Integrand size = 26, antiderivative size = 289 \[ \int \frac {1}{x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=-\frac {a+b x^2}{a x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {d x \left (a+b x^2\right )}{a \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac {\sqrt {c} \sqrt {d} \left (a+b x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}+\frac {\sqrt {c} \sqrt {d} \left (a+b x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )} \]
(-b*x^2-a)/a/x/(e*(b*x^2+a)/(d*x^2+c))^(1/2)+d*x*(b*x^2+a)/a/(d*x^2+c)/(e* (b*x^2+a)/(d*x^2+c))^(1/2)-(b*x^2+a)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/ 2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*c^(1/2 )*d^(1/2)/a/(d*x^2+c)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(e*(b*x^2+a)/(d*x^2+ c))^(1/2)+(b*x^2+a)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^ (1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*c^(1/2)*d^(1/2)/a/(d*x^ 2+c)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)
Time = 2.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.38 \[ \int \frac {1}{x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {\left (a+b x^2\right ) \left (-\frac {1}{x}+\frac {d \sqrt {1+\frac {d x^2}{c}} E\left (\arcsin \left (\sqrt {-\frac {d}{c}} x\right )|\frac {b c}{a d}\right )}{\sqrt {-\frac {d}{c}} \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )}\right )}{a \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]
((a + b*x^2)*(-x^(-1) + (d*Sqrt[1 + (d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c )]*x], (b*c)/(a*d)])/(Sqrt[-(d/c)]*Sqrt[1 + (b*x^2)/a]*(c + d*x^2))))/(a*S qrt[(e*(a + b*x^2))/(c + d*x^2)])
Time = 0.39 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2058, 377, 27, 324, 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.
| \(\displaystyle \int \frac {1}{x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \int \frac {\sqrt {d x^2+c}}{x^2 \sqrt {b x^2+a}}dx}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 377 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {\int \frac {d \sqrt {b x^2+a}}{\sqrt {d x^2+c}}dx}{a}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a x}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {d \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c}}dx}{a}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a x}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 324 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {d \left (a \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+b \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a x}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {d \left (b \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a x}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {d \left (b \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a x}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{a}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{a x}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
(Sqrt[a + b*x^2]*(-((Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*x)) + (d*(b*((x*S qrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[A rcTan[(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])) + (Sqrt[c]*Sqrt[a + b*x^2]*Elliptic F[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])))/a))/(Sqrt[(e*(a + b*x^2))/(c + d* x^2)]*Sqrt[c + d*x^2])
3.4.5.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b Int[x^2/(Sqr t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c ] && PosQ[b/a]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b *c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && 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[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Time = 4.66 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.03
| method | result | size |
| default | \(-\frac {\left (b \,x^{2}+a \right ) \left (\sqrt {-\frac {b}{a}}\, b d \,x^{4}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a d x +b c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, x F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )-b c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, x E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )+\sqrt {-\frac {b}{a}}\, a d \,x^{2}+\sqrt {-\frac {b}{a}}\, b c \,x^{2}+\sqrt {-\frac {b}{a}}\, a c \right )}{\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a x \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(297\) |
| risch | \(-\frac {b \,x^{2}+a}{a x \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {d \left (\frac {a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 b a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{a \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(344\) |
-(b*x^2+a)*((-b/a)^(1/2)*b*d*x^4-((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*E llipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*d*x+b*c*((b*x^2+a)/a)^(1/2)*((d *x^2+c)/c)^(1/2)*x*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))-b*c*((b*x^2+a )/a)^(1/2)*((d*x^2+c)/c)^(1/2)*x*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2)) +(-b/a)^(1/2)*a*d*x^2+(-b/a)^(1/2)*b*c*x^2+(-b/a)^(1/2)*a*c)/(e*(b*x^2+a)/ (d*x^2+c))^(1/2)/((d*x^2+c)*(b*x^2+a))^(1/2)/a/x/(-b/a)^(1/2)/(b*d*x^4+a*d *x^2+b*c*x^2+a*c)^(1/2)
Time = 0.11 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.54 \[ \int \frac {1}{x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {b^{2} c d \sqrt {\frac {a c e}{d^{2}}} x \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (b^{2} c d + a^{2} d^{2}\right )} \sqrt {\frac {a c e}{d^{2}}} x \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (a b c d x^{2} + a b c^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{a^{2} b c e x} \]
(b^2*c*d*sqrt(a*c*e/d^2)*x*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d /(b*c)) - (b^2*c*d + a^2*d^2)*sqrt(a*c*e/d^2)*x*sqrt(-b/a)*elliptic_f(arcs in(x*sqrt(-b/a)), a*d/(b*c)) - (a*b*c*d*x^2 + a*b*c^2)*sqrt((b*e*x^2 + a*e )/(d*x^2 + c)))/(a^2*b*c*e*x)
Timed out. \[ \int \frac {1}{x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int { \frac {1}{\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} x^{2}} \,d x } \]
\[ \int \frac {1}{x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int { \frac {1}{\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int \frac {1}{x^2\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \]