Integrand size = 33, antiderivative size = 109 \[ \int \frac {x \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\left (c d^4+b d^2 e^2+a e^4\right ) \sqrt {d-e x} \sqrt {d+e x}}{e^6}+\frac {\left (2 c d^2+b e^2\right ) (d-e x)^{3/2} (d+e x)^{3/2}}{3 e^6}-\frac {c (d-e x)^{5/2} (d+e x)^{5/2}}{5 e^6} \] Output:
-(a*e^4+b*d^2*e^2+c*d^4)*(-e*x+d)^(1/2)*(e*x+d)^(1/2)/e^6+1/3*(b*e^2+2*c*d ^2)*(-e*x+d)^(3/2)*(e*x+d)^(3/2)/e^6-1/5*c*(-e*x+d)^(5/2)*(e*x+d)^(5/2)/e^ 6
Time = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.73 \[ \int \frac {x \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (5 \left (2 b d^2 e^2+3 a e^4+b e^4 x^2\right )+c \left (8 d^4+4 d^2 e^2 x^2+3 e^4 x^4\right )\right )}{15 e^6} \] Input:
Integrate[(x*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]
Output:
-1/15*(Sqrt[d - e*x]*Sqrt[d + e*x]*(5*(2*b*d^2*e^2 + 3*a*e^4 + b*e^4*x^2) + c*(8*d^4 + 4*d^2*e^2*x^2 + 3*e^4*x^4)))/e^6
Time = 0.33 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.28, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1905, 1576, 1140, 2009}
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.
\(\displaystyle \int \frac {x \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx\) |
\(\Big \downarrow \) 1905 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \int \frac {x \left (c x^4+b x^2+a\right )}{\sqrt {d^2-e^2 x^2}}dx}{\sqrt {d-e x} \sqrt {d+e x}}\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \int \frac {c x^4+b x^2+a}{\sqrt {d^2-e^2 x^2}}dx^2}{2 \sqrt {d-e x} \sqrt {d+e x}}\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \int \left (\frac {c \left (d^2-e^2 x^2\right )^{3/2}}{e^4}+\frac {\left (-2 c d^2-b e^2\right ) \sqrt {d^2-e^2 x^2}}{e^4}+\frac {c d^4+b e^2 d^2+a e^4}{e^4 \sqrt {d^2-e^2 x^2}}\right )dx^2}{2 \sqrt {d-e x} \sqrt {d+e x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^2} \left (-\frac {2 \sqrt {d^2-e^2 x^2} \left (a e^4+b d^2 e^2+c d^4\right )}{e^6}+\frac {2 \left (d^2-e^2 x^2\right )^{3/2} \left (b e^2+2 c d^2\right )}{3 e^6}-\frac {2 c \left (d^2-e^2 x^2\right )^{5/2}}{5 e^6}\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}\) |
Input:
Int[(x*(a + b*x^2 + c*x^4))/(Sqrt[d - e*x]*Sqrt[d + e*x]),x]
Output:
(Sqrt[d^2 - e^2*x^2]*((-2*(c*d^4 + b*d^2*e^2 + a*e^4)*Sqrt[d^2 - e^2*x^2]) /e^6 + (2*(2*c*d^2 + b*e^2)*(d^2 - e^2*x^2)^(3/2))/(3*e^6) - (2*c*(d^2 - e ^2*x^2)^(5/2))/(5*e^6)))/(2*Sqrt[d - e*x]*Sqrt[d + e*x])
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
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[((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_)^(non2_.))^(q_.)*((d2_) + (e2_.) *(x_)^(non2_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_.), x _Symbol] :> Simp[(d1 + e1*x^(n/2))^FracPart[q]*((d2 + e2*x^(n/2))^FracPart[ q]/(d1*d2 + e1*e2*x^n)^FracPart[q]) Int[(f*x)^m*(d1*d2 + e1*e2*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[non2, n/2] && EqQ[d2*e1 + d1*e2, 0]
Time = 0.79 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.67
method | result | size |
gosper | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (3 c \,x^{4} e^{4}+5 b \,e^{4} x^{2}+4 c \,d^{2} e^{2} x^{2}+15 a \,e^{4}+10 b \,d^{2} e^{2}+8 c \,d^{4}\right )}{15 e^{6}}\) | \(73\) |
default | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (3 c \,x^{4} e^{4}+5 b \,e^{4} x^{2}+4 c \,d^{2} e^{2} x^{2}+15 a \,e^{4}+10 b \,d^{2} e^{2}+8 c \,d^{4}\right )}{15 e^{6}}\) | \(73\) |
risch | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (3 c \,x^{4} e^{4}+5 b \,e^{4} x^{2}+4 c \,d^{2} e^{2} x^{2}+15 a \,e^{4}+10 b \,d^{2} e^{2}+8 c \,d^{4}\right )}{15 e^{6}}\) | \(73\) |
orering | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (3 c \,x^{4} e^{4}+5 b \,e^{4} x^{2}+4 c \,d^{2} e^{2} x^{2}+15 a \,e^{4}+10 b \,d^{2} e^{2}+8 c \,d^{4}\right )}{15 e^{6}}\) | \(73\) |
Input:
int(x*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE )
Output:
-1/15*(e*x+d)^(1/2)*(-e*x+d)^(1/2)*(3*c*e^4*x^4+5*b*e^4*x^2+4*c*d^2*e^2*x^ 2+15*a*e^4+10*b*d^2*e^2+8*c*d^4)/e^6
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.65 \[ \int \frac {x \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (3 \, c e^{4} x^{4} + 8 \, c d^{4} + 10 \, b d^{2} e^{2} + 15 \, a e^{4} + {\left (4 \, c d^{2} e^{2} + 5 \, b e^{4}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{15 \, e^{6}} \] Input:
integrate(x*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fri cas")
Output:
-1/15*(3*c*e^4*x^4 + 8*c*d^4 + 10*b*d^2*e^2 + 15*a*e^4 + (4*c*d^2*e^2 + 5* b*e^4)*x^2)*sqrt(e*x + d)*sqrt(-e*x + d)/e^6
Result contains complex when optimal does not.
Time = 10.71 (sec) , antiderivative size = 350, normalized size of antiderivative = 3.21 \[ \int \frac {x \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=- \frac {i a d {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{2}} - \frac {a d {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{2}} - \frac {i b d^{3} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4} & -1, -1, - \frac {1}{2}, 1 \\- \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{4}} - \frac {b d^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, 1 & \\- \frac {7}{4}, - \frac {5}{4} & -2, - \frac {3}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{4}} - \frac {i c d^{5} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {9}{4}, - \frac {7}{4} & -2, -2, - \frac {3}{2}, 1 \\- \frac {5}{2}, - \frac {9}{4}, -2, - \frac {7}{4}, - \frac {3}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{6}} - \frac {c d^{5} {G_{6, 6}^{2, 6}\left (\begin {matrix} -3, - \frac {11}{4}, - \frac {5}{2}, - \frac {9}{4}, -2, 1 & \\- \frac {11}{4}, - \frac {9}{4} & -3, - \frac {5}{2}, - \frac {5}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{6}} \] Input:
integrate(x*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)
Output:
-I*a*d*meijerg(((-1/4, 1/4), (0, 0, 1/2, 1)), ((-1/2, -1/4, 0, 1/4, 1/2, 0 ), ()), d**2/(e**2*x**2))/(4*pi**(3/2)*e**2) - a*d*meijerg(((-1, -3/4, -1/ 2, -1/4, 0, 1), ()), ((-3/4, -1/4), (-1, -1/2, -1/2, 0)), d**2*exp_polar(- 2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e**2) - I*b*d**3*meijerg(((-5/4, -3/4), (-1, -1, -1/2, 1)), ((-3/2, -5/4, -1, -3/4, -1/2, 0), ()), d**2/(e**2*x**2 ))/(4*pi**(3/2)*e**4) - b*d**3*meijerg(((-2, -7/4, -3/2, -5/4, -1, 1), ()) , ((-7/4, -5/4), (-2, -3/2, -3/2, 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2) )/(4*pi**(3/2)*e**4) - I*c*d**5*meijerg(((-9/4, -7/4), (-2, -2, -3/2, 1)), ((-5/2, -9/4, -2, -7/4, -3/2, 0), ()), d**2/(e**2*x**2))/(4*pi**(3/2)*e** 6) - c*d**5*meijerg(((-3, -11/4, -5/2, -9/4, -2, 1), ()), ((-11/4, -9/4), (-3, -5/2, -5/2, 0)), d**2*exp_polar(-2*I*pi)/(e**2*x**2))/(4*pi**(3/2)*e* *6)
