Integrand size = 23, antiderivative size = 93 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=-\frac {e f p}{4 d x^2}-\frac {e (e f-2 d g) p \log (x)}{2 d^2}+\frac {(e f-d g)^2 p \log \left (d+e x^2\right )}{4 d^2 f}-\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 f x^4} \] Output:
-1/4*e*f*p/d/x^2-1/2*e*(-2*d*g+e*f)*p*ln(x)/d^2+1/4*(-d*g+e*f)^2*p*ln(e*x^ 2+d)/d^2/f-1/4*(g*x^2+f)^2*ln(c*(e*x^2+d)^p)/f/x^4
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.13 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\frac {e g p \log (x)}{d}-\frac {e g p \log \left (d+e x^2\right )}{2 d}+\frac {1}{4} e f p \left (-\frac {1}{d x^2}-\frac {2 e \log (x)}{d^2}+\frac {e \log \left (d+e x^2\right )}{d^2}\right )-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2} \] Input:
Integrate[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^5,x]
Output:
(e*g*p*Log[x])/d - (e*g*p*Log[d + e*x^2])/(2*d) + (e*f*p*(-(1/(d*x^2)) - ( 2*e*Log[x])/d^2 + (e*Log[d + e*x^2])/d^2))/4 - (f*Log[c*(d + e*x^2)^p])/(4 *x^4) - (g*Log[c*(d + e*x^2)^p])/(2*x^2)
Time = 0.60 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2925, 2861, 27, 99, 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.
\(\displaystyle \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 2925 |
\(\displaystyle \frac {1}{2} \int \frac {\left (g x^2+f\right ) \log \left (c \left (e x^2+d\right )^p\right )}{x^6}dx^2\) |
\(\Big \downarrow \) 2861 |
\(\displaystyle \frac {1}{2} \left (-e p \int -\frac {\left (g x^2+f\right )^2}{2 f x^4 \left (e x^2+d\right )}dx^2-\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 f x^4}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {e p \int \frac {\left (g x^2+f\right )^2}{x^4 \left (e x^2+d\right )}dx^2}{2 f}-\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 f x^4}\right )\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{2} \left (\frac {e p \int \left (\frac {f^2}{d x^4}+\frac {(2 d g-e f) f}{d^2 x^2}+\frac {(d g-e f)^2}{d^2 \left (e x^2+d\right )}\right )dx^2}{2 f}-\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 f x^4}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {e p \left (-\frac {f \log \left (x^2\right ) (e f-2 d g)}{d^2}+\frac {(e f-d g)^2 \log \left (d+e x^2\right )}{d^2 e}-\frac {f^2}{d x^2}\right )}{2 f}-\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 f x^4}\right )\) |
Input:
Int[((f + g*x^2)*Log[c*(d + e*x^2)^p])/x^5,x]
Output:
((e*p*(-(f^2/(d*x^2)) - (f*(e*f - 2*d*g)*Log[x^2])/d^2 + ((e*f - d*g)^2*Lo g[d + e*x^2])/(d^2*e)))/(2*f) - ((f + g*x^2)^2*Log[c*(d + e*x^2)^p])/(2*f* x^4))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(x_)^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(f + g*x^r)^q, x]}, Simp[(a + b*Log[c*(d + e*x)^n]) u, x] - Simp[b*e*n Int[SimplifyIn tegrand[u/(d + e*x), x], x], x] /; InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q, r}, x] && IntegerQ[m] && IntegerQ[q] && Integer Q[r]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Si mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 ] || IGtQ[q, 0])
Time = 1.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95
method | result | size |
parts | \(-\frac {g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2 x^{2}}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f}{4 x^{4}}-\frac {e p \left (\frac {\left (2 d g -e f \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{2}}+\frac {\left (-2 d g +e f \right ) \ln \left (x \right )}{d^{2}}+\frac {f}{2 d \,x^{2}}\right )}{2}\) | \(88\) |
parallelrisch | \(\frac {4 \ln \left (x \right ) x^{4} d e g \,p^{2}-2 \ln \left (x \right ) x^{4} e^{2} f \,p^{2}-2 x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d e g p +x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{2} f p +x^{4} e^{2} f \,p^{2}-2 x^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{2} g p -x^{2} d e f \,p^{2}-\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{2} f p}{4 x^{4} p \,d^{2}}\) | \(145\) |
