Integrand size = 23, antiderivative size = 124 \[ \int x \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {(e f-d g)^2 p x^2}{6 e^2}-\frac {(e f-d g) p \left (f+g x^2\right )^2}{12 e g}-\frac {p \left (f+g x^2\right )^3}{18 g}-\frac {(e f-d g)^3 p \log \left (d+e x^2\right )}{6 e^3 g}+\frac {\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{6 g} \]
-1/6*(-d*g+e*f)^2*p*x^2/e^2-1/12*(-d*g+e*f)*p*(g*x^2+f)^2/e/g-1/18*p*(g*x^ 2+f)^3/g-1/6*(-d*g+e*f)^3*p*ln(e*x^2+d)/e^3/g+1/6*(g*x^2+f)^3*ln(c*(e*x^2+ d)^p)/g
Time = 0.07 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.09 \[ \int x \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {6 d^2 g (-3 e f+d g) p \log \left (d+e x^2\right )+e \left (-p x^2 \left (6 d^2 g^2-3 d e g \left (6 f+g x^2\right )+e^2 \left (18 f^2+9 f g x^2+2 g^2 x^4\right )\right )+6 e \left (3 d f^2+e x^2 \left (3 f^2+3 f g x^2+g^2 x^4\right )\right ) \log \left (c \left (d+e x^2\right )^p\right )\right )}{36 e^3} \]
(6*d^2*g*(-3*e*f + d*g)*p*Log[d + e*x^2] + e*(-(p*x^2*(6*d^2*g^2 - 3*d*e*g *(6*f + g*x^2) + e^2*(18*f^2 + 9*f*g*x^2 + 2*g^2*x^4))) + 6*e*(3*d*f^2 + e *x^2*(3*f^2 + 3*f*g*x^2 + g^2*x^4))*Log[c*(d + e*x^2)^p]))/(36*e^3)
Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2925, 2842, 49, 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.
| \(\displaystyle \int x \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {1}{2} \int \left (g x^2+f\right )^2 \log \left (c \left (e x^2+d\right )^p\right )dx^2\) |
| \(\Big \downarrow \) 2842 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}-\frac {e p \int \frac {\left (g x^2+f\right )^3}{e x^2+d}dx^2}{3 g}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}-\frac {e p \int \left (\frac {(e f-d g)^3}{e^3 \left (e x^2+d\right )}+\frac {g (e f-d g)^2}{e^3}+\frac {g \left (g x^2+f\right ) (e f-d g)}{e^2}+\frac {g \left (g x^2+f\right )^2}{e}\right )dx^2}{3 g}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (f+g x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}-\frac {e p \left (\frac {(e f-d g)^3 \log \left (d+e x^2\right )}{e^4}+\frac {g x^2 (e f-d g)^2}{e^3}+\frac {\left (f+g x^2\right )^2 (e f-d g)}{2 e^2}+\frac {\left (f+g x^2\right )^3}{3 e}\right )}{3 g}\right )\) |
(-1/3*(e*p*((g*(e*f - d*g)^2*x^2)/e^3 + ((e*f - d*g)*(f + g*x^2)^2)/(2*e^2 ) + (f + g*x^2)^3/(3*e) + ((e*f - d*g)^3*Log[d + e*x^2])/e^4))/g + ((f + g *x^2)^3*Log[c*(d + e*x^2)^p])/(3*g))/2
3.4.25.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ ))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1))) Int[(f + g*x)^(q + 1)/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
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 = 1.81 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.66
| method | result | size |
| parts | \(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) g^{2} x^{6}}{6}+\frac {f g \,x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2}+\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f^{2} x^{2}}{2}+\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f^{3}}{6 g}-\frac {p e \left (\frac {g \left (\frac {1}{3} e^{2} g^{2} x^{6}-\frac {1}{2} d e \,g^{2} x^{4}+\frac {3}{2} e^{2} f g \,x^{4}+d^{2} g^{2} x^{2}-3 d e f g \,x^{2}+3 e^{2} f^{2} x^{2}\right )}{2 e^{3}}+\frac {\left (-d^{3} g^{3}+3 d^{2} e f \,g^{2}-3 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{4}}\right )}{3 g}\) | \(206\) |
