Integrand size = 31, antiderivative size = 57 \[ \int \frac {a g+e x-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {g x}{\sqrt {a+b x^2+c x^4}}-\frac {e \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \] Output:
g*x/(c*x^4+b*x^2+a)^(1/2)-e*(2*c*x^2+b)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)
Time = 10.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {a g+e x-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {-b e+b^2 g x-4 a c g x-2 c e x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[(a*g + e*x - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(-(b*e) + b^2*g*x - 4*a*c*g*x - 2*c*e*x^2)/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4])
Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2202, 27, 1432, 1088, 2021}
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.
\(\displaystyle \int \frac {a g-c g x^4+e x}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2202 |
\(\displaystyle \int \frac {e x}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e \int \frac {x}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 1432 |
\(\displaystyle \frac {1}{2} e \int \frac {1}{\left (c x^4+b x^2+a\right )^{3/2}}dx^2+\int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 1088 |
\(\displaystyle \int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx-\frac {e \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
\(\Big \downarrow \) 2021 |
\(\displaystyle \frac {g x}{\sqrt {a+b x^2+c x^4}}-\frac {e \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
Input:
Int[(a*g + e*x - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(g*x)/Sqrt[a + b*x^2 + c*x^4] - (e*(b + 2*c*x^2))/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4])
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_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x ]}, Simp[Coeff[Pp, x, p]*x^(p - q + 1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x, q]*Pp , Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; Free Q[m, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && NeQ[m, -1]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Time = 1.97 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91
method | result | size |
gosper | \(\frac {4 a c g x -b^{2} g x +2 c e \,x^{2}+b e}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(52\) |
trager | \(\frac {4 a c g x -b^{2} g x +2 c e \,x^{2}+b e}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(52\) |
orering | \(\frac {4 a c g x -b^{2} g x +2 c e \,x^{2}+b e}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(52\) |
elliptic | \(\frac {e \left (2 c \,x^{2}+b \right )}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {g x}{\sqrt {c \,x^{4}+b \,x^{2}+a}}\) | \(55\) |
default | \(a g \left (-\frac {2 c \left (\frac {b \,x^{3}}{2 a \left (4 a c -b^{2}\right )}-\frac {\left (2 a c -b^{2}\right ) x}{2 a \left (4 a c -b^{2}\right ) c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (\frac {1}{a}-\frac {2 a c -b^{2}}{a \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )+\frac {e \left (2 c \,x^{2}+b \right )}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}-c g \left (-\frac {2 c \left (\frac {b \,x^{3}}{2 \left (4 a c -b^{2}\right ) c}+\frac {a x}{c \left (4 a c -b^{2}\right )}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )\) | \(974\) |
Input:
int((-c*g*x^4+a*g+e*x)/(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
(4*a*c*g*x-b^2*g*x+2*c*e*x^2+b*e)/(c*x^4+b*x^2+a)^(1/2)/(4*a*c-b^2)
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.44 \[ \int \frac {a g+e x-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c e x^{2} - {\left (b^{2} - 4 \, a c\right )} g x + b e\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}} \] Input:
integrate((-c*g*x^4+a*g+e*x)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-sqrt(c*x^4 + b*x^2 + a)*(2*c*e*x^2 - (b^2 - 4*a*c)*g*x + b*e)/((b^2*c - 4 *a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)
\[ \int \frac {a g+e x-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=- \int \left (- \frac {a g}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \left (- \frac {e x}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \frac {c g x^{4}}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\, dx \] Input:
integrate((-c*g*x**4+a*g+e*x)/(c*x**4+b*x**2+a)**(3/2),x)
Output:
-Integral(-a*g/(a*sqrt(a + b*x**2 + c*x**4) + b*x**2*sqrt(a + b*x**2 + c*x **4) + c*x**4*sqrt(a + b*x**2 + c*x**4)), x) - Integral(-e*x/(a*sqrt(a + b *x**2 + c*x**4) + b*x**2*sqrt(a + b*x**2 + c*x**4) + c*x**4*sqrt(a + b*x** 2 + c*x**4)), x) - Integral(c*g*x**4/(a*sqrt(a + b*x**2 + c*x**4) + b*x**2 *sqrt(a + b*x**2 + c*x**4) + c*x**4*sqrt(a + b*x**2 + c*x**4)), x)
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {a g+e x-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {2 \, c e x^{2} + b e - {\left (b^{2} g - 4 \, a c g\right )} x}{\sqrt {c x^{4} + b x^{2} + a} {\left (b^{2} - 4 \, a c\right )}} \] Input:
integrate((-c*g*x^4+a*g+e*x)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
-(2*c*e*x^2 + b*e - (b^2*g - 4*a*c*g)*x)/(sqrt(c*x^4 + b*x^2 + a)*(b^2 - 4 *a*c))
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (53) = 106\).
Time = 0.36 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.42 \[ \int \frac {a g+e x-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {{\left (\frac {2 \, {\left (b^{2} c e - 4 \, a c^{2} e\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} - \frac {b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {b^{3} e - 4 \, a b c e}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}}{\sqrt {c x^{4} + b x^{2} + a}} \] Input:
integrate((-c*g*x^4+a*g+e*x)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
-((2*(b^2*c*e - 4*a*c^2*e)*x/(b^4 - 8*a*b^2*c + 16*a^2*c^2) - (b^4*g - 8*a *b^2*c*g + 16*a^2*c^2*g)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x + (b^3*e - 4*a* b*c*e)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))/sqrt(c*x^4 + b*x^2 + a)
Time = 18.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {a g+e x-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {-g\,b^2\,x+e\,b+2\,c\,e\,x^2+4\,a\,c\,g\,x}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^4+b\,x^2+a}} \] Input:
int((a*g + e*x - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x)
Output:
(b*e + 2*c*e*x^2 - b^2*g*x + 4*a*c*g*x)/((4*a*c - b^2)*(a + b*x^2 + c*x^4) ^(1/2))
Time = 0.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.53 \[ \int \frac {a g+e x-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c g x -b^{2} g x +2 c e \,x^{2}+b e \right )}{4 a \,c^{2} x^{4}-b^{2} c \,x^{4}+4 a b c \,x^{2}-b^{3} x^{2}+4 a^{2} c -a \,b^{2}} \] Input:
int((-c*g*x^4+a*g+e*x)/(c*x^4+b*x^2+a)^(3/2),x)
Output:
(sqrt(a + b*x**2 + c*x**4)*(4*a*c*g*x - b**2*g*x + b*e + 2*c*e*x**2))/(4*a **2*c - a*b**2 + 4*a*b*c*x**2 + 4*a*c**2*x**4 - b**3*x**2 - b**2*c*x**4)