Integrand size = 33, antiderivative size = 447 \[ \int \frac {x^6}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\frac {d^2 \left (8 b^2 d^2-9 a b d e+9 a^2 e^2\right ) x \left (a d+(b d-a e) x^2\right )}{15 e^2 (b d-a e)^3 \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}-\frac {4 b d^4 x \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}{15 e^2 (b d-a e)^2}+\frac {d^2 x^3 \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}{5 e (b d-a e)}-\frac {a d^{5/2} \left (8 b^2 d^2-9 a b d e+9 a^2 e^2\right ) \left (d+e x^2\right ) \sqrt {\frac {a d+(b d-a e) x^2}{a \left (d+e x^2\right )}} E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|2-\frac {b d}{a e}\right )}{15 e^{5/2} (b d-a e)^3 \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}+\frac {4 a b d^{7/2} \left (d+e x^2\right ) \sqrt {\frac {a d+(b d-a e) x^2}{a \left (d+e x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),2-\frac {b d}{a e}\right )}{15 e^{5/2} (b d-a e)^2 \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}} \] Output:
1/15*d^2*(9*a^2*e^2-9*a*b*d*e+8*b^2*d^2)*x*(a*d+(-a*e+b*d)*x^2)/e^2/(-a*e+ b*d)^3/(a+b*x^2+e*(-a*e+b*d)*x^4/d^2)^(1/2)-4/15*b*d^4*x*(a+b*x^2+e*(-a*e+ b*d)*x^4/d^2)^(1/2)/e^2/(-a*e+b*d)^2+1/5*d^2*x^3*(a+b*x^2+e*(-a*e+b*d)*x^4 /d^2)^(1/2)/e/(-a*e+b*d)-1/15*a*d^(5/2)*(9*a^2*e^2-9*a*b*d*e+8*b^2*d^2)*(e *x^2+d)*((a*d+(-a*e+b*d)*x^2)/a/(e*x^2+d))^(1/2)*EllipticE(e^(1/2)*x/d^(1/ 2)/(1+e*x^2/d)^(1/2),(2-b*d/a/e)^(1/2))/e^(5/2)/(-a*e+b*d)^3/(a+b*x^2+e*(- a*e+b*d)*x^4/d^2)^(1/2)+4/15*a*b*d^(7/2)*(e*x^2+d)*((a*d+(-a*e+b*d)*x^2)/a /(e*x^2+d))^(1/2)*InverseJacobiAM(arctan(e^(1/2)*x/d^(1/2)),(2-b*d/a/e)^(1 /2))/e^(5/2)/(-a*e+b*d)^2/(a+b*x^2+e*(-a*e+b*d)*x^4/d^2)^(1/2)
Result contains complex when optimal does not.
Time = 4.51 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.77 \[ \int \frac {x^6}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\frac {-\sqrt {\frac {e}{d}} (b d-a e) x \left (d+e x^2\right ) \left (b^2 d^2 x^2 \left (4 d-3 e x^2\right )+3 a^2 e^2 x^2 \left (d-e x^2\right )+a b d \left (4 d^2-7 d e x^2+6 e^2 x^4\right )\right )-i a d^3 \left (8 b^2 d^2-9 a b d e+9 a^2 e^2\right ) \sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )|-1+\frac {b d}{a e}\right )+i a d^3 \left (4 b^2 d^2-5 a b d e+9 a^2 e^2\right ) \sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),-1+\frac {b d}{a e}\right )}{15 d^2 \left (\frac {e}{d}\right )^{5/2} (b d-a e)^3 \sqrt {\frac {\left (d+e x^2\right ) \left (b d x^2+a \left (d-e x^2\right )\right )}{d^2}}} \] Input:
Integrate[x^6/Sqrt[a + b*x^2 + ((b*d*e - a*e^2)*x^4)/d^2],x]
Output:
(-(Sqrt[e/d]*(b*d - a*e)*x*(d + e*x^2)*(b^2*d^2*x^2*(4*d - 3*e*x^2) + 3*a^ 2*e^2*x^2*(d - e*x^2) + a*b*d*(4*d^2 - 7*d*e*x^2 + 6*e^2*x^4))) - I*a*d^3* (8*b^2*d^2 - 9*a*b*d*e + 9*a^2*e^2)*Sqrt[1 + (b*x^2)/a - (e*x^2)/d]*Sqrt[1 + (e*x^2)/d]*EllipticE[I*ArcSinh[Sqrt[e/d]*x], -1 + (b*d)/(a*e)] + I*a*d^ 3*(4*b^2*d^2 - 5*a*b*d*e + 9*a^2*e^2)*Sqrt[1 + (b*x^2)/a - (e*x^2)/d]*Sqrt [1 + (e*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[e/d]*x], -1 + (b*d)/(a*e)])/(15*d ^2*(e/d)^(5/2)*(b*d - a*e)^3*Sqrt[((d + e*x^2)*(b*d*x^2 + a*(d - e*x^2)))/ d^2])
Time = 1.71 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.63, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1442, 1602, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {x^6}{\sqrt {\frac {x^4 \left (b d e-a e^2\right )}{d^2}+a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle \frac {d^2 x^3 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 e (b d-a e)}-\frac {d^2 \int \frac {x^2 \left (4 b x^2+3 a\right )}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{5 e (b d-a e)}\) |
| \(\Big \downarrow \) 1602 |
| \(\displaystyle \frac {d^2 x^3 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 e (b d-a e)}-\frac {d^2 \left (\frac {4 b d^2 x \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 e (b d-a e)}-\frac {d^2 \int \frac {\left (8 b^2-\frac {9 a e (b d-a e)}{d^2}\right ) x^2+4 a b}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{3 e (b d-a e)}\right )}{5 e (b d-a e)}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {d^2 x^3 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 e (b d-a e)}-\frac {d^2 \left (\frac {4 b d^2 x \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 e (b d-a e)}-\frac {d^2 \left (\sqrt {a} \left (\frac {8 b^2 d}{\sqrt {e} \sqrt {b d-a e}}-\frac {9 a \sqrt {e} \sqrt {b d-a e}}{d}+4 \sqrt {a} b\right ) \int \frac {1}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx-\frac {\sqrt {a} d \left (8 b^2-\frac {9 a e (b d-a e)}{d^2}\right ) \int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {a} d \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\right )}{3 e (b d-a e)}\right )}{5 e (b d-a e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 x^3 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 e (b d-a e)}-\frac {d^2 \left (\frac {4 b d^2 x \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 e (b d-a e)}-\frac {d^2 \left (\sqrt {a} \left (\frac {8 b^2 d}{\sqrt {e} \sqrt {b d-a e}}-\frac {9 a \sqrt {e} \sqrt {b d-a e}}{d}+4 \sqrt {a} b\right ) \int \frac {1}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx-\frac {\left (8 b^2-\frac {9 a e (b d-a e)}{d^2}\right ) \int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\right )}{3 e (b d-a e)}\right )}{5 e (b d-a e)}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {d^2 x^3 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 e (b d-a e)}-\frac {d^2 \left (\frac {4 b d^2 x \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 e (b d-a e)}-\frac {d^2 \left (\frac {\sqrt [4]{a} \left (\frac {8 b^2 d}{\sqrt {e} \sqrt {b d-a e}}-\frac {9 a \sqrt {e} \sqrt {b d-a e}}{d}+4 \sqrt {a} b\right ) \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 \sqrt {d} \sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {\left (8 b^2-\frac {9 a e (b d-a e)}{d^2}\right ) \int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\right )}{3 e (b d-a e)}\right )}{5 e (b d-a e)}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {d^2 x^3 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 e (b d-a e)}-\frac {d^2 \left (\frac {4 b d^2 x \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 e (b d-a e)}-\frac {d^2 \left (\frac {\sqrt [4]{a} \left (\frac {8 b^2 d}{\sqrt {e} \sqrt {b d-a e}}-\frac {9 a \sqrt {e} \sqrt {b d-a e}}{d}+4 \sqrt {a} b\right ) \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 \sqrt {d} \sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {\left (8 b^2-\frac {9 a e (b d-a e)}{d^2}\right ) \left (\frac {\sqrt [4]{a} \sqrt {d} \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{\sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {d^2 x \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d}\right )}{\sqrt {e} \sqrt {b d-a e}}\right )}{3 e (b d-a e)}\right )}{5 e (b d-a e)}\) |
Input:
Int[x^6/Sqrt[a + b*x^2 + ((b*d*e - a*e^2)*x^4)/d^2],x]
Output:
(d^2*x^3*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])/(5*e*(b*d - a*e)) - (d ^2*((4*b*d^2*x*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])/(3*e*(b*d - a*e) ) - (d^2*(-(((8*b^2 - (9*a*e*(b*d - a*e))/d^2)*(-((d^2*x*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])/(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2)) + (a^ (1/4)*Sqrt[d]*(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2)*Sqrt[(d^2*(a + b*x ^2 + (e*(b*d - a*e)*x^4)/d^2))/(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2)^2 ]*EllipticE[2*ArcTan[(e^(1/4)*(b*d - a*e)^(1/4)*x)/(a^(1/4)*Sqrt[d])], (2 - (b*d)/(Sqrt[a]*Sqrt[e]*Sqrt[b*d - a*e]))/4])/(e^(1/4)*(b*d - a*e)^(1/4)* Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])))/(Sqrt[e]*Sqrt[b*d - a*e])) + (a^(1/4)*(4*Sqrt[a]*b + (8*b^2*d)/(Sqrt[e]*Sqrt[b*d - a*e]) - (9*a*Sqrt[e] *Sqrt[b*d - a*e])/d)*(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e]*x^2)*Sqrt[(d^2*( a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2))/(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - a*e] *x^2)^2]*EllipticF[2*ArcTan[(e^(1/4)*(b*d - a*e)^(1/4)*x)/(a^(1/4)*Sqrt[d] )], (2 - (b*d)/(Sqrt[a]*Sqrt[e]*Sqrt[b*d - a*e]))/4])/(2*Sqrt[d]*e^(1/4)*( b*d - a*e)^(1/4)*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])))/(3*e*(b*d - a*e))))/(5*e*(b*d - a*e))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[d^4/(c*(m + 4*p + 1)) Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x ] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* p] && (IntegerQ[p] || IntegerQ[m])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3)) Int[(f*x)^(m - 2)* (a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c , 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | | IntegerQ[m])
Time = 5.36 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(\frac {x^{3} \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{-\frac {5 a \,e^{2}}{d^{2}}+\frac {5 e b}{d}}-\frac {4 b x \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{15 \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )^{2}}+\frac {4 b a \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )}{15 \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )^{2} \sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}-\frac {2 \left (-\frac {3 a}{5 \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}+\frac {8 b^{2}}{15 \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )^{2}}\right ) a \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}\, \left (b +\frac {2 a e -b d}{d}\right )}\) | \(505\) |
| elliptic | \(\frac {x^{3} \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{-\frac {5 a \,e^{2}}{d^{2}}+\frac {5 e b}{d}}-\frac {4 b x \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{15 \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )^{2}}+\frac {4 b a \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )}{15 \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )^{2} \sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}-\frac {2 \left (-\frac {3 a}{5 \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}+\frac {8 b^{2}}{15 \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )^{2}}\right ) a \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}\, \left (b +\frac {2 a e -b d}{d}\right )}\) | \(505\) |
