Integrand size = 30, antiderivative size = 145 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{a d-c d x^4} \, dx=-\frac {\sqrt {b-2 \sqrt {a} \sqrt {c}} \text {arctanh}\left (\frac {\sqrt {b-2 \sqrt {a} \sqrt {c}} x}{\sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {a} \sqrt {c} d}+\frac {\sqrt {b+2 \sqrt {a} \sqrt {c}} \text {arctanh}\left (\frac {\sqrt {b+2 \sqrt {a} \sqrt {c}} x}{\sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {a} \sqrt {c} d} \] Output:
-1/4*(b-2*a^(1/2)*c^(1/2))^(1/2)*arctanh((b-2*a^(1/2)*c^(1/2))^(1/2)*x/(c* x^4+b*x^2+a)^(1/2))/a^(1/2)/c^(1/2)/d+1/4*(b+2*a^(1/2)*c^(1/2))^(1/2)*arct anh((b+2*a^(1/2)*c^(1/2))^(1/2)*x/(c*x^4+b*x^2+a)^(1/2))/a^(1/2)/c^(1/2)/d
Time = 0.88 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{a d-c d x^4} \, dx=\frac {-\sqrt {-b-2 \sqrt {a} \sqrt {c}} \arctan \left (\frac {\sqrt {-b-2 \sqrt {a} \sqrt {c}} x}{\sqrt {a+b x^2+c x^4}}\right )+\sqrt {-b+2 \sqrt {a} \sqrt {c}} \arctan \left (\frac {\sqrt {-b+2 \sqrt {a} \sqrt {c}} x}{\sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {a} \sqrt {c} d} \] Input:
Integrate[Sqrt[a + b*x^2 + c*x^4]/(a*d - c*d*x^4),x]
Output:
(-(Sqrt[-b - 2*Sqrt[a]*Sqrt[c]]*ArcTan[(Sqrt[-b - 2*Sqrt[a]*Sqrt[c]]*x)/Sq rt[a + b*x^2 + c*x^4]]) + Sqrt[-b + 2*Sqrt[a]*Sqrt[c]]*ArcTan[(Sqrt[-b + 2 *Sqrt[a]*Sqrt[c]]*x)/Sqrt[a + b*x^2 + c*x^4]])/(4*Sqrt[a]*Sqrt[c]*d)
Time = 0.59 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.32, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2517, 1406, 221}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2+c x^4}}{a d-c d x^4} \, dx\) |
| \(\Big \downarrow \) 2517 |
| \(\displaystyle \frac {\int \frac {1}{\frac {\left (b^2-4 a c\right ) x^4}{\left (c x^4+b x^2+a\right )^2}-\frac {2 b x^2}{c x^4+b x^2+a}+1}d\frac {x}{\sqrt {c x^4+b x^2+a}}}{d}\) |
| \(\Big \downarrow \) 1406 |
| \(\displaystyle \frac {\frac {\left (b^2-4 a c\right ) \int \frac {1}{\frac {\left (b^2-4 a c\right ) x^2}{c x^4+b x^2+a}-b-2 \sqrt {a} \sqrt {c}}d\frac {x}{\sqrt {c x^4+b x^2+a}}}{4 \sqrt {a} \sqrt {c}}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\frac {\left (b^2-4 a c\right ) x^2}{c x^4+b x^2+a}-b+2 \sqrt {a} \sqrt {c}}d\frac {x}{\sqrt {c x^4+b x^2+a}}}{4 \sqrt {a} \sqrt {c}}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {x \sqrt {2 \sqrt {a} \sqrt {c}+b}}{\sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {a} \sqrt {c} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt {2 \sqrt {a} \sqrt {c}+b}}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {x \sqrt {b-2 \sqrt {a} \sqrt {c}}}{\sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {a} \sqrt {c} \sqrt {b-2 \sqrt {a} \sqrt {c}} \left (2 \sqrt {a} \sqrt {c}+b\right )}}{d}\) |
Input:
Int[Sqrt[a + b*x^2 + c*x^4]/(a*d - c*d*x^4),x]
Output:
(-1/4*((b^2 - 4*a*c)*ArcTanh[(Sqrt[b - 2*Sqrt[a]*Sqrt[c]]*x)/Sqrt[a + b*x^ 2 + c*x^4]])/(Sqrt[a]*Sqrt[b - 2*Sqrt[a]*Sqrt[c]]*(b + 2*Sqrt[a]*Sqrt[c])* Sqrt[c]) + ((b^2 - 4*a*c)*ArcTanh[(Sqrt[b + 2*Sqrt[a]*Sqrt[c]]*x)/Sqrt[a + b*x^2 + c*x^4]])/(4*Sqrt[a]*(b - 2*Sqrt[a]*Sqrt[c])*Sqrt[b + 2*Sqrt[a]*Sq rt[c]]*Sqrt[c]))/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Int[Sqrt[v_]/((d_) + (e_.)*(x_)^4), x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coeff[v, x, 4]}, Simp[a/d Subst[Int[1/(1 - 2*b*x^ 2 + (b^2 - 4*a*c)*x^4), x], x, x/Sqrt[v]], x] /; EqQ[c*d + a*e, 0] && PosQ[ a*c]] /; FreeQ[{d, e}, x] && PolyQ[v, x^2, 2]
Time = 1.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.72
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {-2 \sqrt {a c}-b}\, \arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{x \sqrt {-2 \sqrt {a c}-b}}\right )-\sqrt {2 \sqrt {a c}-b}\, \arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{x \sqrt {2 \sqrt {a c}-b}}\right )}{4 \sqrt {a c}\, d}\) | \(105\) |
| default | \(\frac {\left (-\frac {\left (2 \sqrt {a c}-b \right ) \arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x \sqrt {4 \sqrt {a c}-2 b}}\right )}{2 \sqrt {a c}\, \sqrt {4 \sqrt {a c}-2 b}}-\frac {\left (2 \sqrt {a c}+b \right ) \arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x \sqrt {-4 \sqrt {a c}-2 b}}\right )}{2 \sqrt {a c}\, \sqrt {-4 \sqrt {a c}-2 b}}\right ) \sqrt {2}}{2 d}\) | \(140\) |
| elliptic | \(\frac {2 \left (-\frac {\left (2 \sqrt {a c}-b \right ) \arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x \sqrt {4 \sqrt {a c}-2 b}}\right )}{8 \sqrt {a c}\, \sqrt {4 \sqrt {a c}-2 b}}-\frac {\left (2 \sqrt {a c}+b \right ) \arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x \sqrt {-4 \sqrt {a c}-2 b}}\right )}{8 \sqrt {a c}\, \sqrt {-4 \sqrt {a c}-2 b}}\right ) \sqrt {2}}{d}\) | \(140\) |
Input:
int((c*x^4+b*x^2+a)^(1/2)/(-c*d*x^4+a*d),x,method=_RETURNVERBOSE)
Output:
1/4*((-2*(a*c)^(1/2)-b)^(1/2)*arctan(1/x*(c*x^4+b*x^2+a)^(1/2)/(-2*(a*c)^( 1/2)-b)^(1/2))-(2*(a*c)^(1/2)-b)^(1/2)*arctan(1/x*(c*x^4+b*x^2+a)^(1/2)/(2 *(a*c)^(1/2)-b)^(1/2)))/(a*c)^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (105) = 210\).
Time = 1.38 (sec) , antiderivative size = 603, normalized size of antiderivative = 4.16 \[ \int \frac {\sqrt {a+b x^2+c x^4}}{a d-c d x^4} \, dx=\frac {1}{8} \, \sqrt {\frac {2 \, a c d^{2} \sqrt {\frac {1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (a d^{2} \sqrt {\frac {1}{a c d^{4}}} + x^{2}\right )} + {\left (a c d^{3} x^{3} \sqrt {\frac {1}{a c d^{4}}} + a d x\right )} \sqrt {\frac {2 \, a c d^{2} \sqrt {\frac {1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} - a}\right ) - \frac {1}{8} \, \sqrt {\frac {2 \, a c d^{2} \sqrt {\frac {1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (a d^{2} \sqrt {\frac {1}{a c d^{4}}} + x^{2}\right )} - {\left (a c d^{3} x^{3} \sqrt {\frac {1}{a c d^{4}}} + a d x\right )} \sqrt {\frac {2 \, a c d^{2} \sqrt {\frac {1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} - a}\right ) + \frac {1}{8} \, \sqrt {-\frac {2 \, a c d^{2} \sqrt {\frac {1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (a d^{2} \sqrt {\frac {1}{a c d^{4}}} - x^{2}\right )} + {\left (a c d^{3} x^{3} \sqrt {\frac {1}{a c d^{4}}} - a d x\right )} \sqrt {-\frac {2 \, a c d^{2} \sqrt {\frac {1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} - a}\right ) - \frac {1}{8} \, \sqrt {-\frac {2 \, a c d^{2} \sqrt {\frac {1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (a d^{2} \sqrt {\frac {1}{a c d^{4}}} - x^{2}\right )} - {\left (a c d^{3} x^{3} \sqrt {\frac {1}{a c d^{4}}} - a d x\right )} \sqrt {-\frac {2 \, a c d^{2} \sqrt {\frac {1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} - a}\right ) \] Input:
integrate((c*x^4+b*x^2+a)^(1/2)/(-c*d*x^4+a*d),x, algorithm="fricas")
Output:
1/8*sqrt((2*a*c*d^2*sqrt(1/(a*c*d^4)) + b)/(a*c*d^2))*log((sqrt(c*x^4 + b* x^2 + a)*(a*d^2*sqrt(1/(a*c*d^4)) + x^2) + (a*c*d^3*x^3*sqrt(1/(a*c*d^4)) + a*d*x)*sqrt((2*a*c*d^2*sqrt(1/(a*c*d^4)) + b)/(a*c*d^2)))/(c*x^4 - a)) - 1/8*sqrt((2*a*c*d^2*sqrt(1/(a*c*d^4)) + b)/(a*c*d^2))*log((sqrt(c*x^4 + b *x^2 + a)*(a*d^2*sqrt(1/(a*c*d^4)) + x^2) - (a*c*d^3*x^3*sqrt(1/(a*c*d^4)) + a*d*x)*sqrt((2*a*c*d^2*sqrt(1/(a*c*d^4)) + b)/(a*c*d^2)))/(c*x^4 - a)) + 1/8*sqrt(-(2*a*c*d^2*sqrt(1/(a*c*d^4)) - b)/(a*c*d^2))*log(-(sqrt(c*x^4 + b*x^2 + a)*(a*d^2*sqrt(1/(a*c*d^4)) - x^2) + (a*c*d^3*x^3*sqrt(1/(a*c*d^ 4)) - a*d*x)*sqrt(-(2*a*c*d^2*sqrt(1/(a*c*d^4)) - b)/(a*c*d^2)))/(c*x^4 - a)) - 1/8*sqrt(-(2*a*c*d^2*sqrt(1/(a*c*d^4)) - b)/(a*c*d^2))*log(-(sqrt(c* x^4 + b*x^2 + a)*(a*d^2*sqrt(1/(a*c*d^4)) - x^2) - (a*c*d^3*x^3*sqrt(1/(a* c*d^4)) - a*d*x)*sqrt(-(2*a*c*d^2*sqrt(1/(a*c*d^4)) - b)/(a*c*d^2)))/(c*x^ 4 - a))
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{a d-c d x^4} \, dx=- \frac {\int \frac {\sqrt {a + b x^{2} + c x^{4}}}{- a + c x^{4}}\, dx}{d} \] Input:
integrate((c*x**4+b*x**2+a)**(1/2)/(-c*d*x**4+a*d),x)
Output:
-Integral(sqrt(a + b*x**2 + c*x**4)/(-a + c*x**4), x)/d
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{a d-c d x^4} \, dx=\int { -\frac {\sqrt {c x^{4} + b x^{2} + a}}{c d x^{4} - a d} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(1/2)/(-c*d*x^4+a*d),x, algorithm="maxima")
Output:
-integrate(sqrt(c*x^4 + b*x^2 + a)/(c*d*x^4 - a*d), x)
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{a d-c d x^4} \, dx=\int { -\frac {\sqrt {c x^{4} + b x^{2} + a}}{c d x^{4} - a d} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(1/2)/(-c*d*x^4+a*d),x, algorithm="giac")
Output:
integrate(-sqrt(c*x^4 + b*x^2 + a)/(c*d*x^4 - a*d), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2+c x^4}}{a d-c d x^4} \, dx=\int \frac {\sqrt {c\,x^4+b\,x^2+a}}{a\,d-c\,d\,x^4} \,d x \] Input:
int((a + b*x^2 + c*x^4)^(1/2)/(a*d - c*d*x^4),x)
Output:
int((a + b*x^2 + c*x^4)^(1/2)/(a*d - c*d*x^4), x)
\[ \int \frac {\sqrt {a+b x^2+c x^4}}{a d-c d x^4} \, dx=\frac {\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{-c \,x^{4}+a}d x}{d} \] Input:
int((c*x^4+b*x^2+a)^(1/2)/(-c*d*x^4+a*d),x)
Output:
int(sqrt(a + b*x**2 + c*x**4)/(a - c*x**4),x)/d