Integrand size = 33, antiderivative size = 57 \[ \int \frac {a g+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {g x}{\sqrt {a+b x^2+c x^4}}+\frac {f \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
Time = 10.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84 \[ \int \frac {a g+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {b x (b g+f x)+2 a (f-2 c g x)}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2202, 27, 1434, 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.
\(\displaystyle \int \frac {a g-c g x^4+f x^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2202 |
\(\displaystyle \int \frac {f x^3}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle f \int \frac {x^3}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} f \int \frac {x^2}{\left (c x^4+b x^2+a\right )^{3/2}}dx^2+\int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx\) |
\(\Big \downarrow \) 1158 |
\(\displaystyle \int \frac {a g-c g x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\frac {f \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\) |
\(\Big \downarrow \) 2021 |
\(\displaystyle \frac {f \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {g x}{\sqrt {a+b x^2+c x^4}}\) |
3.2.10.3.1 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[((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_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
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 = 1.60 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(\frac {4 a c g x -b^{2} g x -b f \,x^{2}-2 a f}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(53\) |
trager | \(\frac {4 a c g x -b^{2} g x -b f \,x^{2}-2 a f}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(53\) |
elliptic | \(-\frac {f \left (b \,x^{2}+2 a \right )}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {g x}{\sqrt {c \,x^{4}+b \,x^{2}+a}}\) | \(57\) |
default | \(-\frac {f \left (b \,x^{2}+2 a \right )}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+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}}\, F\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 (F\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 )-E\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 )-g c \left (-\frac {2 c \left (\frac {b \,x^{3}}{2 c \left (4 a c -b^{2}\right )}+\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}}\, F\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 (F\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 )-E\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 )\) | \(976\) |
Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.40 \[ \int \frac {a g+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (b f x^{2} + {\left (b^{2} - 4 \, a c\right )} g x + 2 \, a f\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}} \]
sqrt(c*x^4 + b*x^2 + a)*(b*f*x^2 + (b^2 - 4*a*c)*g*x + 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)
\[ \int \frac {a g+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=- \int \left (- \frac {a g}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \left (- \frac {f x^{3}}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \frac {c g x^{4}}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\, dx \]
-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(-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) - 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 = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {a g+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {b f x^{2} + 2 \, a f + {\left (b^{2} g - 4 \, a c g\right )} x}{\sqrt {c x^{4} + b x^{2} + a} {\left (b^{2} - 4 \, a c\right )}} \]
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (53) = 106\).
Time = 0.62 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.39 \[ \int \frac {a g+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {{\left (\frac {{\left (b^{3} f - 4 \, a b c f\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {2 \, {\left (a b^{2} f - 4 \, a^{2} c f\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}}{\sqrt {c x^{4} + b x^{2} + a}} \]
(((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 + 2*(a*b^2*f - 4*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 = 7.94 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {a g+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {g\,b^2\,x+f\,b\,x^2-4\,a\,c\,g\,x+2\,a\,f}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^4+b\,x^2+a}} \]