Integrand size = 24, antiderivative size = 518 \[ \int \frac {1}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx=\frac {e \arctan \left (\frac {\sqrt {-c d^4-b d^2 e^2-a e^4} x}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {-c d^4-b d^2 e^2-a e^4}}-\frac {e \text {arctanh}\left (\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{2 \sqrt {c d^4+b d^2 e^2+a e^4} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {c d^4+b d^2 e^2+a e^4}}+\frac {\sqrt [4]{c} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+b x^2+c x^4}} \]
1/2*e*arctan(x*(-a*e^4-b*d^2*e^2-c*d^4)^(1/2)/d/e/(c*x^4+b*x^2+a)^(1/2))/( -a*e^4-b*d^2*e^2-c*d^4)^(1/2)-1/2*e*arctanh(1/2*(b*d^2+2*a*e^2+(b*e^2+2*c* d^2)*x^2)/(a*e^4+b*d^2*e^2+c*d^4)^(1/2)/(c*x^4+b*x^2+a)^(1/2))/(a*e^4+b*d^ 2*e^2+c*d^4)^(1/2)+1/2*c^(1/4)*d*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2 )/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4 ))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+b*x^2+a )/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/(e^2*a^(1/2)+d^2*c^(1/2))/(c*x^4+ b*x^2+a)^(1/2)-1/4*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan (c^(1/4)*x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2* a^(1/2)+d^2*c^(1/2))^2/d^2/e^2/a^(1/2)/c^(1/2),1/2*(2-b/a^(1/2)/c^(1/2))^( 1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+b*x^2+a)/(a ^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/c^(1/4)/d/(e^2*a^(1/2)+d^2*c^(1/2))/( c*x^4+b*x^2+a)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1407\) vs. \(2(518)=1036\).
Time = 15.38 (sec) , antiderivative size = 1407, normalized size of antiderivative = 2.72 \[ \int \frac {1}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx=-\frac {\sqrt {2} c \sqrt {\frac {\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}} \left (\sqrt {2} \sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}-2 x\right )}{\left (\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right ) \left (\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}-2 x\right )}} \left (\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}-2 x\right ) \sqrt {-\frac {\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}} \left (\sqrt {2} \sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+2 x\right )}{\left (-\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right ) \left (\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}-2 x\right )}} \left (\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}+2 x\right ) \left (-\left (\left (2 d-\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (-\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right ) \left (\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}+2 x\right )}{\left (\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right ) \left (\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}-2 x\right )}}\right ),\frac {\left (\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right )^2}{\left (\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}-\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right )^2}\right )\right )-2 \sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}} e \operatorname {EllipticPi}\left (\frac {\left (\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right ) \left (2 d+\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}} e\right )}{\left (-\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right ) \left (-2 d+\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}} e\right )},\arcsin \left (\sqrt {\frac {\left (-\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right ) \left (\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}+2 x\right )}{\left (\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right ) \left (\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}-2 x\right )}}\right ),\frac {\left (\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right )^2}{\left (\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}-\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right )^2}\right )\right )}{\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}} \left (4 c d^2+2 \left (b+\sqrt {b^2-4 a c}\right ) e^2\right ) \sqrt {\frac {\left (-\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right ) \left (\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}+2 x\right )}{\left (\sqrt {\frac {-b+\sqrt {b^2-4 a c}}{c}}+\sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}\right ) \left (\sqrt {2} \sqrt {-\frac {b+\sqrt {b^2-4 a c}}{c}}-2 x\right )}} \sqrt {a+b x^2+c x^4}} \]
-((Sqrt[2]*c*Sqrt[(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*(Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c] - 2*x))/((Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c] + Sqrt[-( (b + Sqrt[b^2 - 4*a*c])/c)])*(Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)] - 2*x))]*(Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)] - 2*x)*Sqrt[-((Sqrt[-( (b + Sqrt[b^2 - 4*a*c])/c)]*(Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c] + 2* x))/((-Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c] + Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c )])*(Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)] - 2*x)))]*(Sqrt[2]*Sqrt[-( (b + Sqrt[b^2 - 4*a*c])/c)] + 2*x)*(-((2*d - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e)*EllipticF[ArcSin[Sqrt[((-Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c] + Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)])*(Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a *c])/c)] + 2*x))/((Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c] + Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)])*(Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)] - 2*x))]], (Sq rt[(-b + Sqrt[b^2 - 4*a*c])/c] + Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)])^2/(Sq rt[(-b + Sqrt[b^2 - 4*a*c])/c] - Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)])^2]) - 2*Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e*EllipticPi[((Sqrt[(-b + Sq rt[b^2 - 4*a*c])/c] + Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)])*(2*d + Sqrt[2]*S qrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e))/((-Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c] + Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)])*(-2*d + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*e)), ArcSin[Sqrt[((-Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c] + Sqrt [-((b + Sqrt[b^2 - 4*a*c])/c)])*(Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])...
Time = 0.95 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2262, 1540, 27, 1416, 1576, 1154, 219, 2222}
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.
| \(\displaystyle \int \frac {1}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx\) |
| \(\Big \downarrow \) 2262 |
| \(\displaystyle d \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx-e \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle d \left (\frac {\sqrt {a} e^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\sqrt {c} \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}\right )-e \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {e^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\sqrt {c} \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}\right )-e \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle d \left (\frac {e^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt {a+b x^2+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}\right )-e \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle d \left (\frac {e^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt {a+b x^2+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}\right )-\frac {1}{2} e \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx^2\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle d \left (\frac {e^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt {a+b x^2+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}\right )+e \int \frac {1}{4 \left (c d^4+b e^2 d^2+a e^4\right )-x^4}d\left (-\frac {b d^2+2 a e^2+\left (2 c d^2+b e^2\right ) x^2}{\sqrt {c x^4+b x^2+a}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d \left (\frac {e^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt {a+b x^2+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}\right )-\frac {e \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {a e^4+b d^2 e^2+c d^4}}\) |
| \(\Big \downarrow \) 2222 |
| \(\displaystyle d \left (\frac {e^2 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+b d^2 e^2+c d^4}}{d e \sqrt {a+b x^2+c x^4}}\right )}{2 d e \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{\sqrt {a} e^2+\sqrt {c} d^2}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt {a+b x^2+c x^4} \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}\right )-\frac {e \text {arctanh}\left (\frac {2 a e^2+x^2 \left (b e^2+2 c d^2\right )+b d^2}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^4+b d^2 e^2+c d^4}}\right )}{2 \sqrt {a e^4+b d^2 e^2+c d^4}}\) |
-1/2*(e*ArcTanh[(b*d^2 + 2*a*e^2 + (2*c*d^2 + b*e^2)*x^2)/(2*Sqrt[c*d^4 + b*d^2*e^2 + a*e^4]*Sqrt[a + b*x^2 + c*x^4])])/Sqrt[c*d^4 + b*d^2*e^2 + a*e ^4] + d*((c^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a ] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[ a]*Sqrt[c]))/4])/(2*a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*Sqrt[a + b*x^2 + c *x^4]) + (e^2*(((Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTanh[(Sqrt[c*d^4 + b*d^2*e^ 2 + a*e^4]*x)/(d*e*Sqrt[a + b*x^2 + c*x^4])])/(2*d*e*Sqrt[c*d^4 + b*d^2*e^ 2 + a*e^4]) + ((Sqrt[a]/d^2 - Sqrt[c]/e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[c]*d^2 + Sqr t[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(4*a^(1/4)*c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/ (Sqrt[c]*d^2 + Sqrt[a]*e^2))
3.1.3.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, 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[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2) Int[1 /Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2) I nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Sym bol] :> Simp[d Int[1/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] - Simp[e Int[x/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ[{ a, b, c, d, e}, x]
Time = 0.75 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.54
| method | result | size |
| default | \(\frac {-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \Pi \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}}{e}\) | \(281\) |
| elliptic | \(\frac {-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+\frac {b \,d^{2}}{e^{2}}+b \,x^{2}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+\frac {b \,d^{2}}{e^{2}}+a}}+\frac {\sqrt {2}\, e \sqrt {1-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \sqrt {1+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{2 a}}\, \Pi \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {2 a \,e^{2}}{\left (-b +\sqrt {-4 a c +b^{2}}\right ) d^{2}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, d \sqrt {c \,x^{4}+b \,x^{2}+a}}}{e}\) | \(281\) |
1/e*(-1/2/(c/e^4*d^4+b/e^2*d^2+a)^(1/2)*arctanh(1/2*(2*c*x^2/e^2*d^2+b/e^2 *d^2+b*x^2+2*a)/(c/e^4*d^4+b/e^2*d^2+a)^(1/2)/(c*x^4+b*x^2+a)^(1/2))+2^(1/ 2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*e/d*(1-1/2*(-b+(-4*a*c+b^2)^(1/2))/a* x^2)^(1/2)*(1+1/2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2 )*EllipticPi(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),2/(-b+(-4*a*c +b^2)^(1/2))*a*e^2/d^2,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+ (-4*a*c+b^2)^(1/2))/a)^(1/2)))
Timed out. \[ \int \frac {1}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {1}{\left (d + e x\right ) \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
\[ \int \frac {1}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x + d\right )}} \,d x } \]
\[ \int \frac {1}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} + a} {\left (e x + d\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x) \sqrt {a+b x^2+c x^4}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \]