Integrand size = 33, antiderivative size = 396 \[ \int \frac {1}{x^6 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=-\frac {\sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}{5 a x^5}+\frac {4 b \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}{15 a^2 x^3}-\frac {d \left (8 b^2-\frac {9 a e (b d-a e)}{d^2}\right ) \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}{15 a^3 x \left (d+e x^2\right )}-\frac {\sqrt {e} \left (8 b^2 d^2-9 a b d e+9 a^2 e^2\right ) \left (d+e x^2\right ) \sqrt {\frac {a d+(b d-a e) x^2}{a \left (d+e x^2\right )}} E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|2-\frac {b d}{a e}\right )}{15 a^2 d^{7/2} \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}+\frac {4 b \sqrt {e} (b d-a e) \left (d+e x^2\right ) \sqrt {\frac {a d+(b d-a e) x^2}{a \left (d+e x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),2-\frac {b d}{a e}\right )}{15 a^2 d^{5/2} \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}} \] Output:
-1/5*(a+b*x^2+e*(-a*e+b*d)*x^4/d^2)^(1/2)/a/x^5+4/15*b*(a+b*x^2+e*(-a*e+b* d)*x^4/d^2)^(1/2)/a^2/x^3-1/15*d*(8*b^2-9*a*e*(-a*e+b*d)/d^2)*(a+b*x^2+e*( -a*e+b*d)*x^4/d^2)^(1/2)/a^3/x/(e*x^2+d)-1/15*e^(1/2)*(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))/a^2/d^(7/2)/(a+b*x^2 +e*(-a*e+b*d)*x^4/d^2)^(1/2)+4/15*b*e^(1/2)*(-a*e+b*d)*(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))/a^2/d^(5/2)/(a+b*x^2+e*(-a*e+b*d)*x^4/d^2)^(1/2)
Result contains complex when optimal does not.
Time = 4.53 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^6 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\frac {\sqrt {\frac {e}{d}} \left (d+e x^2\right ) \left (-8 b^3 d^3 x^6+a b^2 d^2 x^4 \left (-4 d+17 e x^2\right )+a^2 b d x^2 \left (d^2+5 d e x^2-18 e^2 x^4\right )-3 a^3 \left (d^3-d^2 e x^2+3 d e^2 x^4-3 e^3 x^6\right )\right )-i a d e \left (8 b^2 d^2-9 a b d e+9 a^2 e^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )|-1+\frac {b d}{a e}\right )+i a d e \left (4 b^2 d^2-5 a b d e+9 a^2 e^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),-1+\frac {b d}{a e}\right )}{15 a^3 d^4 \sqrt {\frac {e}{d}} x^5 \sqrt {\frac {\left (d+e x^2\right ) \left (b d x^2+a \left (d-e x^2\right )\right )}{d^2}}} \] Input:
Integrate[1/(x^6*Sqrt[a + b*x^2 + ((b*d*e - a*e^2)*x^4)/d^2]),x]
Output:
(Sqrt[e/d]*(d + e*x^2)*(-8*b^3*d^3*x^6 + a*b^2*d^2*x^4*(-4*d + 17*e*x^2) + a^2*b*d*x^2*(d^2 + 5*d*e*x^2 - 18*e^2*x^4) - 3*a^3*(d^3 - d^2*e*x^2 + 3*d *e^2*x^4 - 3*e^3*x^6)) - I*a*d*e*(8*b^2*d^2 - 9*a*b*d*e + 9*a^2*e^2)*x^5*S qrt[1 + (b*x^2)/a - (e*x^2)/d]*Sqrt[1 + (e*x^2)/d]*EllipticE[I*ArcSinh[Sqr t[e/d]*x], -1 + (b*d)/(a*e)] + I*a*d*e*(4*b^2*d^2 - 5*a*b*d*e + 9*a^2*e^2) *x^5*Sqrt[1 + (b*x^2)/a - (e*x^2)/d]*Sqrt[1 + (e*x^2)/d]*EllipticF[I*ArcSi nh[Sqrt[e/d]*x], -1 + (b*d)/(a*e)])/(15*a^3*d^4*Sqrt[e/d]*x^5*Sqrt[((d + e *x^2)*(b*d*x^2 + a*(d - e*x^2)))/d^2])
Time = 1.83 (sec) , antiderivative size = 756, normalized size of antiderivative = 1.91, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1443, 25, 27, 1604, 1604, 25, 27, 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.