| parallelrisch | \(\frac {6 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{3} g^{2}-2 x^{6} e^{3} g^{2} p +18 x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{3} f g +3 x^{4} d \,e^{2} g^{2} p -9 x^{4} e^{3} f g p +18 x^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{3} f^{2}-6 x^{2} d^{2} e \,g^{2} p +18 x^{2} d \,e^{2} f g p -18 x^{2} e^{3} f^{2} p +6 \ln \left (e \,x^{2}+d \right ) d^{3} g^{2} p -18 \ln \left (e \,x^{2}+d \right ) d^{2} e f g p +36 \ln \left (e \,x^{2}+d \right ) d \,e^{2} f^{2} p -18 \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d \,e^{2} f^{2}+6 d^{3} g^{2} p -18 d^{2} e f g p +18 d \,e^{2} f^{2} p}{36 e^{3}}\) | \(249\) |
| risch | \(\frac {\ln \left (c \right ) f^{2} x^{2}}{2}+\frac {g^{2} \ln \left (c \right ) x^{6}}{6}+\frac {g^{2} d p \,x^{4}}{12 e}+\frac {i g^{2} \pi \,x^{6} \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}}{12}+\frac {i g^{2} \pi \,x^{6} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{12}-\frac {i g \pi f \,x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{4}+\frac {i \pi \,f^{2} 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}}{4}+\frac {i \pi \,f^{2} x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{4}-\frac {i \pi \,f^{2} x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{4}-\frac {i g^{2} \pi \,x^{6} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{12}+\frac {\left (g \,x^{2}+f \right )^{3} \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{6 g}-\frac {g^{2} d^{2} p \,x^{2}}{6 e^{2}}-\frac {f^{2} p \,x^{2}}{2}+\frac {\ln \left (e \,x^{2}+d \right ) d \,f^{2} p}{2 e}+\frac {g^{2} \ln \left (e \,x^{2}+d \right ) d^{3} p}{6 e^{3}}-\frac {g^{2} p \,x^{6}}{18}+\frac {g \ln \left (c \right ) f \,x^{4}}{2}-\frac {\ln \left (e \,x^{2}+d \right ) f^{3} p}{6 g}-\frac {i g \pi f \,x^{4} \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 )}{4}-\frac {i g^{2} \pi \,x^{6} \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 )}{12}+\frac {i g \pi f \,x^{4} \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}}{4}+\frac {i g \pi f \,x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{4}-\frac {i \pi \,f^{2} 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 )}{4}+\frac {d f g p \,x^{2}}{2 e}-\frac {d^{2} f g p \ln \left (e \,x^{2}+d \right )}{2 e^{2}}-\frac {f g p \,x^{4}}{4}\) | \(605\) |
1/6*ln(c*(e*x^2+d)^p)*g^2*x^6+1/2*f*g*x^4*ln(c*(e*x^2+d)^p)+1/2*ln(c*(e*x^ 2+d)^p)*f^2*x^2+1/6*ln(c*(e*x^2+d)^p)/g*f^3-1/3/g*p*e*(1/2*g/e^3*(1/3*e^2* g^2*x^6-1/2*d*e*g^2*x^4+3/2*e^2*f*g*x^4+d^2*g^2*x^2-3*d*e*f*g*x^2+3*e^2*f^ 2*x^2)+1/2*(-d^3*g^3+3*d^2*e*f*g^2-3*d*e^2*f^2*g+e^3*f^3)/e^4*ln(e*x^2+d))
Time = 0.32 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.45 \[ \int x \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {2 \, e^{3} g^{2} p x^{6} + 3 \, {\left (3 \, e^{3} f g - d e^{2} g^{2}\right )} p x^{4} + 6 \, {\left (3 \, e^{3} f^{2} - 3 \, d e^{2} f g + d^{2} e g^{2}\right )} p x^{2} - 6 \, {\left (e^{3} g^{2} p x^{6} + 3 \, e^{3} f g p x^{4} + 3 \, e^{3} f^{2} p x^{2} + {\left (3 \, d e^{2} f^{2} - 3 \, d^{2} e f g + d^{3} g^{2}\right )} p\right )} \log \left (e x^{2} + d\right ) - 6 \, {\left (e^{3} g^{2} x^{6} + 3 \, e^{3} f g x^{4} + 3 \, e^{3} f^{2} x^{2}\right )} \log \left (c\right )}{36 \, e^{3}} \]
-1/36*(2*e^3*g^2*p*x^6 + 3*(3*e^3*f*g - d*e^2*g^2)*p*x^4 + 6*(3*e^3*f^2 - 3*d*e^2*f*g + d^2*e*g^2)*p*x^2 - 6*(e^3*g^2*p*x^6 + 3*e^3*f*g*p*x^4 + 3*e^ 3*f^2*p*x^2 + (3*d*e^2*f^2 - 3*d^2*e*f*g + d^3*g^2)*p)*log(e*x^2 + d) - 6* (e^3*g^2*x^6 + 3*e^3*f*g*x^4 + 3*e^3*f^2*x^2)*log(c))/e^3
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (104) = 208\).
