Integrand size = 36, antiderivative size = 69 \[ \int \frac {a g+e x+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {g x}{\sqrt {a+b x^2+c x^4}}-\frac {b e-2 a f+(2 c e-b f) x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \] Output:
g*x/(c*x^4+b*x^2+a)^(1/2)-(b*e-2*a*f+(-b*f+2*c*e)*x^2)/(-4*a*c+b^2)/(c*x^4 +b*x^2+a)^(1/2)
Time = 10.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {a g+e x+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {-b e+2 a f+b^2 g x-4 a c g x-2 c e x^2+b f x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[(a*g + e*x + f*x^3 - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(-(b*e) + 2*a*f + b^2*g*x - 4*a*c*g*x - 2*c*e*x^2 + b*f*x^2)/((b^2 - 4*a*c )*Sqrt[a + b*x^2 + c*x^4])
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2202, 1576, 1158, 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.
\(\displaystyle \int \frac {a g-c g x^4+e x+f x^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2202 |
\(\displaystyle \int \frac {x \left (f x^2+e\right )}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle \frac {1}{2} \int \frac {f x^2+e}{\left (c x^4+b x^2+a\right )^{3/2}}dx^2+\int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 1158 |
\(\displaystyle \int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx-\frac {-2 a f+x^2 (2 c e-b f)+b e}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
\(\Big \downarrow \) 2021 |
\(\displaystyle \frac {g x}{\sqrt {a+b x^2+c x^4}}-\frac {-2 a f+x^2 (2 c e-b f)+b e}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
Input:
Int[(a*g + e*x + f*x^3 - 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] - (b*e - 2*a*f + (2*c*e - b*f)*x^2)/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4])
Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
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[(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 = 2.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91
method | result | size |
gosper | \(\frac {4 a c g x -b^{2} g x -b f \,x^{2}+2 c e \,x^{2}-2 a f +b e}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(63\) |
trager | \(\frac {4 a c g x -b^{2} g x -b f \,x^{2}+2 c e \,x^{2}-2 a f +b e}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(63\) |
orering | \(\frac {4 a c g x -b^{2} g x -b f \,x^{2}+2 c e \,x^{2}-2 a f +b e}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(63\) |
elliptic | \(-\frac {b f \,x^{2}-2 c e \,x^{2}+2 a f -b e}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {g x}{\sqrt {c \,x^{4}+b \,x^{2}+a}}\) | \(69\) |
default | \(\text {Expression too large to display}\) | \(1012\) |
Input:
int((-c*g*x^4+f*x^3+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-b*f*x^2+2*c*e*x^2-2*a*f+b*e)/(c*x^4+b*x^2+a)^(1/2)/(4*a *c-b^2)
Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.33 \[ \int \frac {a g+e x+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {c x^{4} + b x^{2} + a} {\left ({\left (b^{2} - 4 \, a c\right )} g x - {\left (2 \, c e - b f\right )} x^{2} - b e + 2 \, a f\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}} \] Input:
integrate((-c*g*x^4+f*x^3+a*g+e*x)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fri cas")
Output:
sqrt(c*x^4 + b*x^2 + a)*((b^2 - 4*a*c)*g*x - (2*c*e - b*f)*x^2 - b*e + 2*a *f)/((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)
\[ \int \frac {a g+e x+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=- \int \left (- \frac {a g}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \left (- \frac {e x}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \left (- \frac {f x^{3}}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \frac {c g x^{4}}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\, dx \] Input:
integrate((-c*g*x**4+f*x**3+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(-f*x**3/(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) - Integr al(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 = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.36 \[ \int \frac {a g+e x+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {\sqrt {c x^{4} + b x^{2} + a} {\left ({\left (2 \, c e - b f\right )} x^{2} + b e - 2 \, a f - {\left (b^{2} g - 4 \, a c g\right )} x\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}} \] Input:
integrate((-c*g*x^4+f*x^3+a*g+e*x)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="max ima")
Output:
-sqrt(c*x^4 + b*x^2 + a)*((2*c*e - b*f)*x^2 + b*e - 2*a*f - (b^2*g - 4*a*c *g)*x)/((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (65) = 130\).
Time = 0.38 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.38 \[ \int \frac {a g+e x+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {{\left (\frac {{\left (2 \, b^{2} c e - 8 \, a c^{2} e - b^{3} f + 4 \, a b c f\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} - \frac {b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {b^{3} e - 4 \, a b c e - 2 \, a b^{2} f + 8 \, a^{2} c f}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}}{\sqrt {c x^{4} + b x^{2} + a}} \] Input:
integrate((-c*g*x^4+f*x^3+a*g+e*x)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="gia c")
Output:
-(((2*b^2*c*e - 8*a*c^2*e - b^3*f + 4*a*b*c*f)*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 - 2*a*b^2*f + 8*a^2*c*f)/(b^4 - 8*a*b^2*c + 16*a ^2*c^2))/sqrt(c*x^4 + b*x^2 + a)
Time = 18.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {a g+e x+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {g\,b^2\,x+f\,b\,x^2-e\,b-2\,c\,e\,x^2-4\,a\,c\,g\,x+2\,a\,f}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^4+b\,x^2+a}} \] Input:
int((a*g + e*x + f*x^3 - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x)
Output:
-(2*a*f - b*e + b*f*x^2 - 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 = 98, normalized size of antiderivative = 1.42 \[ \int \frac {a g+e x+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c g x -b^{2} g x -b f \,x^{2}+2 c e \,x^{2}-2 a f +b e \right )}{4 a \,c^{2} x^{4}-b^{2} c \,x^{4}+4 a b c \,x^{2}-b^{3} x^{2}+4 a^{2} c -a \,b^{2}} \] Input:
int((-c*g*x^4+f*x^3+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 - 2*a*f - b**2*g*x + b*e - b*f*x**2 + 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)