Integrand size = 42, antiderivative size = 60 \[ \int \frac {\left (a e+c d x^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\frac {\left (c d^2-b d e+a e^2\right ) x^3}{3 \left (a+b x^2+c x^4\right )^{3/2}}+\frac {d e x}{\sqrt {a+b x^2+c x^4}} \] Output:
1/3*(a*e^2-b*d*e+c*d^2)*x^3/(c*x^4+b*x^2+a)^(3/2)+d*e*x/(c*x^4+b*x^2+a)^(1 /2)
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a e+c d x^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\frac {a e x \left (3 d+e x^2\right )+d x^3 \left (2 b e+c \left (d+3 e x^2\right )\right )}{3 \left (a+b x^2+c x^4\right )^{3/2}} \] Input:
Integrate[((a*e + c*d*x^2)*(d + e*x^2)*(a - c*x^4))/(a + b*x^2 + c*x^4)^(5 /2),x]
Output:
(a*e*x*(3*d + e*x^2) + d*x^3*(2*b*e + c*(d + 3*e*x^2)))/(3*(a + b*x^2 + c* x^4)^(3/2))
Time = 0.62 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2206, 27, 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.
\(\displaystyle \int \frac {\left (a-c x^4\right ) \left (d+e x^2\right ) \left (a e+c d x^2\right )}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 2206 |
\(\displaystyle \frac {x^3 \left (a e^2-b d e+c d^2\right )}{3 \left (a+b x^2+c x^4\right )^{3/2}}-\frac {\int -\frac {3 a \left (b^2-4 a c\right ) d e \left (a-c x^4\right )}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{3 a \left (b^2-4 a c\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle d e \int \frac {a-c x^4}{\left (c x^4+b x^2+a\right )^{3/2}}dx+\frac {x^3 \left (a e^2-b d e+c d^2\right )}{3 \left (a+b x^2+c x^4\right )^{3/2}}\) |
\(\Big \downarrow \) 2021 |
\(\displaystyle \frac {x^3 \left (a e^2-b d e+c d^2\right )}{3 \left (a+b x^2+c x^4\right )^{3/2}}+\frac {d e x}{\sqrt {a+b x^2+c x^4}}\) |
Input:
Int[((a*e + c*d*x^2)*(d + e*x^2)*(a - c*x^4))/(a + b*x^2 + c*x^4)^(5/2),x]
Output:
((c*d^2 - b*d*e + a*e^2)*x^3)/(3*(a + b*x^2 + c*x^4)^(3/2)) + (d*e*x)/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[(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[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Time = 0.65 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {x \left (c d e \,x^{4}+\frac {\left (a \,e^{2}+2 b d e +c \,d^{2}\right ) x^{2}}{3}+a d e \right )}{\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(50\) |
pseudoelliptic | \(\frac {x \left (c d e \,x^{4}+\frac {\left (a \,e^{2}+2 b d e +c \,d^{2}\right ) x^{2}}{3}+a d e \right )}{\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(50\) |
gosper | \(\frac {x \left (3 c d e \,x^{4}+a \,e^{2} x^{2}+2 b d e \,x^{2}+c \,d^{2} x^{2}+3 a d e \right )}{3 \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(56\) |
trager | \(\frac {x \left (3 c d e \,x^{4}+a \,e^{2} x^{2}+2 b d e \,x^{2}+c \,d^{2} x^{2}+3 a d e \right )}{3 \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(56\) |
orering | \(\frac {x \left (3 c d e \,x^{4}+a \,e^{2} x^{2}+2 b d e \,x^{2}+c \,d^{2} x^{2}+3 a d e \right )}{3 \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(56\) |
elliptic | \(\frac {\left (\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {2}\, x^{3}}{3 \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {d e \sqrt {2}\, x}{\sqrt {c \,x^{4}+b \,x^{2}+a}}\right ) \sqrt {2}}{2}\) | \(66\) |
Input:
int((c*d*x^2+a*e)*(e*x^2+d)*(-c*x^4+a)/(c*x^4+b*x^2+a)^(5/2),x,method=_RET URNVERBOSE)
Output:
1/(c*x^4+b*x^2+a)^(3/2)*x*(c*d*e*x^4+1/3*(a*e^2+2*b*d*e+c*d^2)*x^2+a*d*e)
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a e+c d x^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\frac {{\left (3 \, c d e x^{5} + 3 \, a d e x + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{3 \, {\left (c^{2} x^{8} + 2 \, b c x^{6} + {\left (b^{2} + 2 \, a c\right )} x^{4} + 2 \, a b x^{2} + a^{2}\right )}} \] Input:
integrate((c*d*x^2+a*e)*(e*x^2+d)*(-c*x^4+a)/(c*x^4+b*x^2+a)^(5/2),x, algo rithm="fricas")
Output:
1/3*(3*c*d*e*x^5 + 3*a*d*e*x + (c*d^2 + 2*b*d*e + a*e^2)*x^3)*sqrt(c*x^4 + b*x^2 + a)/(c^2*x^8 + 2*b*c*x^6 + (b^2 + 2*a*c)*x^4 + 2*a*b*x^2 + a^2)
Timed out. \[ \int \frac {\left (a e+c d x^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((c*d*x**2+a*e)*(e*x**2+d)*(-c*x**4+a)/(c*x**4+b*x**2+a)**(5/2),x )
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a e+c d x^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\frac {{\left (3 \, c d e x^{5} + 3 \, a d e x + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{3}\right )} \sqrt {c x^{4} + b x^{2} + a}}{3 \, {\left (c^{2} x^{8} + 2 \, b c x^{6} + {\left (b^{2} + 2 \, a c\right )} x^{4} + 2 \, a b x^{2} + a^{2}\right )}} \] Input:
integrate((c*d*x^2+a*e)*(e*x^2+d)*(-c*x^4+a)/(c*x^4+b*x^2+a)^(5/2),x, algo rithm="maxima")
Output:
1/3*(3*c*d*e*x^5 + 3*a*d*e*x + (c*d^2 + 2*b*d*e + a*e^2)*x^3)*sqrt(c*x^4 + b*x^2 + a)/(c^2*x^8 + 2*b*c*x^6 + (b^2 + 2*a*c)*x^4 + 2*a*b*x^2 + a^2)
Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (54) = 108\).
Time = 0.42 (sec) , antiderivative size = 489, normalized size of antiderivative = 8.15 \[ \int \frac {\left (a e+c d x^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\frac {{\left ({\left (\frac {3 \, {\left (a b^{8} c^{2} d e - 16 \, a^{2} b^{6} c^{3} d e + 96 \, a^{3} b^{4} c^{4} d e - 256 \, a^{4} b^{2} c^{5} d e + 256 \, a^{5} c^{6} d e\right )} x^{2}}{a b^{8} c - 16 \, a^{2} b^{6} c^{2} + 96 \, a^{3} b^{4} c^{3} - 256 \, a^{4} b^{2} c^{4} + 256 \, a^{5} c^{5}} + \frac {a b^{8} c^{2} d^{2} - 16 \, a^{2} b^{6} c^{3} d^{2} + 96 \, a^{3} b^{4} c^{4} d^{2} - 256 \, a^{4} b^{2} c^{5} d^{2} + 256 \, a^{5} c^{6} d^{2} + 2 \, a b^{9} c d e - 32 \, a^{2} b^{7} c^{2} d e + 192 \, a^{3} b^{5} c^{3} d e - 512 \, a^{4} b^{3} c^{4} d e + 512 \, a^{5} b c^{5} d e + a^{2} b^{8} c e^{2} - 16 \, a^{3} b^{6} c^{2} e^{2} + 96 \, a^{4} b^{4} c^{3} e^{2} - 256 \, a^{5} b^{2} c^{4} e^{2} + 256 \, a^{6} c^{5} e^{2}}{a b^{8} c - 16 \, a^{2} b^{6} c^{2} + 96 \, a^{3} b^{4} c^{3} - 256 \, a^{4} b^{2} c^{4} + 256 \, a^{5} c^{5}}\right )} x^{2} + \frac {3 \, {\left (a^{2} b^{8} c d e - 16 \, a^{3} b^{6} c^{2} d e + 96 \, a^{4} b^{4} c^{3} d e - 256 \, a^{5} b^{2} c^{4} d e + 256 \, a^{6} c^{5} d e\right )}}{a b^{8} c - 16 \, a^{2} b^{6} c^{2} + 96 \, a^{3} b^{4} c^{3} - 256 \, a^{4} b^{2} c^{4} + 256 \, a^{5} c^{5}}\right )} x}{3 \, {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \] Input:
integrate((c*d*x^2+a*e)*(e*x^2+d)*(-c*x^4+a)/(c*x^4+b*x^2+a)^(5/2),x, algo rithm="giac")
Output:
1/3*((3*(a*b^8*c^2*d*e - 16*a^2*b^6*c^3*d*e + 96*a^3*b^4*c^4*d*e - 256*a^4 *b^2*c^5*d*e + 256*a^5*c^6*d*e)*x^2/(a*b^8*c - 16*a^2*b^6*c^2 + 96*a^3*b^4 *c^3 - 256*a^4*b^2*c^4 + 256*a^5*c^5) + (a*b^8*c^2*d^2 - 16*a^2*b^6*c^3*d^ 2 + 96*a^3*b^4*c^4*d^2 - 256*a^4*b^2*c^5*d^2 + 256*a^5*c^6*d^2 + 2*a*b^9*c *d*e - 32*a^2*b^7*c^2*d*e + 192*a^3*b^5*c^3*d*e - 512*a^4*b^3*c^4*d*e + 51 2*a^5*b*c^5*d*e + a^2*b^8*c*e^2 - 16*a^3*b^6*c^2*e^2 + 96*a^4*b^4*c^3*e^2 - 256*a^5*b^2*c^4*e^2 + 256*a^6*c^5*e^2)/(a*b^8*c - 16*a^2*b^6*c^2 + 96*a^ 3*b^4*c^3 - 256*a^4*b^2*c^4 + 256*a^5*c^5))*x^2 + 3*(a^2*b^8*c*d*e - 16*a^ 3*b^6*c^2*d*e + 96*a^4*b^4*c^3*d*e - 256*a^5*b^2*c^4*d*e + 256*a^6*c^5*d*e )/(a*b^8*c - 16*a^2*b^6*c^2 + 96*a^3*b^4*c^3 - 256*a^4*b^2*c^4 + 256*a^5*c ^5))*x/(c*x^4 + b*x^2 + a)^(3/2)
Time = 23.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a e+c d x^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\frac {a\,e^2\,x^3+c\,d^2\,x^3-b\,d\,e\,x^3+3\,d\,e\,x\,\left (c\,x^4+b\,x^2+a\right )}{\left (3\,c\,x^4+3\,b\,x^2+3\,a\right )\,\sqrt {c\,x^4+b\,x^2+a}} \] Input:
int(((a - c*x^4)*(d + e*x^2)*(a*e + c*d*x^2))/(a + b*x^2 + c*x^4)^(5/2),x)
Output:
(a*e^2*x^3 + c*d^2*x^3 - b*d*e*x^3 + 3*d*e*x*(a + b*x^2 + c*x^4))/((3*a + 3*b*x^2 + 3*c*x^4)*(a + b*x^2 + c*x^4)^(1/2))
Time = 0.18 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a e+c d x^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x \left (3 c d e \,x^{4}+a \,e^{2} x^{2}+2 b d e \,x^{2}+c \,d^{2} x^{2}+3 a d e \right )}{3 c^{2} x^{8}+6 b c \,x^{6}+6 a c \,x^{4}+3 b^{2} x^{4}+6 a b \,x^{2}+3 a^{2}} \] Input:
int((c*d*x^2+a*e)*(e*x^2+d)*(-c*x^4+a)/(c*x^4+b*x^2+a)^(5/2),x)
Output:
(sqrt(a + b*x**2 + c*x**4)*x*(3*a*d*e + a*e**2*x**2 + 2*b*d*e*x**2 + c*d** 2*x**2 + 3*c*d*e*x**4))/(3*(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2 *b*c*x**6 + c**2*x**8))