Integrand size = 22, antiderivative size = 155 \[ \int \frac {x^5}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {-a e-c d x^2}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {d \left (c d^2-a e^2\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 \sqrt {a} \sqrt {c} \left (c d^2+a e^2\right )^2}+\frac {d^2 e \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}-\frac {d^2 e \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )^2} \]
1/4*(-c*d*x^2-a*e)/c/(a*e^2+c*d^2)/(c*x^4+a)+1/2*d^2*e*ln(e*x^2+d)/(a*e^2+ c*d^2)^2-1/4*d^2*e*ln(c*x^4+a)/(a*e^2+c*d^2)^2+1/4*d*(-a*e^2+c*d^2)*arctan (x^2*c^(1/2)/a^(1/2))/(a*e^2+c*d^2)^2/a^(1/2)/c^(1/2)
Time = 0.07 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.96 \[ \int \frac {x^5}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {\sqrt {c} d \left (c d^2-a e^2\right ) \left (a+c x^4\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )-\sqrt {a} \left (\left (c d^2+a e^2\right ) \left (a e+c d x^2\right )-2 c d^2 e \left (a+c x^4\right ) \log \left (d+e x^2\right )+c d^2 e \left (a+c x^4\right ) \log \left (a+c x^4\right )\right )}{4 \sqrt {a} c \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )} \]
(Sqrt[c]*d*(c*d^2 - a*e^2)*(a + c*x^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]] - Sqr t[a]*((c*d^2 + a*e^2)*(a*e + c*d*x^2) - 2*c*d^2*e*(a + c*x^4)*Log[d + e*x^ 2] + c*d^2*e*(a + c*x^4)*Log[a + c*x^4]))/(4*Sqrt[a]*c*(c*d^2 + a*e^2)^2*( a + c*x^4))
Time = 0.35 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1579, 601, 25, 27, 657, 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.
| \(\displaystyle \int \frac {x^5}{\left (a+c x^4\right )^2 \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1579 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (e x^2+d\right ) \left (c x^4+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int -\frac {a d \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (e x^2+d\right ) \left (c x^4+a\right )}dx^2}{2 a}-\frac {a e+c d x^2}{2 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {a d \left (d-e x^2\right )}{\left (c d^2+a e^2\right ) \left (e x^2+d\right ) \left (c x^4+a\right )}dx^2}{2 a}-\frac {a e+c d x^2}{2 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {d \int \frac {d-e x^2}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx^2}{2 \left (a e^2+c d^2\right )}-\frac {a e+c d x^2}{2 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}\right )\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {1}{2} \left (\frac {d \int \left (\frac {2 d e^2}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}+\frac {c d^2-2 c e x^2 d-a e^2}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}\right )dx^2}{2 \left (a e^2+c d^2\right )}-\frac {a e+c d x^2}{2 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (c d^2-a e^2\right )}{\sqrt {a} \sqrt {c} \left (a e^2+c d^2\right )}-\frac {d e \log \left (a+c x^4\right )}{a e^2+c d^2}+\frac {2 d e \log \left (d+e x^2\right )}{a e^2+c d^2}\right )}{2 \left (a e^2+c d^2\right )}-\frac {a e+c d x^2}{2 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}\right )\) |