\(\displaystyle \int \frac {1}{x^6 \sqrt {\frac {x^4 \left (b d e-a e^2\right )}{d^2}+a+b x^2}} \, dx\) |
\(\Big \downarrow \) 1443 |
\(\displaystyle \frac {\int -\frac {4 b d^2+3 e (b d-a e) x^2}{d^2 x^4 \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{5 a}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {4 b d^2+3 e (b d-a e) x^2}{d^2 x^4 \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{5 a}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {4 b d^2+3 e (b d-a e) x^2}{x^4 \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{5 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
\(\Big \downarrow \) 1604 |
\(\displaystyle -\frac {-\frac {\int \frac {8 b^2 d^2+4 b e (b d-a e) x^2-9 a e (b d-a e)}{x^2 \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{3 a}-\frac {4 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}}{5 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
\(\Big \downarrow \) 1604 |
\(\displaystyle -\frac {-\frac {-\frac {\int -\frac {e \left (4 a b (b d-a e) d^2+(b d-a e) \left (8 b^2 d^2-9 a e (b d-a e)\right ) x^2\right )}{d^2 \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{a}-\frac {\left (8 b^2 d^2-9 a e (b d-a e)\right ) \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a}-\frac {4 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}}{5 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {-\frac {\frac {\int \frac {e (b d-a e) \left (4 a b d^2+\left (8 b^2 d^2-9 a e (b d-a e)\right ) x^2\right )}{d^2 \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{a}-\frac {\left (8 b^2 d^2-9 a e (b d-a e)\right ) \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a}-\frac {4 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}}{5 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {-\frac {\frac {e (b d-a e) \int \frac {4 a b d^2+\left (8 b^2 d^2-9 a e (b d-a e)\right ) x^2}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{a d^2}-\frac {\left (8 b^2 d^2-9 a e (b d-a e)\right ) \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a}-\frac {4 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}}{5 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
\(\Big \downarrow \) 1511 |
\(\displaystyle -\frac {-\frac {\frac {e (b d-a e) \left (\sqrt {a} d \left (\frac {8 b^2 d^2}{\sqrt {e} \sqrt {b d-a e}}-9 a \sqrt {e} \sqrt {b d-a e}+4 \sqrt {a} b d\right ) \int \frac {1}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx-\frac {\sqrt {a} d \left (8 b^2 d^2-9 a e (b d-a e)\right ) \int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {a} d \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\right )}{a d^2}-\frac {\left (8 b^2 d^2-9 a e (b d-a e)\right ) \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a}-\frac {4 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}}{5 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {-\frac {\frac {e (b d-a e) \left (\sqrt {a} d \left (\frac {8 b^2 d^2}{\sqrt {e} \sqrt {b d-a e}}-9 a \sqrt {e} \sqrt {b d-a e}+4 \sqrt {a} b d\right ) \int \frac {1}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx-\frac {\left (8 b^2 d^2-9 a e (b d-a e)\right ) \int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\right )}{a d^2}-\frac {\left (8 b^2 d^2-9 a e (b d-a e)\right ) \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a}-\frac {4 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}}{5 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle -\frac {-\frac {\frac {e (b d-a e) \left (\frac {\sqrt [4]{a} \sqrt {d} \left (\frac {8 b^2 d^2}{\sqrt {e} \sqrt {b d-a e}}-9 a \sqrt {e} \sqrt {b d-a e}+4 \sqrt {a} b d\right ) \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 \sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {\left (8 b^2 d^2-9 a e (b d-a e)\right ) \int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\right )}{a d^2}-\frac {\left (8 b^2 d^2-9 a e (b d-a e)\right ) \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a}-\frac {4 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}}{5 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
\(\Big \downarrow \) 1509 |
\(\displaystyle -\frac {-\frac {\frac {e (b d-a e) \left (\frac {\sqrt [4]{a} \sqrt {d} \left (\frac {8 b^2 d^2}{\sqrt {e} \sqrt {b d-a e}}-9 a \sqrt {e} \sqrt {b d-a e}+4 \sqrt {a} b d\right ) \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 \sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {\left (8 b^2 d^2-9 a e (b d-a e)\right ) \left (\frac {\sqrt [4]{a} \sqrt {d} \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{\sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {d^2 x \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d}\right )}{\sqrt {e} \sqrt {b d-a e}}\right )}{a d^2}-\frac {\left (8 b^2 d^2-9 a e (b d-a e)\right ) \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{a x}}{3 a}-\frac {4 b d^2 \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{3 a x^3}}{5 a d^2}-\frac {\sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{5 a x^5}\) |
Input:
Int[1/(x^6*Sqrt[a + b*x^2 + ((b*d*e - a*e^2)*x^4)/d^2]),x]
Output:
-1/5*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2]/(a*x^5) - ((-4*b*d^2*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])/(3*a*x^3) - (-(((8*b^2*d^2 - 9*a*e*(b *d - a*e))*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])/(a*x)) + (e*(b*d - a *e)*(-(((8*b^2*d^2 - 9*a*e*(b*d - a*e))*(-((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)*S qrt[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]*Ellip ticE[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 )*Sqrt[d]*(4*Sqrt[a]*b*d + (8*b^2*d^2)/(Sqrt[e]*Sqrt[b*d - a*e]) - 9*a*Sqr t[e]*Sqrt[b*d - a*e])*(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*e^(1/4)*(b*d - a *e)^(1/4)*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])))/(a*d^2))/(3*a))/(5* a*d^2)
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*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Sim p[1/(a*d^2*(m + 1)) Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p + 5)* x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && 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[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) /(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1)) Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Time = 5.39 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.35
method | result | size |
default | \(-\frac {\sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{5 a \,x^{5}}+\frac {4 b \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{15 a^{2} x^{3}}-\frac {\left (9 a^{2} e^{2}-9 a b d e +8 b^{2} d^{2}\right ) \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{15 a^{3} d^{2} x}-\frac {4 b e \left (a e -b d \right ) \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )}{15 a^{2} d^{2} \sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}+\frac {2 e \left (9 a^{3} e^{3}-18 a^{2} b d \,e^{2}+17 a \,b^{2} d^{2} e -8 b^{3} d^{3}\right ) \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )\right )}{15 a^{2} d^{4} \sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}\, \left (b +\frac {2 a e -b d}{d}\right )}\) | \(535\) |
elliptic | \(-\frac {\sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{5 a \,x^{5}}+\frac {4 b \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{15 a^{2} x^{3}}-\frac {\left (9 a^{2} e^{2}-9 a b d e +8 b^{2} d^{2}\right ) \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}{15 a^{3} d^{2} x}-\frac {4 b e \left (a e -b d \right ) \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )}{15 a^{2} d^{2} \sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}}+\frac {2 e \left (9 a^{3} e^{3}-18 a^{2} b d \,e^{2}+17 a \,b^{2} d^{2} e -8 b^{3} d^{3}\right ) \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )\right )}{15 a^{2} d^{4} \sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}\, \left (b +\frac {2 a e -b d}{d}\right )}\) | \(535\) |
risch | \(-\frac {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right ) \left (9 a^{2} e^{2} x^{4}-9 a b d e \,x^{4}+8 b^{2} d^{2} x^{4}-4 a \,d^{2} b \,x^{2}+3 a^{2} d^{2}\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}}{15 a^{3} d^{4} x^{5} \sqrt {-\left (e \,x^{2}+d \right ) \left (a e \,x^{2}-b d \,x^{2}-a d \right )}\, \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}{d^{2}}}}-\frac {e \left (-\frac {2 \left (9 a^{3} e^{3}-18 a^{2} b d \,e^{2}+17 a \,b^{2} d^{2} e -8 b^{3} d^{3}\right ) a \,d^{2} \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}\, \left (b \,d^{2}+d \left (2 a e -b d \right )\right )}-\frac {4 a \,b^{2} d^{3} \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}+\frac {4 a^{2} b \,d^{2} e \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b d e}{-a \,e^{2}+b d e}}\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}}\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}}{15 a^{3} d^{4} \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (-a e \,x^{2}+b d \,x^{2}+a d \right )}{d^{2}}}}\) | \(740\) |
Input:
int(1/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*(-x^4*a/d^2*e^2+e/d*x^4*b+b*x^2+a)^(1/2)/x^5+4/15*b/a^2*(-x^4*a/d^2 *e^2+e/d*x^4*b+b*x^2+a)^(1/2)/x^3-1/15*(9*a^2*e^2-9*a*b*d*e+8*b^2*d^2)/a^3 /d^2*(-x^4*a/d^2*e^2+e/d*x^4*b+b*x^2+a)^(1/2)/x-4/15*b*e*(a*e-b*d)/a^2/d^2 /((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))+2/15*e*(9*a^3*e^3-18*a^2*b*d*e^2+17*a*b^2 *d^2*e-8*b^3*d^3)/a^2/d^4/((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 = 0.09 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^6 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\frac {{\left (8 \, b^{3} d^{3} - 17 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 9 \, a^{3} e^{3}\right )} \sqrt {a d^{2}} x^{5} \sqrt {-\frac {b d - a e}{a d}} E(\arcsin \left (x \sqrt {-\frac {b d - a e}{a d}}\right )\,|\,\frac {a e}{b d - a e}) - {\left (8 \, b^{3} d^{3} + 18 \, a^{2} b d e^{2} - 9 \, a^{3} e^{3} + {\left (4 \, a^{2} b - 17 \, a b^{2}\right )} d^{2} e\right )} \sqrt {a d^{2}} x^{5} \sqrt {-\frac {b d - a e}{a d}} F(\arcsin \left (x \sqrt {-\frac {b d - a e}{a d}}\right )\,|\,\frac {a e}{b d - a e}) + {\left (4 \, a^{2} b d^{4} x^{2} - 3 \, a^{3} d^{4} - {\left (8 \, a b^{2} d^{4} - 9 \, a^{2} b d^{3} e + 9 \, a^{3} d^{2} e^{2}\right )} x^{4}\right )} \sqrt {\frac {b d^{2} x^{2} + {\left (b d e - a e^{2}\right )} x^{4} + a d^{2}}{d^{2}}}}{15 \, a^{4} d^{4} x^{5}} \] Input:
integrate(1/x^6/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="frica s")
Output:
1/15*((8*b^3*d^3 - 17*a*b^2*d^2*e + 18*a^2*b*d*e^2 - 9*a^3*e^3)*sqrt(a*d^2 )*x^5*sqrt(-(b*d - a*e)/(a*d))*elliptic_e(arcsin(x*sqrt(-(b*d - a*e)/(a*d) )), a*e/(b*d - a*e)) - (8*b^3*d^3 + 18*a^2*b*d*e^2 - 9*a^3*e^3 + (4*a^2*b - 17*a*b^2)*d^2*e)*sqrt(a*d^2)*x^5*sqrt(-(b*d - a*e)/(a*d))*elliptic_f(arc sin(x*sqrt(-(b*d - a*e)/(a*d))), a*e/(b*d - a*e)) + (4*a^2*b*d^4*x^2 - 3*a ^3*d^4 - (8*a*b^2*d^4 - 9*a^2*b*d^3*e + 9*a^3*d^2*e^2)*x^4)*sqrt((b*d^2*x^ 2 + (b*d*e - a*e^2)*x^4 + a*d^2)/d^2))/(a^4*d^4*x^5)
\[ \int \frac {1}{x^6 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int \frac {1}{x^{6} \sqrt {- \left (1 + \frac {e x^{2}}{d}\right ) \left (- a + \frac {a e x^{2}}{d} - b x^{2}\right )}}\, dx \] Input:
integrate(1/x**6/(a+b*x**2+(-a*e**2+b*d*e)*x**4/d**2)**(1/2),x)
Output:
Integral(1/(x**6*sqrt(-(1 + e*x**2/d)*(-a + a*e*x**2/d - b*x**2))), x)
\[ \int \frac {1}{x^6 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + \frac {{\left (b d e - a e^{2}\right )} x^{4}}{d^{2}} + a} x^{6}} \,d x } \] Input:
integrate(1/x^6/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="maxim a")
Output:
integrate(1/(sqrt(b*x^2 + (b*d*e - a*e^2)*x^4/d^2 + a)*x^6), x)
\[ \int \frac {1}{x^6 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + \frac {{\left (b d e - a e^{2}\right )} x^{4}}{d^{2}} + a} x^{6}} \,d x } \] Input:
integrate(1/x^6/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x, algorithm="giac" )
Output:
integrate(1/(sqrt(b*x^2 + (b*d*e - a*e^2)*x^4/d^2 + a)*x^6), x)
Timed out. \[ \int \frac {1}{x^6 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\int \frac {1}{x^6\,\sqrt {a+b\,x^2-\frac {x^4\,\left (a\,e^2-b\,d\,e\right )}{d^2}}} \,d x \] Input:
int(1/(x^6*(a + b*x^2 - (x^4*(a*e^2 - b*d*e))/d^2)^(1/2)),x)
Output:
int(1/(x^6*(a + b*x^2 - (x^4*(a*e^2 - b*d*e))/d^2)^(1/2)), x)
\[ \int \frac {1}{x^6 \sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, \sqrt {-a e \,x^{2}+b d \,x^{2}+a d}\, a \,d^{2}-3 \sqrt {e \,x^{2}+d}\, \sqrt {-a e \,x^{2}+b d \,x^{2}+a d}\, a \,e^{2} x^{4}+3 \sqrt {e \,x^{2}+d}\, \sqrt {-a e \,x^{2}+b d \,x^{2}+a d}\, b d e \,x^{4}-3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-a e \,x^{2}+b d \,x^{2}+a d}\, x^{2}}{-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}d x \right ) a^{2} e^{4} x^{5}+6 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-a e \,x^{2}+b d \,x^{2}+a d}\, x^{2}}{-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}d x \right ) a b d \,e^{3} x^{5}-3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-a e \,x^{2}+b d \,x^{2}+a d}\, x^{2}}{-a \,e^{2} x^{4}+b d e \,x^{4}+b \,d^{2} x^{2}+a \,d^{2}}d x \right ) b^{2} d^{2} e^{2} x^{5}-4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-a e \,x^{2}+b d \,x^{2}+a d}}{-a \,e^{2} x^{8}+b d e \,x^{8}+b \,d^{2} x^{6}+a \,d^{2} x^{4}}d x \right ) a b \,d^{4} x^{5}}{5 a^{2} d^{3} x^{5}} \] Input:
int(1/x^6/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x)
Output:
( - sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2)*a*d**2 - 3*sqrt(d + e *x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2)*a*e**2*x**4 + 3*sqrt(d + e*x**2)*sq rt(a*d - a*e*x**2 + b*d*x**2)*b*d*e*x**4 - 3*int((sqrt(d + e*x**2)*sqrt(a* d - a*e*x**2 + b*d*x**2)*x**2)/(a*d**2 - a*e**2*x**4 + b*d**2*x**2 + b*d*e *x**4),x)*a**2*e**4*x**5 + 6*int((sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b *d*x**2)*x**2)/(a*d**2 - a*e**2*x**4 + b*d**2*x**2 + b*d*e*x**4),x)*a*b*d* e**3*x**5 - 3*int((sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2)*x**2)/ (a*d**2 - a*e**2*x**4 + b*d**2*x**2 + b*d*e*x**4),x)*b**2*d**2*e**2*x**5 - 4*int((sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2))/(a*d**2*x**4 - a *e**2*x**8 + b*d**2*x**6 + b*d*e*x**8),x)*a*b*d**4*x**5)/(5*a**2*d**3*x**5 )