Integrand size = 26, antiderivative size = 308 \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=-\frac {2 (b c+2 a d) x \sqrt {c+d x^2}}{3 b^2 d^2 \sqrt {a+b x^2}}+\frac {x^3 \sqrt {c+d x^2}}{3 b d \sqrt {a+b x^2}}+\frac {\sqrt {a} \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 b^{5/2} d^2 (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (b c-4 a d) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{5/2} d (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-2/3*(2*a*d+b*c)*x*(d*x^2+c)^(1/2)/b^2/d^2/(b*x^2+a)^(1/2)+1/3*x^3*(d*x^2+ c)^(1/2)/b/d/(b*x^2+a)^(1/2)+1/3*a^(1/2)*(-8*a^2*d^2+3*a*b*c*d+2*b^2*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^2/(-a*d+b*c)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1 /2)-1/3*a^(3/2)*(-4*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/(-a*d+b*c)/(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 = 3.37 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.85 \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (c+d x^2\right ) \left (-4 a^2 d+b^2 c x^2+a b \left (c-d x^2\right )\right )-i c \left (-2 b^2 c^2-3 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+2 i c \left (-b^2 c^2-a b c d+2 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \left (\frac {b}{a}\right )^{5/2} d^2 (b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[x^6/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
(Sqrt[b/a]*d*x*(c + d*x^2)*(-4*a^2*d + b^2*c*x^2 + a*b*(c - d*x^2)) - I*c* (-2*b^2*c^2 - 3*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)] + (2*I)*c*(-(b^2*c^2) - a*b*c*d + 2*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*A rcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^2*(b/a)^(5/2)*d^2*(b*c - a*d)*Sqrt [a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.49 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {372, 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.
| \(\displaystyle \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {a x^3 \sqrt {c+d x^2}}{b \sqrt {a+b x^2} (b c-a d)}-\frac {\int \frac {x^2 \left (3 a c-(b c-4 a d) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b (b c-a d)}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {a x^3 \sqrt {c+d x^2}}{b \sqrt {a+b x^2} (b c-a d)}-\frac {-\frac {\int -\frac {\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{b (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a x^3 \sqrt {c+d x^2}}{b \sqrt {a+b x^2} (b c-a d)}-\frac {\frac {\int \frac {\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{b (b c-a d)}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {a x^3 \sqrt {c+d x^2}}{b \sqrt {a+b x^2} (b c-a d)}-\frac {\frac {\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (b c-4 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{b (b c-a d)}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {a x^3 \sqrt {c+d x^2}}{b \sqrt {a+b x^2} (b c-a d)}-\frac {\frac {\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{b (b c-a d)}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {a x^3 \sqrt {c+d x^2}}{b \sqrt {a+b x^2} (b c-a d)}-\frac {\frac {\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{b (b c-a d)}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {a x^3 \sqrt {c+d x^2}}{b \sqrt {a+b x^2} (b c-a d)}-\frac {\frac {\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{b (b c-a d)}\) |
Input:
Int[x^6/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
(a*x^3*Sqrt[c + d*x^2])/(b*(b*c - a*d)*Sqrt[a + b*x^2]) - (-1/3*((b*c - 4* a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(b*d) + ((2*b^2*c^2 + 3*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]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*S qrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(b*c - 4 *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))/(b*(b*c - a*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[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && 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 = 20.04 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.31
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+b c \right ) a^{2} x}{b^{3} \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b^{2} d}+\frac {\left (-\frac {c \,a^{2}}{b^{2} \left (a d -b c \right )}-\frac {a c}{3 b^{2} d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {a}{b^{2}}-\frac {d \,a^{2}}{b^{2} \left (a d -b c \right )}-\frac {2 a d +2 b c}{3 b^{2} d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(403\) |
| default | \(\frac {\left (\sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}-\sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+4 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}-\sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+4 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x -\sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{3 b^{2} \left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, d^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(519\) |
| risch | \(\frac {x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 b^{2} d}-\frac {\left (\frac {-\frac {b \left (5 a d +2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {3 d \,a^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}}{b}+\frac {3 a^{3} d \left (-\frac {\left (b d \,x^{2}+b c \right ) x}{a \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {1}{a}+\frac {b c}{\left (a d -b c \right ) a}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b^{2} d \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(689\) |
Input:
int(x^6/(b*x^2+a)^(3/2)/(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)*((b*d*x^2+b*c) /b^3*a^2/(a*d-b*c)*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+1/3/b^2/d*x*(b*d*x^4+ a*d*x^2+b*c*x^2+a*c)^(1/2)+(-1/b^2*c*a^2/(a*d-b*c)-1/3/b^2/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/b^2-1/b^2*d*a^2/ (a*d-b*c)-1/3/b^2/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 = 395, normalized size of antiderivative = 1.28 \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {{\left ({\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 8 \, a^{2} b c d^{2}\right )} x^{3} + {\left (2 \, a b^{2} c^{3} + 3 \, a^{2} b c^{2} d - 8 \, a^{3} c d^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 4 \, a^{2} b d^{3} - {\left (8 \, a^{2} b - a b^{2}\right )} c d^{2}\right )} x^{3} + {\left (2 \, a b^{2} c^{3} + 3 \, a^{2} b c^{2} d - 4 \, a^{3} d^{3} - {\left (8 \, a^{3} - a^{2} b\right )} c d^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 8 \, a^{3} d^{3} - {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 2 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left ({\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} + {\left (a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x\right )}} \] Input:
integrate(x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
1/3*(((2*b^3*c^3 + 3*a*b^2*c^2*d - 8*a^2*b*c*d^2)*x^3 + (2*a*b^2*c^3 + 3*a ^2*b*c^2*d - 8*a^3*c*d^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(- c/d)/x), a*d/(b*c)) - ((2*b^3*c^3 + 3*a*b^2*c^2*d - 4*a^2*b*d^3 - (8*a^2*b - a*b^2)*c*d^2)*x^3 + (2*a*b^2*c^3 + 3*a^2*b*c^2*d - 4*a^3*d^3 - (8*a^3 - a^2*b)*c*d^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a* d/(b*c)) - (2*a*b^2*c^2*d + 3*a^2*b*c*d^2 - 8*a^3*d^3 - (b^3*c*d^2 - a*b^2 *d^3)*x^4 + 2*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*x^2)*sqrt(b*x^2 + a) *sqrt(d*x^2 + c))/((b^5*c*d^3 - a*b^4*d^4)*x^3 + (a*b^4*c*d^3 - a^2*b^3*d^ 4)*x)
\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{6}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate(x**6/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)
Output:
Integral(x**6/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)), x)
\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate(x^6/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate(x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate(x^6/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^6}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int(x^6/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)),x)
Output:
int(x^6/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {-3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, c x +2 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, d \,x^{3}-8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} d^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b c d -8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b \,d^{2} x^{2}-\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} c d \,x^{2}+3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} c^{2}+3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b \,c^{2} x^{2}}{6 b \,d^{2} \left (b \,x^{2}+a \right )} \] Input:
int(x^6/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*c*x + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*d*x**3 - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)* a**2*d**2 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d* x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b*c*d - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b*d**2*x**2 - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a *b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*b**2*c*d*x**2 + 3 *int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b*c*x **2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*c**2 + 3*int((sqrt (c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a* b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b*c**2*x**2)/(6*b*d**2*(a + b*x **2))