(-1/2*(a*e + c*d*x^2)/(c*(c*d^2 + a*e^2)*(a + c*x^4)) + (d*(((c*d^2 - a*e^ 2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]*(c*d^2 + a*e^2)) + (2*d *e*Log[d + e*x^2])/(c*d^2 + a*e^2) - (d*e*Log[a + c*x^4])/(c*d^2 + a*e^2)) )/(2*(c*d^2 + a*e^2)))/2
3.3.46.3.1 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[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* (2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]
Time = 0.42 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(-\frac {\frac {\left (\frac {1}{2} d \,e^{2} a +\frac {1}{2} d^{3} c \right ) x^{2}+\frac {a e \left (a \,e^{2}+c \,d^{2}\right )}{2 c}}{c \,x^{4}+a}+\frac {d \left (d e \ln \left (c \,x^{4}+a \right )+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{\sqrt {a c}}\right )}{2}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}+\frac {d^{2} e \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}\) | \(136\) |
| risch | \(\frac {-\frac {d \,x^{2}}{4 \left (a \,e^{2}+c \,d^{2}\right )}-\frac {a e}{4 c \left (a \,e^{2}+c \,d^{2}\right )}}{c \,x^{4}+a}+\frac {d^{2} e \ln \left (e \,x^{2}+d \right )}{2 a^{2} e^{4}+4 a c \,d^{2} e^{2}+2 c^{2} d^{4}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} c \,e^{4}+2 a^{2} c^{2} d^{2} e^{2}+a \,c^{3} d^{4}\right ) \textit {\_Z}^{2}+4 a c \,d^{2} e \textit {\_Z} +d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (2 a^{4} c \,e^{7}+2 a^{3} c^{2} d^{2} e^{5}-2 a^{2} c^{3} d^{4} e^{3}-2 a \,c^{4} d^{6} e \right ) \textit {\_R}^{2}+\left (7 a^{2} c \,d^{2} e^{4}+6 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}\right ) \textit {\_R} +2 a \,d^{2} e^{3}+4 c \,d^{4} e \right ) x^{2}+\left (3 a^{4} c d \,e^{6}+5 a^{3} c^{2} d^{3} e^{4}+a^{2} c^{3} d^{5} e^{2}-a \,c^{4} d^{7}\right ) \textit {\_R}^{2}+\left (4 a^{2} c \,d^{3} e^{3}+4 a \,c^{2} d^{5} e \right ) \textit {\_R} +2 a \,d^{3} e^{2}\right )\right )}{8}\) | \(344\) |
-1/2/(a*e^2+c*d^2)^2*(((1/2*d*e^2*a+1/2*d^3*c)*x^2+1/2*a*e*(a*e^2+c*d^2)/c )/(c*x^4+a)+1/2*d*(d*e*ln(c*x^4+a)+(a*e^2-c*d^2)/(a*c)^(1/2)*arctan(c*x^2/ (a*c)^(1/2))))+1/2*d^2*e*ln(e*x^2+d)/(a*e^2+c*d^2)^2
Time = 1.79 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.14 \[ \int \frac {x^5}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\left [-\frac {2 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} + 2 \, {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} x^{2} - {\left (a c d^{3} - a^{2} d e^{2} + {\left (c^{2} d^{3} - a c d e^{2}\right )} x^{4}\right )} \sqrt {-a c} \log \left (\frac {c x^{4} + 2 \, \sqrt {-a c} x^{2} - a}{c x^{4} + a}\right ) + 2 \, {\left (a c^{2} d^{2} e x^{4} + a^{2} c d^{2} e\right )} \log \left (c x^{4} + a\right ) - 4 \, {\left (a c^{2} d^{2} e x^{4} + a^{2} c d^{2} e\right )} \log \left (e x^{2} + d\right )}{8 \, {\left (a^{2} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4} + {\left (a c^{4} d^{4} + 2 \, a^{2} c^{3} d^{2} e^{2} + a^{3} c^{2} e^{4}\right )} x^{4}\right )}}, -\frac {a^{2} c d^{2} e + a^{3} e^{3} + {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} x^{2} + {\left (a c d^{3} - a^{2} d e^{2} + {\left (c^{2} d^{3} - a c d e^{2}\right )} x^{4}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c}}{c x^{2}}\right ) + {\left (a c^{2} d^{2} e x^{4} + a^{2} c d^{2} e\right )} \log \left (c x^{4} + a\right ) - 2 \, {\left (a c^{2} d^{2} e x^{4} + a^{2} c d^{2} e\right )} \log \left (e x^{2} + d\right )}{4 \, {\left (a^{2} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4} + {\left (a c^{4} d^{4} + 2 \, a^{2} c^{3} d^{2} e^{2} + a^{3} c^{2} e^{4}\right )} x^{4}\right )}}\right ] \]
[-1/8*(2*a^2*c*d^2*e + 2*a^3*e^3 + 2*(a*c^2*d^3 + a^2*c*d*e^2)*x^2 - (a*c* d^3 - a^2*d*e^2 + (c^2*d^3 - a*c*d*e^2)*x^4)*sqrt(-a*c)*log((c*x^4 + 2*sqr t(-a*c)*x^2 - a)/(c*x^4 + a)) + 2*(a*c^2*d^2*e*x^4 + a^2*c*d^2*e)*log(c*x^ 4 + a) - 4*(a*c^2*d^2*e*x^4 + a^2*c*d^2*e)*log(e*x^2 + d))/(a^2*c^3*d^4 + 2*a^3*c^2*d^2*e^2 + a^4*c*e^4 + (a*c^4*d^4 + 2*a^2*c^3*d^2*e^2 + a^3*c^2*e ^4)*x^4), -1/4*(a^2*c*d^2*e + a^3*e^3 + (a*c^2*d^3 + a^2*c*d*e^2)*x^2 + (a *c*d^3 - a^2*d*e^2 + (c^2*d^3 - a*c*d*e^2)*x^4)*sqrt(a*c)*arctan(sqrt(a*c) /(c*x^2)) + (a*c^2*d^2*e*x^4 + a^2*c*d^2*e)*log(c*x^4 + a) - 2*(a*c^2*d^2* e*x^4 + a^2*c*d^2*e)*log(e*x^2 + d))/(a^2*c^3*d^4 + 2*a^3*c^2*d^2*e^2 + a^ 4*c*e^4 + (a*c^4*d^4 + 2*a^2*c^3*d^2*e^2 + a^3*c^2*e^4)*x^4)]
Timed out. \[ \int \frac {x^5}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.24 \[ \int \frac {x^5}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=-\frac {d^{2} e \log \left (c x^{4} + a\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {d^{2} e \log \left (e x^{2} + d\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {{\left (c d^{3} - a d e^{2}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c}} - \frac {c d x^{2} + a e}{4 \, {\left (a c^{2} d^{2} + a^{2} c e^{2} + {\left (c^{3} d^{2} + a c^{2} e^{2}\right )} x^{4}\right )}} \]
-1/4*d^2*e*log(c*x^4 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + 1/2*d^2*e* log(e*x^2 + d)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + 1/4*(c*d^3 - a*d*e^2) *arctan(c*x^2/sqrt(a*c))/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(a*c)) - 1/4*(c*d*x^2 + a*e)/(a*c^2*d^2 + a^2*c*e^2 + (c^3*d^2 + a*c^2*e^2)*x^4)
Time = 0.28 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.47 \[ \int \frac {x^5}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {d^{2} e^{2} \log \left ({\left | e x^{2} + d \right |}\right )}{2 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} - \frac {d^{2} e \log \left (c x^{4} + a\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {{\left (c d^{3} - a d e^{2}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c}} + \frac {c^{2} d^{2} e x^{4} - c^{2} d^{3} x^{2} - a c d e^{2} x^{2} - a^{2} e^{3}}{4 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (c x^{4} + a\right )}} \]
