Integrand size = 18, antiderivative size = 82 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \, dx=-\frac {2 e f p (f x)^{-1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{2},\frac {3-m}{2},-\frac {e}{d x^2}\right )}{d \left (1-m^2\right )}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (1+m)} \]
-2*e*f*p*(f*x)^(-1+m)*hypergeom([1, -1/2*m+1/2],[3/2-1/2*m],-e/d/x^2)/d/(- m^2+1)+(f*x)^(1+m)*ln(c*(d+e/x^2)^p)/f/(1+m)
Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \, dx=\frac {(f x)^m \left (2 e p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {m}{2},\frac {3}{2}-\frac {m}{2},-\frac {e}{d x^2}\right )+d (-1+m) x^2 \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )\right )}{d (-1+m) (1+m) x} \]
((f*x)^m*(2*e*p*Hypergeometric2F1[1, 1/2 - m/2, 3/2 - m/2, -(e/(d*x^2))] + d*(-1 + m)*x^2*Log[c*(d + e/x^2)^p]))/(d*(-1 + m)*(1 + m)*x)
Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2905, 8, 862, 278}
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.
| \(\displaystyle \int (f x)^m \log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2905 |
| \(\displaystyle \frac {2 e p \int \frac {(f x)^{m+1}}{\left (d+\frac {e}{x^2}\right ) x^3}dx}{f (m+1)}+\frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (m+1)}\) |
| \(\Big \downarrow \) 8 |
| \(\displaystyle \frac {2 e f^2 p \int \frac {(f x)^{m-2}}{d+\frac {e}{x^2}}dx}{m+1}+\frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (m+1)}\) |
| \(\Big \downarrow \) 862 |
| \(\displaystyle \frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (m+1)}-\frac {2 e f p \left (\frac {1}{x}\right )^{m-1} (f x)^{m-1} \int \frac {\left (\frac {1}{x}\right )^{-m}}{d+\frac {e}{x^2}}d\frac {1}{x}}{m+1}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x^2}\right )^p\right )}{f (m+1)}-\frac {2 e f p (f x)^{m-1} \operatorname {Hypergeometric2F1}\left (1,\frac {1-m}{2},\frac {3-m}{2},-\frac {e}{d x^2}\right )}{d (1-m) (m+1)}\) |
(-2*e*f*p*(f*x)^(-1 + m)*Hypergeometric2F1[1, (1 - m)/2, (3 - m)/2, -(e/(d *x^2))])/(d*(1 - m)*(1 + m)) + ((f*x)^(1 + m)*Log[c*(d + e/x^2)^p])/(f*(1 + m))
3.1.59.3.1 Defintions of rubi rules used
Int[(u_.)*(x_)^(m_.)*((a_.)*(x_))^(p_), x_Symbol] :> Simp[1/a^m Int[u*(a* x)^(m + p), x], x] /; FreeQ[{a, m, p}, x] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-c^ (-1))*(c*x)^(m + 1)*(1/x)^(m + 1) Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] && !RationalQ[m]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^ (m_.), x_Symbol] :> Simp[(f*x)^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Simp[b*e*n*(p/(f*(m + 1))) Int[x^(n - 1)*((f*x)^(m + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]
\[\int \left (f x \right )^{m} \ln \left (c \left (d +\frac {e}{x^{2}}\right )^{p}\right )d x\]
\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x^{2}}\right )}^{p}\right ) \,d x } \]
Time = 28.76 (sec) , antiderivative size = 366, normalized size of antiderivative = 4.46 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \, dx=2 e p \left (\begin {cases} - \frac {0^{m} \sqrt {- \frac {1}{d e}} \log {\left (- e \sqrt {- \frac {1}{d e}} + x \right )}}{2} + \frac {0^{m} \sqrt {- \frac {1}{d e}} \log {\left (e \sqrt {- \frac {1}{d e}} + x \right )}}{2} & \text {for}\: \left (f = 0 \wedge m \neq -1\right ) \vee f = 0 \\\frac {f^{m + 1} m x^{m - 1} \Phi \left (\frac {e e^{i \pi }}{d x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 d f m \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) + 4 d f \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} - \frac {f^{m + 1} x^{m - 1} \Phi \left (\frac {e e^{i \pi }}{d x^{2}}, 1, \frac {1}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )}{4 d f m \Gamma \left (\frac {3}{2} - \frac {m}{2}\right ) + 4 d f \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq -1 \\\frac {\begin {cases} \frac {\operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x^{2}}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} + \frac {\operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x^{2}}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x^{2}}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + \frac {\operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x^{2}}\right )}{2} & \text {otherwise} \end {cases}}{2 e f} - \frac {\log {\left (f x \right )} \log {\left (d + \frac {e}{x^{2}} \right )}}{2 e f} & \text {otherwise} \end {cases}\right ) + \left (\begin {cases} 0^{m} x & \text {for}\: f = 0 \\\frac {\begin {cases} \frac {\left (f x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (f x \right )} & \text {otherwise} \end {cases}}{f} & \text {otherwise} \end {cases}\right ) \log {\left (c \left (d + \frac {e}{x^{2}}\right )^{p} \right )} \]
2*e*p*Piecewise((-0**m*sqrt(-1/(d*e))*log(-e*sqrt(-1/(d*e)) + x)/2 + 0**m* sqrt(-1/(d*e))*log(e*sqrt(-1/(d*e)) + x)/2, Eq(f, 0) | (Eq(f, 0) & Ne(m, - 1))), (f**(m + 1)*m*x**(m - 1)*lerchphi(e*exp_polar(I*pi)/(d*x**2), 1, 1/2 - m/2)*gamma(1/2 - m/2)/(4*d*f*m*gamma(3/2 - m/2) + 4*d*f*gamma(3/2 - m/2 )) - f**(m + 1)*x**(m - 1)*lerchphi(e*exp_polar(I*pi)/(d*x**2), 1, 1/2 - m /2)*gamma(1/2 - m/2)/(4*d*f*m*gamma(3/2 - m/2) + 4*d*f*gamma(3/2 - m/2)), (m > -oo) & (m < oo) & Ne(m, -1)), (Piecewise((polylog(2, e*exp_polar(I*pi )/(d*x**2))/2, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log(x) + polylog(2, e*exp_polar(I*pi)/(d*x**2))/2, Abs(x) < 1), (-log(d)*log(1/x) + polylog(2 , e*exp_polar(I*pi)/(d*x**2))/2, 1/Abs(x) < 1), (-meijerg(((), (1, 1)), (( 0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) + po lylog(2, e*exp_polar(I*pi)/(d*x**2))/2, True))/(2*e*f) - log(f*x)*log(d + e/x**2)/(2*e*f), True)) + Piecewise((0**m*x, Eq(f, 0)), (Piecewise(((f*x)* *(m + 1)/(m + 1), Ne(m, -1)), (log(f*x), True))/f, True))*log(c*(d + e/x** 2)**p)
\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x^{2}}\right )}^{p}\right ) \,d x } \]
(f^m*x*x^m*log((d*x^2 + e)^p) - 2*f^m*x*x^m*log(x^p))/(m + 1) + integrate( (d*f^m*(m + 1)*x^2*log(c) + e*f^m*(m + 1)*log(c) + 2*e*f^m*p)*x^m/(d*(m + 1)*x^2 + e*(m + 1)), x)
\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x^{2}}\right )}^{p}\right ) \,d x } \]
Timed out. \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^2}\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+\frac {e}{x^2}\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]