| risch | \(-\frac {\left (-a e \,x^{2}+b d \,x^{2}+a d \right ) \left (e \,x^{2}+d \right ) x \left (3 a \,e^{2} x^{2}-3 b d e \,x^{2}+4 b \,d^{2}\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}}{15 e^{2} \sqrt {-\left (e \,x^{2}+d \right ) \left (a e \,x^{2}-b d \,x^{2}-a d \right )}\, \left (a e -b d \right )^{2} \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}{d^{2}}}}+\frac {d^{2} \left (-\frac {2 \left (9 a^{2} e^{2}-9 a b d e +8 b^{2} d^{2}\right ) a \,d^{2} \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}\, \left (b \,d^{2}+d \left (2 a e -b d \right )\right )}+\frac {4 a b \,d^{2} \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}}{15 e^{2} \left (a e -b d \right )^{2} \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}{d^{2}}}}\) | \(577\) |
Input:
int(x^6/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/5/(-a/d^2*e^2+e/d*b)*x^3*(-x^4*a/d^2*e^2+e/d*x^4*b+b*x^2+a)^(1/2)-4/15/( -a/d^2*e^2+e/d*b)^2*b*x*(-x^4*a/d^2*e^2+e/d*x^4*b+b*x^2+a)^(1/2)+4/15/(-a/ d^2*e^2+e/d*b)^2*b*a/((a*e-b*d)/a/d)^(1/2)*(1-(a*e-b*d)/a/d*x^2)^(1/2)*(1+ e*x^2/d)^(1/2)/(-x^4*a/d^2*e^2+e/d*x^4*b+b*x^2+a)^(1/2)*EllipticF(x*((a*e- b*d)/a/d)^(1/2),(-1+b*e/d/(-a/d^2*e^2+e/d*b))^(1/2))-2*(-3/5/(-a/d^2*e^2+e /d*b)*a+8/15/(-a/d^2*e^2+e/d*b)^2*b^2)*a/((a*e-b*d)/a/d)^(1/2)*(1-(a*e-b*d )/a/d*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(-x^4*a/d^2*e^2+e/d*x^4*b+b*x^2+a)^(1/2 )/(b+(2*a*e-b*d)/d)*(EllipticF(x*((a*e-b*d)/a/d)^(1/2),(-1+b*e/d/(-a/d^2*e ^2+e/d*b))^(1/2))-EllipticE(x*((a*e-b*d)/a/d)^(1/2),(-1+b*e/d/(-a/d^2*e^2+ e/d*b))^(1/2)))
Time = 0.10 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.01 \[ \int \frac {x^6}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=-\frac {{\left (8 \, a b^{2} d^{6} - 9 \, a^{2} b d^{5} e + 9 \, a^{3} d^{4} e^{2}\right )} \sqrt {b d e - a e^{2}} \sqrt {-\frac {a d}{b d - a e}} x E(\arcsin \left (\frac {\sqrt {-\frac {a d}{b d - a e}}}{x}\right )\,|\,\frac {b d - a e}{a e}) - {\left (4 \, {\left (2 \, a b^{2} + b^{3}\right )} d^{6} - {\left (9 \, a^{2} b + 8 \, a b^{2}\right )} d^{5} e + {\left (9 \, a^{3} + 4 \, a^{2} b\right )} d^{4} e^{2}\right )} \sqrt {b d e - a e^{2}} \sqrt {-\frac {a d}{b d - a e}} x F(\arcsin \left (\frac {\sqrt {-\frac {a d}{b d - a e}}}{x}\right )\,|\,\frac {b d - a e}{a e}) - {\left (8 \, b^{3} d^{7} - 17 \, a b^{2} d^{6} e + 18 \, a^{2} b d^{5} e^{2} - 9 \, a^{3} d^{4} e^{3} + 3 \, {\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x^{4} - 4 \, {\left (b^{3} d^{6} e - 2 \, a b^{2} d^{5} e^{2} + a^{2} b d^{4} e^{3}\right )} x^{2}\right )} \sqrt {\frac {b d^{2} x^{2} + {\left (b d e - a e^{2}\right )} x^{4} + a d^{2}}{d^{2}}}}{15 \, {\left (b^{4} d^{4} e^{3} - 4 \, a b^{3} d^{3} e^{4} + 6 \, a^{2} b^{2} d^{2} e^{5} - 4 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x} \] Input:
integrate(x^6/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="fricas" )
Output:
-1/15*((8*a*b^2*d^6 - 9*a^2*b*d^5*e + 9*a^3*d^4*e^2)*sqrt(b*d*e - a*e^2)*s qrt(-a*d/(b*d - a*e))*x*elliptic_e(arcsin(sqrt(-a*d/(b*d - a*e))/x), (b*d - a*e)/(a*e)) - (4*(2*a*b^2 + b^3)*d^6 - (9*a^2*b + 8*a*b^2)*d^5*e + (9*a^ 3 + 4*a^2*b)*d^4*e^2)*sqrt(b*d*e - a*e^2)*sqrt(-a*d/(b*d - a*e))*x*ellipti c_f(arcsin(sqrt(-a*d/(b*d - a*e))/x), (b*d - a*e)/(a*e)) - (8*b^3*d^7 - 17 *a*b^2*d^6*e + 18*a^2*b*d^5*e^2 - 9*a^3*d^4*e^3 + 3*(b^3*d^5*e^2 - 3*a*b^2 *d^4*e^3 + 3*a^2*b*d^3*e^4 - a^3*d^2*e^5)*x^4 - 4*(b^3*d^6*e - 2*a*b^2*d^5 *e^2 + a^2*b*d^4*e^3)*x^2)*sqrt((b*d^2*x^2 + (b*d*e - a*e^2)*x^4 + a*d^2)/ d^2))/((b^4*d^4*e^3 - 4*a*b^3*d^3*e^4 + 6*a^2*b^2*d^2*e^5 - 4*a^3*b*d*e^6 + a^4*e^7)*x)
\[ \int \frac {x^6}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int \frac {x^{6}}{\sqrt {- \left (1 + \frac {e x^{2}}{d}\right ) \left (- a + \frac {a e x^{2}}{d} - b x^{2}\right )}}\, dx \] Input:
integrate(x**6/(a+b*x**2+(-a*e**2+b*d*e)*x**4/d**2)**(1/2),x)
Output:
Integral(x**6/sqrt(-(1 + e*x**2/d)*(-a + a*e*x**2/d - b*x**2)), x)
\[ \int \frac {x^6}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int { \frac {x^{6}}{\sqrt {b x^{2} + \frac {{\left (b d e - a e^{2}\right )} x^{4}}{d^{2}} + a}} \,d x } \] Input:
integrate(x^6/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="maxima" )
Output:
integrate(x^6/sqrt(b*x^2 + (b*d*e - a*e^2)*x^4/d^2 + a), x)
\[ \int \frac {x^6}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int { \frac {x^{6}}{\sqrt {b x^{2} + \frac {{\left (b d e - a e^{2}\right )} x^{4}}{d^{2}} + a}} \,d x } \] Input:
integrate(x^6/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="giac")
Output:
integrate(x^6/sqrt(b*x^2 + (b*d*e - a*e^2)*x^4/d^2 + a), x)
Timed out. \[ \int \frac {x^6}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int \frac {x^6}{\sqrt {a+b\,x^2-\frac {x^4\,\left (a\,e^2-b\,d\,e\right )}{d^2}}} \,d x \] Input:
int(x^6/(a + b*x^2 - (x^4*(a*e^2 - b*d*e))/d^2)^(1/2),x)
Output:
int(x^6/(a + b*x^2 - (x^4*(a*e^2 - b*d*e))/d^2)^(1/2), x)
\[ \int \frac {x^6}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx =\text {Too large to display} \] Input:
int(x^6/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x)
Output:
(d*( - 3*sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2)*a*e**2*x**3 - 4* sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2)*b*d**2*x + 3*sqrt(d + e*x **2)*sqrt(a*d - a*e*x**2 + b*d*x**2)*b*d*e*x**3 + 9*int((sqrt(d + e*x**2)* sqrt(a*d - a*e*x**2 + b*d*x**2)*x**2)/(a**3*d**2*e**2 - a**3*e**4*x**4 - 2 *a**2*b*d**3*e + a**2*b*d**2*e**2*x**2 + 3*a**2*b*d*e**3*x**4 + a*b**2*d** 4 - 2*a*b**2*d**3*e*x**2 - 3*a*b**2*d**2*e**2*x**4 + b**3*d**4*x**2 + b**3 *d**3*e*x**4),x)*a**4*d**2*e**4 - 27*int((sqrt(d + e*x**2)*sqrt(a*d - a*e* x**2 + b*d*x**2)*x**2)/(a**3*d**2*e**2 - a**3*e**4*x**4 - 2*a**2*b*d**3*e + a**2*b*d**2*e**2*x**2 + 3*a**2*b*d*e**3*x**4 + a*b**2*d**4 - 2*a*b**2*d* *3*e*x**2 - 3*a*b**2*d**2*e**2*x**4 + b**3*d**4*x**2 + b**3*d**3*e*x**4),x )*a**3*b*d**3*e**3 + 35*int((sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x* *2)*x**2)/(a**3*d**2*e**2 - a**3*e**4*x**4 - 2*a**2*b*d**3*e + a**2*b*d**2 *e**2*x**2 + 3*a**2*b*d*e**3*x**4 + a*b**2*d**4 - 2*a*b**2*d**3*e*x**2 - 3 *a*b**2*d**2*e**2*x**4 + b**3*d**4*x**2 + b**3*d**3*e*x**4),x)*a**2*b**2*d **4*e**2 - 25*int((sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2)*x**2)/ (a**3*d**2*e**2 - a**3*e**4*x**4 - 2*a**2*b*d**3*e + a**2*b*d**2*e**2*x**2 + 3*a**2*b*d*e**3*x**4 + a*b**2*d**4 - 2*a*b**2*d**3*e*x**2 - 3*a*b**2*d* *2*e**2*x**4 + b**3*d**4*x**2 + b**3*d**3*e*x**4),x)*a*b**3*d**5*e + 8*int ((sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2)*x**2)/(a**3*d**2*e**2 - a**3*e**4*x**4 - 2*a**2*b*d**3*e + a**2*b*d**2*e**2*x**2 + 3*a**2*b*d*...