Integrand size = 18, antiderivative size = 81 \[ \int (f x)^m \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {2 e p (f x)^{3+m} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )}{d f^3 (1+m) (3+m)}+\frac {(f x)^{1+m} \log \left (c \left (d+e x^2\right )^p\right )}{f (1+m)} \]
-2*e*p*(f*x)^(3+m)*hypergeom([1, 3/2+1/2*m],[5/2+1/2*m],-e*x^2/d)/d/f^3/(1 +m)/(3+m)+(f*x)^(1+m)*ln(c*(e*x^2+d)^p)/f/(1+m)
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int (f x)^m \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {x (f x)^m \left (-2 e p x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{2},\frac {5+m}{2},-\frac {e x^2}{d}\right )+d (3+m) \log \left (c \left (d+e x^2\right )^p\right )\right )}{d (1+m) (3+m)} \]
(x*(f*x)^m*(-2*e*p*x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2 )/d)] + d*(3 + m)*Log[c*(d + e*x^2)^p]))/(d*(1 + m)*(3 + m))
Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2905, 8, 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.
\(\displaystyle \int (f x)^m \log \left (c \left (d+e x^2\right )^p\right ) \, dx\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle \frac {(f x)^{m+1} \log \left (c \left (d+e x^2\right )^p\right )}{f (m+1)}-\frac {2 e p \int \frac {x (f x)^{m+1}}{e x^2+d}dx}{f (m+1)}\) |
\(\Big \downarrow \) 8 |
\(\displaystyle \frac {(f x)^{m+1} \log \left (c \left (d+e x^2\right )^p\right )}{f (m+1)}-\frac {2 e p \int \frac {(f x)^{m+2}}{e x^2+d}dx}{f^2 (m+1)}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {(f x)^{m+1} \log \left (c \left (d+e x^2\right )^p\right )}{f (m+1)}-\frac {2 e p (f x)^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{2},\frac {m+5}{2},-\frac {e x^2}{d}\right )}{d f^3 (m+1) (m+3)}\) |
(-2*e*p*(f*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -((e*x^2) /d)])/(d*f^3*(1 + m)*(3 + m)) + ((f*x)^(1 + m)*Log[c*(d + e*x^2)^p])/(f*(1 + m))
3.1.56.3.1 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[((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]
\[\int \left (f x \right )^{m} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )d x\]
\[ \int (f x)^m \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \,d x } \]
Time = 29.45 (sec) , antiderivative size = 377, normalized size of antiderivative = 4.65 \[ \int (f x)^m \log \left (c \left (d+e x^2\right )^p\right ) \, dx=- 2 e p \left (\begin {cases} \frac {0^{m} \sqrt {- \frac {d}{e^{3}}} \log {\left (- e \sqrt {- \frac {d}{e^{3}}} + x \right )}}{2} - \frac {0^{m} \sqrt {- \frac {d}{e^{3}}} \log {\left (e \sqrt {- \frac {d}{e^{3}}} + x \right )}}{2} + \frac {0^{m} x}{e} & \text {for}\: \left (f = 0 \wedge m \neq -1\right ) \vee f = 0 \\\frac {f^{m + 1} m x^{m + 3} \Phi \left (\frac {e x^{2} e^{i \pi }}{d}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 d f m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 4 d f \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 f^{m + 1} x^{m + 3} \Phi \left (\frac {e x^{2} e^{i \pi }}{d}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 d f m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 4 d f \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq -1 \\- \frac {\begin {cases} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\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 x^{2} e^{i \pi }}{d}\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 x^{2} e^{i \pi }}{d}\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 x^{2} e^{i \pi }}{d}\right )}{2} & \text {otherwise} \end {cases}}{2 e f} + \frac {\log {\left (f x \right )} \log {\left (d + 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 + e x^{2}\right )^{p} \right )} \]
-2*e*p*Piecewise((0**m*sqrt(-d/e**3)*log(-e*sqrt(-d/e**3) + x)/2 - 0**m*sq rt(-d/e**3)*log(e*sqrt(-d/e**3) + x)/2 + 0**m*x/e, Eq(f, 0) | (Eq(f, 0) & Ne(m, -1))), (f**(m + 1)*m*x**(m + 3)*lerchphi(e*x**2*exp_polar(I*pi)/d, 1 , m/2 + 3/2)*gamma(m/2 + 3/2)/(4*d*f*m*gamma(m/2 + 5/2) + 4*d*f*gamma(m/2 + 5/2)) + 3*f**(m + 1)*x**(m + 3)*lerchphi(e*x**2*exp_polar(I*pi)/d, 1, m/ 2 + 3/2)*gamma(m/2 + 3/2)/(4*d*f*m*gamma(m/2 + 5/2) + 4*d*f*gamma(m/2 + 5/ 2)), (m > -oo) & (m < oo) & Ne(m, -1)), (-Piecewise((-polylog(2, e*x**2*ex p_polar(I*pi)/d)/2, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log(x) - polyl og(2, e*x**2*exp_polar(I*pi)/d)/2, Abs(x) < 1), (-log(d)*log(1/x) - polylo g(2, e*x**2*exp_polar(I*pi)/d)/2, 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ( (0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - p olylog(2, e*x**2*exp_polar(I*pi)/d)/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)
\[ \int (f x)^m \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \,d x } \]
f^m*x*x^m*log((e*x^2 + d)^p)/(m + 1) + integrate((d*f^m*(m + 1)*log(c) + ( e*f^m*(m + 1)*log(c) - 2*e*f^m*p)*x^2)*x^m/(e*(m + 1)*x^2 + d*(m + 1)), x)
\[ \int (f x)^m \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \,d x } \]
Timed out. \[ \int (f x)^m \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\int \ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]