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.28 \[ \int \frac {x \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{4}}{5 \, e^{2}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2} x^{2}}{15 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b x^{2}}{3 \, e^{2}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{4}}{15 \, e^{6}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} b d^{2}}{3 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{e^{2}} \] Input:
integrate(x*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="max ima")
Output:
-1/5*sqrt(-e^2*x^2 + d^2)*c*x^4/e^2 - 4/15*sqrt(-e^2*x^2 + d^2)*c*d^2*x^2/ e^4 - 1/3*sqrt(-e^2*x^2 + d^2)*b*x^2/e^2 - 8/15*sqrt(-e^2*x^2 + d^2)*c*d^4 /e^6 - 2/3*sqrt(-e^2*x^2 + d^2)*b*d^2/e^4 - sqrt(-e^2*x^2 + d^2)*a/e^2
Time = 0.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {{\left (15 \, c d^{4} + 15 \, b d^{2} e^{2} + 15 \, a e^{4} - {\left (20 \, c d^{3} + 10 \, b d e^{2} - {\left (22 \, c d^{2} + 5 \, b e^{2} + 3 \, {\left ({\left (e x + d\right )} c - 4 \, c d\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} {\left (e x + d\right )}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{15 \, e^{6}} \] Input:
integrate(x*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="gia c")
Output:
-1/15*(15*c*d^4 + 15*b*d^2*e^2 + 15*a*e^4 - (20*c*d^3 + 10*b*d*e^2 - (22*c *d^2 + 5*b*e^2 + 3*((e*x + d)*c - 4*c*d)*(e*x + d))*(e*x + d))*(e*x + d))* sqrt(e*x + d)*sqrt(-e*x + d)/e^6
Time = 3.92 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.31 \[ \int \frac {x \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=-\frac {\sqrt {d-e\,x}\,\left (\frac {8\,c\,d^5+10\,b\,d^3\,e^2+15\,a\,d\,e^4}{15\,e^6}+\frac {x^3\,\left (4\,c\,d^2\,e^3+5\,b\,e^5\right )}{15\,e^6}+\frac {c\,x^5}{5\,e}+\frac {x^2\,\left (4\,c\,d^3\,e^2+5\,b\,d\,e^4\right )}{15\,e^6}+\frac {x\,\left (8\,c\,d^4\,e+10\,b\,d^2\,e^3+15\,a\,e^5\right )}{15\,e^6}+\frac {c\,d\,x^4}{5\,e^2}\right )}{\sqrt {d+e\,x}} \] Input:
int((x*(a + b*x^2 + c*x^4))/((d + e*x)^(1/2)*(d - e*x)^(1/2)),x)
Output:
-((d - e*x)^(1/2)*((8*c*d^5 + 10*b*d^3*e^2 + 15*a*d*e^4)/(15*e^6) + (x^3*( 5*b*e^5 + 4*c*d^2*e^3))/(15*e^6) + (c*x^5)/(5*e) + (x^2*(4*c*d^3*e^2 + 5*b *d*e^4))/(15*e^6) + (x*(15*a*e^5 + 10*b*d^2*e^3 + 8*c*d^4*e))/(15*e^6) + ( c*d*x^4)/(5*e^2)))/(d + e*x)^(1/2)
Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64 \[ \int \frac {x \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (-3 c \,e^{4} x^{4}-5 b \,e^{4} x^{2}-4 c \,d^{2} e^{2} x^{2}-15 a \,e^{4}-10 b \,d^{2} e^{2}-8 c \,d^{4}\right )}{15 e^{6}} \] Input:
int(x*(c*x^4+b*x^2+a)/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)
Output:
(sqrt(d + e*x)*sqrt(d - e*x)*( - 15*a*e**4 - 10*b*d**2*e**2 - 5*b*e**4*x** 2 - 8*c*d**4 - 4*c*d**2*e**2*x**2 - 3*c*e**4*x**4))/(15*e**6)