Integrand size = 56, antiderivative size = 42 \[ \int \frac {x^m \left (e (1+m)+2 f (1+m-n) x^n\right )}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx=\frac {\text {arctanh}\left (\frac {2 \sqrt {d} \sqrt {f} x^{1+m}}{e+2 f x^n}\right )}{2 \sqrt {d} \sqrt {f}} \]
Time = 0.57 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \left (e (1+m)+2 f (1+m-n) x^n\right )}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx=\frac {\text {arctanh}\left (\frac {2 \sqrt {d} \sqrt {f} x^{1+m}}{e+2 f x^n}\right )}{2 \sqrt {d} \sqrt {f}} \]
Integrate[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 - 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)),x]
Time = 0.39 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.67, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2520, 221}
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.
\(\displaystyle \int \frac {x^m \left (e (m+1)+2 f (m-n+1) x^n\right )}{-4 d f x^{2 m+2}+e^2+4 e f x^n+4 f^2 x^{2 n}} \, dx\) |
\(\Big \downarrow \) 2520 |
\(\displaystyle e^2 (m+1) (m-n+1) \int \frac {1}{e^2-\frac {4 d e^2 f (m+1)^2 (m-n+1)^2 x^{2 m+2}}{\left (2 f (m+1) (m-n+1) x^n+e (m+1) (m-n+1)\right )^2}}d\frac {x^{m+1}}{2 f (m+1) (m-n+1) x^n+e (m+1) (m-n+1)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {2 \sqrt {d} \sqrt {f} (m+1) (m-n+1) x^{m+1}}{e (m+1) (m-n+1)+2 f (m+1) (m-n+1) x^n}\right )}{2 \sqrt {d} \sqrt {f}}\) |
Int[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 - 4*d*f*x^(2 + 2*m) + 4*e *f*x^n + 4*f^2*x^(2*n)),x]
ArcTanh[(2*Sqrt[d]*Sqrt[f]*(1 + m)*(1 + m - n)*x^(1 + m))/(e*(1 + m)*(1 + m - n) + 2*f*(1 + m)*(1 + m - n)*x^n)]/(2*Sqrt[d]*Sqrt[f])
3.6.43.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(n_.)))/((a_) + (b_.)*(x_)^(k_.) + (c_.) *(x_)^(n_.) + (d_.)*(x_)^(n2_)), x_Symbol] :> Simp[A^2*((m - n + 1)/(m + 1) ) Subst[Int[1/(a + A^2*b*(m - n + 1)^2*x^2), x], x, x^(m + 1)/(A*(m - n + 1) + B*(m + 1)*x^n)], x] /; FreeQ[{a, b, c, d, A, B, m, n}, x] && EqQ[n2, 2*n] && EqQ[k, 2*(m + 1)] && EqQ[a*B^2*(m + 1)^2 - A^2*d*(m - n + 1)^2, 0] && EqQ[B*c*(m + 1) - 2*A*d*(m - n + 1), 0]
Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs. \(2(32)=64\).
Time = 3.46 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.86
method | result | size |
risch | \(\frac {\ln \left (x^{n}+\frac {2 x^{m} d f x +\sqrt {d f}\, e}{2 \sqrt {d f}\, f}\right )}{4 \sqrt {d f}}-\frac {\ln \left (x^{n}+\frac {-2 x^{m} d f x +\sqrt {d f}\, e}{2 \sqrt {d f}\, f}\right )}{4 \sqrt {d f}}\) | \(78\) |
int(x^m*(e*(1+m)+2*f*(1+m-n)*x^n)/(e^2-4*d*f*x^(2+2*m)+4*e*f*x^n+4*f^2*x^( 2*n)),x,method=_RETURNVERBOSE)
1/4/(d*f)^(1/2)*ln(x^n+1/2*(2*x^m*d*f*x+(d*f)^(1/2)*e)/(d*f)^(1/2)/f)-1/4/ (d*f)^(1/2)*ln(x^n+1/2*(-2*x^m*d*f*x+(d*f)^(1/2)*e)/(d*f)^(1/2)/f)
Time = 0.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 3.93 \[ \int \frac {x^m \left (e (1+m)+2 f (1+m-n) x^n\right )}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx=\left [\frac {\sqrt {d f} \log \left (-\frac {4 \, d f x^{2} x^{2 \, m} + 4 \, \sqrt {d f} e x x^{m} + 4 \, f^{2} x^{2 \, n} + e^{2} + 4 \, {\left (2 \, \sqrt {d f} f x x^{m} + e f\right )} x^{n}}{4 \, d f x^{2} x^{2 \, m} - 4 \, f^{2} x^{2 \, n} - 4 \, e f x^{n} - e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {-d f} \arctan \left (\frac {2 \, \sqrt {-d f} f x^{n} + \sqrt {-d f} e}{2 \, d f x x^{m}}\right )}{2 \, d f}\right ] \]
integrate(x^m*(e*(1+m)+2*f*(1+m-n)*x^n)/(e^2-4*d*f*x^(2+2*m)+4*e*f*x^n+4*f ^2*x^(2*n)),x, algorithm="fricas")
[1/4*sqrt(d*f)*log(-(4*d*f*x^2*x^(2*m) + 4*sqrt(d*f)*e*x*x^m + 4*f^2*x^(2* n) + e^2 + 4*(2*sqrt(d*f)*f*x*x^m + e*f)*x^n)/(4*d*f*x^2*x^(2*m) - 4*f^2*x ^(2*n) - 4*e*f*x^n - e^2))/(d*f), -1/2*sqrt(-d*f)*arctan(1/2*(2*sqrt(-d*f) *f*x^n + sqrt(-d*f)*e)/(d*f*x*x^m))/(d*f)]
\[ \int \frac {x^m \left (e (1+m)+2 f (1+m-n) x^n\right )}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx=- \int \frac {e x^{m}}{4 d f x^{2 m + 2} - e^{2} - 4 e f x^{n} - 4 f^{2} x^{2 n}}\, dx - \int \frac {e m x^{m}}{4 d f x^{2 m + 2} - e^{2} - 4 e f x^{n} - 4 f^{2} x^{2 n}}\, dx - \int \frac {2 f x^{m} x^{n}}{4 d f x^{2 m + 2} - e^{2} - 4 e f x^{n} - 4 f^{2} x^{2 n}}\, dx - \int \frac {2 f m x^{m} x^{n}}{4 d f x^{2 m + 2} - e^{2} - 4 e f x^{n} - 4 f^{2} x^{2 n}}\, dx - \int \left (- \frac {2 f n x^{m} x^{n}}{4 d f x^{2 m + 2} - e^{2} - 4 e f x^{n} - 4 f^{2} x^{2 n}}\right )\, dx \]
integrate(x**m*(e*(1+m)+2*f*(1+m-n)*x**n)/(e**2-4*d*f*x**(2+2*m)+4*e*f*x** n+4*f**2*x**(2*n)),x)
-Integral(e*x**m/(4*d*f*x**(2*m + 2) - e**2 - 4*e*f*x**n - 4*f**2*x**(2*n) ), x) - Integral(e*m*x**m/(4*d*f*x**(2*m + 2) - e**2 - 4*e*f*x**n - 4*f**2 *x**(2*n)), x) - Integral(2*f*x**m*x**n/(4*d*f*x**(2*m + 2) - e**2 - 4*e*f *x**n - 4*f**2*x**(2*n)), x) - Integral(2*f*m*x**m*x**n/(4*d*f*x**(2*m + 2 ) - e**2 - 4*e*f*x**n - 4*f**2*x**(2*n)), x) - Integral(-2*f*n*x**m*x**n/( 4*d*f*x**(2*m + 2) - e**2 - 4*e*f*x**n - 4*f**2*x**(2*n)), x)
\[ \int \frac {x^m \left (e (1+m)+2 f (1+m-n) x^n\right )}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx=\int { -\frac {{\left (2 \, f {\left (m - n + 1\right )} x^{n} + e {\left (m + 1\right )}\right )} x^{m}}{4 \, d f x^{2 \, m + 2} - 4 \, f^{2} x^{2 \, n} - 4 \, e f x^{n} - e^{2}} \,d x } \]
integrate(x^m*(e*(1+m)+2*f*(1+m-n)*x^n)/(e^2-4*d*f*x^(2+2*m)+4*e*f*x^n+4*f ^2*x^(2*n)),x, algorithm="maxima")
-integrate((2*f*(m - n + 1)*x^n + e*(m + 1))*x^m/(4*d*f*x^(2*m + 2) - 4*f^ 2*x^(2*n) - 4*e*f*x^n - e^2), x)
\[ \int \frac {x^m \left (e (1+m)+2 f (1+m-n) x^n\right )}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx=\int { -\frac {{\left (2 \, f {\left (m - n + 1\right )} x^{n} + e {\left (m + 1\right )}\right )} x^{m}}{4 \, d f x^{2 \, m + 2} - 4 \, f^{2} x^{2 \, n} - 4 \, e f x^{n} - e^{2}} \,d x } \]
integrate(x^m*(e*(1+m)+2*f*(1+m-n)*x^n)/(e^2-4*d*f*x^(2+2*m)+4*e*f*x^n+4*f ^2*x^(2*n)),x, algorithm="giac")
integrate(-(2*f*(m - n + 1)*x^n + e*(m + 1))*x^m/(4*d*f*x^(2*m + 2) - 4*f^ 2*x^(2*n) - 4*e*f*x^n - e^2), x)
Timed out. \[ \int \frac {x^m \left (e (1+m)+2 f (1+m-n) x^n\right )}{e^2-4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx=\int \frac {x^m\,\left (e\,\left (m+1\right )+2\,f\,x^n\,\left (m-n+1\right )\right )}{e^2+4\,f^2\,x^{2\,n}-4\,d\,f\,x^{2\,m+2}+4\,e\,f\,x^n} \,d x \]
int((x^m*(e*(m + 1) + 2*f*x^n*(m - n + 1)))/(e^2 + 4*f^2*x^(2*n) - 4*d*f*x ^(2*m + 2) + 4*e*f*x^n),x)