risch | \(-\frac {\left (2 g \,x^{2}+f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{4 x^{4}}-\frac {2 i \pi \,d^{2} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-2 i \pi \,d^{2} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-2 i \pi \,d^{2} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+2 i \pi \,d^{2} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-8 \ln \left (x \right ) d e g p \,x^{4}+4 \ln \left (x \right ) e^{2} f p \,x^{4}+4 \ln \left (e \,x^{2}+d \right ) d e g p \,x^{4}-2 \ln \left (e \,x^{2}+d \right ) e^{2} f p \,x^{4}+i \pi \,d^{2} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-i \pi \,d^{2} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi \,d^{2} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+i \pi \,d^{2} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+4 \ln \left (c \right ) d^{2} g \,x^{2}+2 d e f p \,x^{2}+2 \ln \left (c \right ) d^{2} f}{8 d^{2} x^{4}}\) | \(392\) |
Input:
int((g*x^2+f)*ln(c*(e*x^2+d)^p)/x^5,x,method=_RETURNVERBOSE)
Output:
-1/2*g/x^2*ln(c*(e*x^2+d)^p)-1/4*ln(c*(e*x^2+d)^p)/x^4*f-1/2*e*p*(1/2*(2*d *g-e*f)/d^2*ln(e*x^2+d)+(-2*d*g+e*f)/d^2*ln(x)+1/2/d*f/x^2)
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.04 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=-\frac {2 \, {\left (e^{2} f - 2 \, d e g\right )} p x^{4} \log \left (x\right ) + d e f p x^{2} + {\left (2 \, d^{2} g p x^{2} - {\left (e^{2} f - 2 \, d e g\right )} p x^{4} + d^{2} f p\right )} \log \left (e x^{2} + d\right ) + {\left (2 \, d^{2} g x^{2} + d^{2} f\right )} \log \left (c\right )}{4 \, d^{2} x^{4}} \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^5,x, algorithm="fricas")
Output:
-1/4*(2*(e^2*f - 2*d*e*g)*p*x^4*log(x) + d*e*f*p*x^2 + (2*d^2*g*p*x^2 - (e ^2*f - 2*d*e*g)*p*x^4 + d^2*f*p)*log(e*x^2 + d) + (2*d^2*g*x^2 + d^2*f)*lo g(c))/(d^2*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (83) = 166\).
Time = 78.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.80 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\begin {cases} - \frac {f \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{4 x^{4}} - \frac {g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2 x^{2}} - \frac {e f p}{4 d x^{2}} + \frac {e g p \log {\left (x \right )}}{d} - \frac {e g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2 d} - \frac {e^{2} f p \log {\left (x \right )}}{2 d^{2}} + \frac {e^{2} f \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{4 d^{2}} & \text {for}\: d \neq 0 \\- \frac {f p}{8 x^{4}} - \frac {f \log {\left (c \left (e x^{2}\right )^{p} \right )}}{4 x^{4}} - \frac {g p}{2 x^{2}} - \frac {g \log {\left (c \left (e x^{2}\right )^{p} \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \] Input:
integrate((g*x**2+f)*ln(c*(e*x**2+d)**p)/x**5,x)
Output:
Piecewise((-f*log(c*(d + e*x**2)**p)/(4*x**4) - g*log(c*(d + e*x**2)**p)/( 2*x**2) - e*f*p/(4*d*x**2) + e*g*p*log(x)/d - e*g*log(c*(d + e*x**2)**p)/( 2*d) - e**2*f*p*log(x)/(2*d**2) + e**2*f*log(c*(d + e*x**2)**p)/(4*d**2), Ne(d, 0)), (-f*p/(8*x**4) - f*log(c*(e*x**2)**p)/(4*x**4) - g*p/(2*x**2) - g*log(c*(e*x**2)**p)/(2*x**2), True))
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.83 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\frac {1}{4} \, e p {\left (\frac {{\left (e f - 2 \, d g\right )} \log \left (e x^{2} + d\right )}{d^{2}} - \frac {{\left (e f - 2 \, d g\right )} \log \left (x^{2}\right )}{d^{2}} - \frac {f}{d x^{2}}\right )} - \frac {{\left (2 \, g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{4 \, x^{4}} \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^5,x, algorithm="maxima")
Output:
1/4*e*p*((e*f - 2*d*g)*log(e*x^2 + d)/d^2 - (e*f - 2*d*g)*log(x^2)/d^2 - f /(d*x^2)) - 1/4*(2*g*x^2 + f)*log((e*x^2 + d)^p*c)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (85) = 170\).
Time = 0.13 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.25 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=-\frac {\frac {{\left (e^{3} f p + 2 \, {\left (e x^{2} + d\right )} e^{2} g p - 2 \, d e^{2} g p\right )} \log \left (e x^{2} + d\right )}{{\left (e x^{2} + d\right )}^{2} - 2 \, {\left (e x^{2} + d\right )} d + d^{2}} + \frac {{\left (e x^{2} + d\right )} e^{3} f p - d e^{3} f p + d e^{3} f \log \left (c\right ) + 2 \, {\left (e x^{2} + d\right )} d e^{2} g \log \left (c\right ) - 2 \, d^{2} e^{2} g \log \left (c\right )}{{\left (e x^{2} + d\right )}^{2} d - 2 \, {\left (e x^{2} + d\right )} d^{2} + d^{3}} - \frac {{\left (e^{3} f p - 2 \, d e^{2} g p\right )} \log \left (e x^{2} + d\right )}{d^{2}} + \frac {{\left (e^{3} f p - 2 \, d e^{2} g p\right )} \log \left (e x^{2}\right )}{d^{2}}}{4 \, e} \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)/x^5,x, algorithm="giac")
Output:
-1/4*((e^3*f*p + 2*(e*x^2 + d)*e^2*g*p - 2*d*e^2*g*p)*log(e*x^2 + d)/((e*x ^2 + d)^2 - 2*(e*x^2 + d)*d + d^2) + ((e*x^2 + d)*e^3*f*p - d*e^3*f*p + d* e^3*f*log(c) + 2*(e*x^2 + d)*d*e^2*g*log(c) - 2*d^2*e^2*g*log(c))/((e*x^2 + d)^2*d - 2*(e*x^2 + d)*d^2 + d^3) - (e^3*f*p - 2*d*e^2*g*p)*log(e*x^2 + d)/d^2 + (e^3*f*p - 2*d*e^2*g*p)*log(e*x^2)/d^2)/e
Time = 26.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.91 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\frac {\ln \left (e\,x^2+d\right )\,\left (e^2\,f\,p-2\,d\,e\,g\,p\right )}{4\,d^2}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {g\,x^2}{2}+\frac {f}{4}\right )}{x^4}-\frac {\ln \left (x\right )\,\left (e^2\,f\,p-2\,d\,e\,g\,p\right )}{2\,d^2}-\frac {e\,f\,p}{4\,d\,x^2} \] Input:
int((log(c*(d + e*x^2)^p)*(f + g*x^2))/x^5,x)
Output:
(log(d + e*x^2)*(e^2*f*p - 2*d*e*g*p))/(4*d^2) - (log(c*(d + e*x^2)^p)*(f/ 4 + (g*x^2)/2))/x^4 - (log(x)*(e^2*f*p - 2*d*e*g*p))/(2*d^2) - (e*f*p)/(4* d*x^2)
Time = 0.17 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.29 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^5} \, dx=\frac {-\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{2} f -2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{2} g \,x^{2}-2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d e g \,x^{4}+\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{2} f \,x^{4}+4 \,\mathrm {log}\left (x \right ) d e g p \,x^{4}-2 \,\mathrm {log}\left (x \right ) e^{2} f p \,x^{4}-d e f p \,x^{2}}{4 d^{2} x^{4}} \] Input:
int((g*x^2+f)*log(c*(e*x^2+d)^p)/x^5,x)
Output:
( - log((d + e*x**2)**p*c)*d**2*f - 2*log((d + e*x**2)**p*c)*d**2*g*x**2 - 2*log((d + e*x**2)**p*c)*d*e*g*x**4 + log((d + e*x**2)**p*c)*e**2*f*x**4 + 4*log(x)*d*e*g*p*x**4 - 2*log(x)*e**2*f*p*x**4 - d*e*f*p*x**2)/(4*d**2*x **4)