Integrand size = 22, antiderivative size = 150 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac {\sqrt {a} e \left (3 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 c^{3/2} \left (c d^2+a e^2\right )^2}-\frac {d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac {d^3 \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )^2} \]
1/4*a*(-e*x^2+d)/c/(a*e^2+c*d^2)/(c*x^4+a)-1/2*d^3*ln(e*x^2+d)/(a*e^2+c*d^ 2)^2+1/4*d^3*ln(c*x^4+a)/(a*e^2+c*d^2)^2+1/4*e*(a*e^2+3*c*d^2)*arctan(x^2* c^(1/2)/a^(1/2))*a^(1/2)/c^(3/2)/(a*e^2+c*d^2)^2
Time = 0.08 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.95 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {\sqrt {a} e \left (3 c d^2+a e^2\right ) \left (a+c x^4\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )+\sqrt {c} \left (a \left (c d^2+a e^2\right ) \left (d-e x^2\right )-2 c d^3 \left (a+c x^4\right ) \log \left (d+e x^2\right )+c d^3 \left (a+c x^4\right ) \log \left (a+c x^4\right )\right )}{4 c^{3/2} \left (c d^2+a e^2\right )^2 \left (a+c x^4\right )} \]
(Sqrt[a]*e*(3*c*d^2 + a*e^2)*(a + c*x^4)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]] + S qrt[c]*(a*(c*d^2 + a*e^2)*(d - e*x^2) - 2*c*d^3*(a + c*x^4)*Log[d + e*x^2] + c*d^3*(a + c*x^4)*Log[a + c*x^4]))/(4*c^(3/2)*(c*d^2 + a*e^2)^2*(a + c* x^4))
Time = 0.38 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.13, 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.
| \(\displaystyle \int \frac {x^7}{\left (a+c x^4\right )^2 \left (d+e x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1579 |
| \(\displaystyle \frac {1}{2} \int \frac {x^6}{\left (e x^2+d\right ) \left (c x^4+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle \frac {1}{2} \left (\frac {a \left (d-e x^2\right )}{2 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac {\int -\frac {a \left (c \left (2 d^2+\frac {a e^2}{c}\right ) x^2+a d e\right )}{c \left (c d^2+a e^2\right ) \left (e x^2+d\right ) \left (c x^4+a\right )}dx^2}{2 a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {a \left (\left (2 c d^2+a e^2\right ) x^2+a d e\right )}{c \left (c d^2+a e^2\right ) \left (e x^2+d\right ) \left (c x^4+a\right )}dx^2}{2 a}+\frac {a \left (d-e x^2\right )}{2 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\left (2 c d^2+a e^2\right ) x^2+a d e}{\left (e x^2+d\right ) \left (c x^4+a\right )}dx^2}{2 c \left (a e^2+c d^2\right )}+\frac {a \left (d-e x^2\right )}{2 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}\right )\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \left (\frac {2 c^2 x^2 d^3+3 a c e d^2+a^2 e^3}{\left (c d^2+a e^2\right ) \left (c x^4+a\right )}-\frac {2 c d^3 e}{\left (c d^2+a e^2\right ) \left (e x^2+d\right )}\right )dx^2}{2 c \left (a e^2+c d^2\right )}+\frac {a \left (d-e x^2\right )}{2 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\sqrt {a} e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (a e^2+3 c d^2\right )}{\sqrt {c} \left (a e^2+c d^2\right )}+\frac {c d^3 \log \left (a+c x^4\right )}{a e^2+c d^2}-\frac {2 c d^3 \log \left (d+e x^2\right )}{a e^2+c d^2}}{2 c \left (a e^2+c d^2\right )}+\frac {a \left (d-e x^2\right )}{2 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}\right )\) |
((a*(d - e*x^2))/(2*c*(c*d^2 + a*e^2)*(a + c*x^4)) + ((Sqrt[a]*e*(3*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[c]*(c*d^2 + a*e^2)) - (2*c*d ^3*Log[d + e*x^2])/(c*d^2 + a*e^2) + (c*d^3*Log[a + c*x^4])/(c*d^2 + a*e^2 ))/(2*c*(c*d^2 + a*e^2)))/2
