(e*x)^(m - 4)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + b*c*(m + 2*p - 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q }, x] && NeQ[b*c - a*d, 0] && GtQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2 , p, q, x]
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 = 7.19 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.14
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 d b}-\frac {\left (4 a d +4 b c \right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{15 d^{2} b^{2}}+\frac {\left (4 a d +4 b c \right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{15 d^{2} b^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {3 a c}{5 d b}+\frac {\left (4 a d +4 b c \right ) \left (2 a d +2 b c \right )}{15 d^{2} b^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(379\) |
| risch | \(-\frac {x \left (-3 b d \,x^{2}+4 a d +4 b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b^{2} d^{2}}+\frac {\left (-\frac {\left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {4 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {4 b \,c^{2} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b^{2} d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(419\) |
| default | \(-\frac {\left (-3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+\sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}+\sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+4 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}+5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x^{3}+4 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+4 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x +4 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 d^{3} b^{2} \sqrt {-\frac {b}{a}}\, \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(546\) |
Input:
int(x^6/(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/d/b*x^3*( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)-1/15/d^2/b^2*(4*a*d+4*b*c)*x*(b*d*x^4+a *d*x^2+b*c*x^2+a*c)^(1/2)+1/15/d^2/b^2*(4*a*d+4*b*c)*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)*Ellipt icF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(-3/5*a*c/d/b+1/15/d^2/b^2*(4 *a*d+4*b*c)*(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 = 232, normalized size of antiderivative = 0.70 \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {{\left (8 \, b^{2} c^{3} + 7 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, b^{2} c^{3} + 7 \, a b c^{2} d + 4 \, a^{2} d^{3} + 4 \, {\left (2 \, a^{2} + a b\right )} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{2} d^{3} x^{4} + 8 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 8 \, a^{2} d^{3} - 4 \, {\left (b^{2} c d^{2} + a b d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b^{3} d^{4} x} \] Input:
integrate(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
-1/15*((8*b^2*c^3 + 7*a*b*c^2*d + 8*a^2*c*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elli ptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (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)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqr t(-c/d)/x), a*d/(b*c)) - (3*b^2*d^3*x^4 + 8*b^2*c^2*d + 7*a*b*c*d^2 + 8*a^ 2*d^3 - 4*(b^2*c*d^2 + a*b*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^3 *d^4*x)
\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{6}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate(x**6/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(x**6/(sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)
\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {x^{6}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)
\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {x^{6}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate(x^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {x^6}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int(x^6/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
Output:
int(x^6/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {x^6}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {-4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d x -4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d \,x^{3}+8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2}+7 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d +8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2}+4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c d +4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2}}{15 b^{2} d^{2}} \] Input:
int(x^6/(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*x - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*x**3 + 8*int((sq rt(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 + 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 + 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 + 4*in t((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 + 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)/(15*b**2*d**2)