1/2*d^2*e^2*log(abs(e*x^2 + d))/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) - 1/ 4*d^2*e*log(c*x^4 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + 1/4*(c*d^3 - a*d*e^2)*arctan(c*x^2/sqrt(a*c))/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*sqrt (a*c)) + 1/4*(c^2*d^2*e*x^4 - c^2*d^3*x^2 - a*c*d*e^2*x^2 - a^2*e^3)/((c^3 *d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)*(c*x^4 + a))
Time = 8.28 (sec) , antiderivative size = 528, normalized size of antiderivative = 3.41 \[ \int \frac {x^5}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {\ln \left (a^4\,e^8\,\sqrt {-a\,c}+c^4\,d^8\,\sqrt {-a\,c}+70\,d^4\,e^4\,{\left (-a\,c\right )}^{5/2}+c^5\,d^8\,x^2+a^4\,c\,e^8\,x^2-36\,a^2\,d^2\,e^6\,{\left (-a\,c\right )}^{3/2}-36\,c^2\,d^6\,e^2\,{\left (-a\,c\right )}^{3/2}+70\,a^2\,c^3\,d^4\,e^4\,x^2+36\,a^3\,c^2\,d^2\,e^6\,x^2+36\,a\,c^4\,d^6\,e^2\,x^2\right )\,\left (c\,\left (\frac {d^3\,\sqrt {-a\,c}}{8}-\frac {a\,d^2\,e}{4}\right )-\frac {a\,d\,e^2\,\sqrt {-a\,c}}{8}\right )}{a^3\,c\,e^4+2\,a^2\,c^2\,d^2\,e^2+a\,c^3\,d^4}-\frac {\frac {d\,x^2}{4\,\left (c\,d^2+a\,e^2\right )}+\frac {a\,e}{4\,c\,\left (c\,d^2+a\,e^2\right )}}{c\,x^4+a}-\frac {\ln \left (c^5\,d^8\,x^2-c^4\,d^8\,\sqrt {-a\,c}-70\,d^4\,e^4\,{\left (-a\,c\right )}^{5/2}-a^4\,e^8\,\sqrt {-a\,c}+a^4\,c\,e^8\,x^2+36\,a^2\,d^2\,e^6\,{\left (-a\,c\right )}^{3/2}+36\,c^2\,d^6\,e^2\,{\left (-a\,c\right )}^{3/2}+70\,a^2\,c^3\,d^4\,e^4\,x^2+36\,a^3\,c^2\,d^2\,e^6\,x^2+36\,a\,c^4\,d^6\,e^2\,x^2\right )\,\left (c\,\left (\frac {d^3\,\sqrt {-a\,c}}{8}+\frac {a\,d^2\,e}{4}\right )-\frac {a\,d\,e^2\,\sqrt {-a\,c}}{8}\right )}{a^3\,c\,e^4+2\,a^2\,c^2\,d^2\,e^2+a\,c^3\,d^4}+\frac {d^2\,e\,\ln \left (e\,x^2+d\right )}{2\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )} \]
(log(a^4*e^8*(-a*c)^(1/2) + c^4*d^8*(-a*c)^(1/2) + 70*d^4*e^4*(-a*c)^(5/2) + c^5*d^8*x^2 + a^4*c*e^8*x^2 - 36*a^2*d^2*e^6*(-a*c)^(3/2) - 36*c^2*d^6* e^2*(-a*c)^(3/2) + 70*a^2*c^3*d^4*e^4*x^2 + 36*a^3*c^2*d^2*e^6*x^2 + 36*a* c^4*d^6*e^2*x^2)*(c*((d^3*(-a*c)^(1/2))/8 - (a*d^2*e)/4) - (a*d*e^2*(-a*c) ^(1/2))/8))/(a*c^3*d^4 + a^3*c*e^4 + 2*a^2*c^2*d^2*e^2) - ((d*x^2)/(4*(a*e ^2 + c*d^2)) + (a*e)/(4*c*(a*e^2 + c*d^2)))/(a + c*x^4) - (log(c^5*d^8*x^2 - c^4*d^8*(-a*c)^(1/2) - 70*d^4*e^4*(-a*c)^(5/2) - a^4*e^8*(-a*c)^(1/2) + a^4*c*e^8*x^2 + 36*a^2*d^2*e^6*(-a*c)^(3/2) + 36*c^2*d^6*e^2*(-a*c)^(3/2) + 70*a^2*c^3*d^4*e^4*x^2 + 36*a^3*c^2*d^2*e^6*x^2 + 36*a*c^4*d^6*e^2*x^2) *(c*((d^3*(-a*c)^(1/2))/8 + (a*d^2*e)/4) - (a*d*e^2*(-a*c)^(1/2))/8))/(a*c ^3*d^4 + a^3*c*e^4 + 2*a^2*c^2*d^2*e^2) + (d^2*e*log(d + e*x^2))/(2*(a^2*e ^4 + c^2*d^4 + 2*a*c*d^2*e^2))