3.3.45.3.1 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 = 0.50 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(\frac {\frac {-\frac {a e \left (a \,e^{2}+c \,d^{2}\right ) x^{2}}{2 c}+\frac {d a \left (a \,e^{2}+c \,d^{2}\right )}{2 c}}{c \,x^{4}+a}+\frac {c \,d^{3} \ln \left (c \,x^{4}+a \right )+\frac {\left (e^{3} a^{2}+3 a c \,d^{2} e \right ) \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 c}}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}-\frac {d^{3} \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}+c \,d^{2}\right )^{2}}\) | \(146\) |
| risch | \(\text {Expression too large to display}\) | \(1428\) |
1/2/(a*e^2+c*d^2)^2*((-1/2*a*e*(a*e^2+c*d^2)/c*x^2+1/2*d*a*(a*e^2+c*d^2)/c )/(c*x^4+a)+1/2/c*(c*d^3*ln(c*x^4+a)+(a^2*e^3+3*a*c*d^2*e)/(a*c)^(1/2)*arc tan(c*x^2/(a*c)^(1/2))))-1/2*d^3*ln(e*x^2+d)/(a*e^2+c*d^2)^2
Time = 4.23 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.05 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\left [\frac {2 \, a c d^{3} + 2 \, a^{2} d e^{2} - 2 \, {\left (a c d^{2} e + a^{2} e^{3}\right )} x^{2} + {\left (3 \, a c d^{2} e + a^{2} e^{3} + {\left (3 \, c^{2} d^{2} e + a c e^{3}\right )} x^{4}\right )} \sqrt {-\frac {a}{c}} \log \left (\frac {c x^{4} + 2 \, c x^{2} \sqrt {-\frac {a}{c}} - a}{c x^{4} + a}\right ) + 2 \, {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) - 4 \, {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (e x^{2} + d\right )}{8 \, {\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} + {\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4}\right )}}, \frac {a c d^{3} + a^{2} d e^{2} - {\left (a c d^{2} e + a^{2} e^{3}\right )} x^{2} + {\left (3 \, a c d^{2} e + a^{2} e^{3} + {\left (3 \, c^{2} d^{2} e + a c e^{3}\right )} x^{4}\right )} \sqrt {\frac {a}{c}} \arctan \left (\frac {c x^{2} \sqrt {\frac {a}{c}}}{a}\right ) + {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) - 2 \, {\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (e x^{2} + d\right )}{4 \, {\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} + {\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4}\right )}}\right ] \]
[1/8*(2*a*c*d^3 + 2*a^2*d*e^2 - 2*(a*c*d^2*e + a^2*e^3)*x^2 + (3*a*c*d^2*e + a^2*e^3 + (3*c^2*d^2*e + a*c*e^3)*x^4)*sqrt(-a/c)*log((c*x^4 + 2*c*x^2* sqrt(-a/c) - a)/(c*x^4 + a)) + 2*(c^2*d^3*x^4 + a*c*d^3)*log(c*x^4 + a) - 4*(c^2*d^3*x^4 + a*c*d^3)*log(e*x^2 + d))/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4), 1/4*(a*c*d^3 + a^2*d*e^2 - (a*c*d^2*e + a^2*e^3)*x^2 + (3*a*c*d^2*e + a^2*e^3 + (3*c^2* d^2*e + a*c*e^3)*x^4)*sqrt(a/c)*arctan(c*x^2*sqrt(a/c)/a) + (c^2*d^3*x^4 + a*c*d^3)*log(c*x^4 + a) - 2*(c^2*d^3*x^4 + a*c*d^3)*log(e*x^2 + d))/(a*c^ 3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c ^2*e^4)*x^4)]
Timed out. \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.31 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {d^{3} \log \left (c x^{4} + a\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {d^{3} \log \left (e x^{2} + d\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {{\left (3 \, a c d^{2} e + a^{2} e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt {a c}} - \frac {a e x^{2} - a d}{4 \, {\left (a c^{2} d^{2} + a^{2} c e^{2} + {\left (c^{3} d^{2} + a c^{2} e^{2}\right )} x^{4}\right )}} \]
1/4*d^3*log(c*x^4 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) - 1/2*d^3*log(e *x^2 + d)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + 1/4*(3*a*c*d^2*e + a^2*e^3 )*arctan(c*x^2/sqrt(a*c))/((c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)*sqrt(a* c)) - 1/4*(a*e*x^2 - a*d)/(a*c^2*d^2 + a^2*c*e^2 + (c^3*d^2 + a*c^2*e^2)*x ^4)
Time = 0.29 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.53 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=-\frac {d^{3} e \log \left ({\left | e x^{2} + d \right |}\right )}{2 \, {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac {d^{3} \log \left (c x^{4} + a\right )}{4 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac {{\left (3 \, a c d^{2} e + a^{2} e^{3}\right )} \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{4 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt {a c}} - \frac {c^{2} d^{3} x^{4} + a c d^{2} e x^{2} + a^{2} e^{3} x^{2} - a^{2} d e^{2}}{4 \, {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (c x^{4} + a\right )}} \]
-1/2*d^3*e*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^3*log(c*x^4 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + 1/4*(3*a*c*d^2*e + a^2*e^3)*arctan(c*x^2/sqrt(a*c))/((c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^ 4)*sqrt(a*c)) - 1/4*(c^2*d^3*x^4 + a*c*d^2*e*x^2 + a^2*e^3*x^2 - a^2*d*e^2 )/((c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)*(c*x^4 + a))
Time = 8.26 (sec) , antiderivative size = 647, normalized size of antiderivative = 4.31 \[ \int \frac {x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx=\frac {\frac {a\,d}{4\,c\,\left (c\,d^2+a\,e^2\right )}-\frac {a\,e\,x^2}{4\,c\,\left (c\,d^2+a\,e^2\right )}}{c\,x^4+a}-\frac {d^3\,\ln \left (e\,x^2+d\right )}{2\,\left (a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}+\frac {\ln \left (36\,c^8\,d^{10}\,x^2+36\,c^6\,d^{10}\,\sqrt {-a\,c^3}+a^5\,c\,e^{10}\,\sqrt {-a\,c^3}+a^5\,c^3\,e^{10}\,x^2-22\,a^2\,d^4\,e^6\,{\left (-a\,c^3\right )}^{3/2}-81\,c^2\,d^8\,e^2\,{\left (-a\,c^3\right )}^{3/2}+60\,a^2\,c^6\,d^6\,e^4\,x^2+22\,a^3\,c^5\,d^4\,e^6\,x^2+8\,a^4\,c^4\,d^2\,e^8\,x^2+8\,a^4\,c^2\,d^2\,e^8\,\sqrt {-a\,c^3}-60\,a\,c\,d^6\,e^4\,{\left (-a\,c^3\right )}^{3/2}+81\,a\,c^7\,d^8\,e^2\,x^2\right )\,\left (2\,c^3\,d^3+a\,e^3\,\sqrt {-a\,c^3}+3\,c\,d^2\,e\,\sqrt {-a\,c^3}\right )}{8\,\left (a^2\,c^3\,e^4+2\,a\,c^4\,d^2\,e^2+c^5\,d^4\right )}-\frac {\ln \left (36\,c^8\,d^{10}\,x^2-36\,c^6\,d^{10}\,\sqrt {-a\,c^3}-a^5\,c\,e^{10}\,\sqrt {-a\,c^3}+a^5\,c^3\,e^{10}\,x^2+22\,a^2\,d^4\,e^6\,{\left (-a\,c^3\right )}^{3/2}+81\,c^2\,d^8\,e^2\,{\left (-a\,c^3\right )}^{3/2}+60\,a^2\,c^6\,d^6\,e^4\,x^2+22\,a^3\,c^5\,d^4\,e^6\,x^2+8\,a^4\,c^4\,d^2\,e^8\,x^2-8\,a^4\,c^2\,d^2\,e^8\,\sqrt {-a\,c^3}+60\,a\,c\,d^6\,e^4\,{\left (-a\,c^3\right )}^{3/2}+81\,a\,c^7\,d^8\,e^2\,x^2\right )\,\left (a\,e^3\,\sqrt {-a\,c^3}-2\,c^3\,d^3+3\,c\,d^2\,e\,\sqrt {-a\,c^3}\right )}{8\,\left (a^2\,c^3\,e^4+2\,a\,c^4\,d^2\,e^2+c^5\,d^4\right )} \]
((a*d)/(4*c*(a*e^2 + c*d^2)) - (a*e*x^2)/(4*c*(a*e^2 + c*d^2)))/(a + c*x^4 ) - (d^3*log(d + e*x^2))/(2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (log(36 *c^8*d^10*x^2 + 36*c^6*d^10*(-a*c^3)^(1/2) + a^5*c*e^10*(-a*c^3)^(1/2) + a ^5*c^3*e^10*x^2 - 22*a^2*d^4*e^6*(-a*c^3)^(3/2) - 81*c^2*d^8*e^2*(-a*c^3)^ (3/2) + 60*a^2*c^6*d^6*e^4*x^2 + 22*a^3*c^5*d^4*e^6*x^2 + 8*a^4*c^4*d^2*e^ 8*x^2 + 8*a^4*c^2*d^2*e^8*(-a*c^3)^(1/2) - 60*a*c*d^6*e^4*(-a*c^3)^(3/2) + 81*a*c^7*d^8*e^2*x^2)*(2*c^3*d^3 + a*e^3*(-a*c^3)^(1/2) + 3*c*d^2*e*(-a*c ^3)^(1/2)))/(8*(c^5*d^4 + a^2*c^3*e^4 + 2*a*c^4*d^2*e^2)) - (log(36*c^8*d^ 10*x^2 - 36*c^6*d^10*(-a*c^3)^(1/2) - a^5*c*e^10*(-a*c^3)^(1/2) + a^5*c^3* e^10*x^2 + 22*a^2*d^4*e^6*(-a*c^3)^(3/2) + 81*c^2*d^8*e^2*(-a*c^3)^(3/2) + 60*a^2*c^6*d^6*e^4*x^2 + 22*a^3*c^5*d^4*e^6*x^2 + 8*a^4*c^4*d^2*e^8*x^2 - 8*a^4*c^2*d^2*e^8*(-a*c^3)^(1/2) + 60*a*c*d^6*e^4*(-a*c^3)^(3/2) + 81*a*c ^7*d^8*e^2*x^2)*(a*e^3*(-a*c^3)^(1/2) - 2*c^3*d^3 + 3*c*d^2*e*(-a*c^3)^(1/ 2)))/(8*(c^5*d^4 + a^2*c^3*e^4 + 2*a*c^4*d^2*e^2))