Time = 60.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.90 \[ \int x \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\begin {cases} \frac {d^{3} g^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{6 e^{3}} - \frac {d^{2} f g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2 e^{2}} - \frac {d^{2} g^{2} p x^{2}}{6 e^{2}} + \frac {d f^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2 e} + \frac {d f g p x^{2}}{2 e} + \frac {d g^{2} p x^{4}}{12 e} - \frac {f^{2} p x^{2}}{2} + \frac {f^{2} x^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2} - \frac {f g p x^{4}}{4} + \frac {f g x^{4} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2} - \frac {g^{2} p x^{6}}{18} + \frac {g^{2} x^{6} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{6} & \text {for}\: e \neq 0 \\\left (\frac {f^{2} x^{2}}{2} + \frac {f g x^{4}}{2} + \frac {g^{2} x^{6}}{6}\right ) \log {\left (c d^{p} \right )} & \text {otherwise} \end {cases} \]
Piecewise((d**3*g**2*log(c*(d + e*x**2)**p)/(6*e**3) - d**2*f*g*log(c*(d + e*x**2)**p)/(2*e**2) - d**2*g**2*p*x**2/(6*e**2) + d*f**2*log(c*(d + e*x* *2)**p)/(2*e) + d*f*g*p*x**2/(2*e) + d*g**2*p*x**4/(12*e) - f**2*p*x**2/2 + f**2*x**2*log(c*(d + e*x**2)**p)/2 - f*g*p*x**4/4 + f*g*x**4*log(c*(d + e*x**2)**p)/2 - g**2*p*x**6/18 + g**2*x**6*log(c*(d + e*x**2)**p)/6, Ne(e, 0)), ((f**2*x**2/2 + f*g*x**4/2 + g**2*x**6/6)*log(c*d**p), True))
Time = 0.19 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.23 \[ \int x \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {{\left (g x^{2} + f\right )}^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{6 \, g} - \frac {e p {\left (\frac {2 \, e^{2} g^{3} x^{6} + 3 \, {\left (3 \, e^{2} f g^{2} - d e g^{3}\right )} x^{4} + 6 \, {\left (3 \, e^{2} f^{2} g - 3 \, d e f g^{2} + d^{2} g^{3}\right )} x^{2}}{e^{3}} + \frac {6 \, {\left (e^{3} f^{3} - 3 \, d e^{2} f^{2} g + 3 \, d^{2} e f g^{2} - d^{3} g^{3}\right )} \log \left (e x^{2} + d\right )}{e^{4}}\right )}}{36 \, g} \]
1/6*(g*x^2 + f)^3*log((e*x^2 + d)^p*c)/g - 1/36*e*p*((2*e^2*g^3*x^6 + 3*(3 *e^2*f*g^2 - d*e*g^3)*x^4 + 6*(3*e^2*f^2*g - 3*d*e*f*g^2 + d^2*g^3)*x^2)/e ^3 + 6*(e^3*f^3 - 3*d*e^2*f^2*g + 3*d^2*e*f*g^2 - d^3*g^3)*log(e*x^2 + d)/ e^4)/g
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (114) = 228\).
Time = 0.30 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.74 \[ \int x \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {{\left (e x^{2} + d\right )}^{2} f g p \log \left (e x^{2} + d\right )}{2 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{3} g^{2} p \log \left (e x^{2} + d\right )}{6 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} d g^{2} p \log \left (e x^{2} + d\right )}{2 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} f g p}{4 \, e^{2}} - \frac {{\left (e x^{2} + d\right )}^{3} g^{2} p}{18 \, e^{3}} + \frac {{\left (e x^{2} + d\right )}^{2} d g^{2} p}{4 \, e^{3}} + \frac {{\left (e x^{2} + d\right )}^{2} f g \log \left (c\right )}{2 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{3} g^{2} \log \left (c\right )}{6 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} d g^{2} \log \left (c\right )}{2 \, e^{3}} - \frac {{\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} e^{2} f^{2} p - 2 \, {\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d e f g p + {\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d^{2} g^{2} p - {\left (e x^{2} + d\right )} e^{2} f^{2} \log \left (c\right ) + 2 \, {\left (e x^{2} + d\right )} d e f g \log \left (c\right ) - {\left (e x^{2} + d\right )} d^{2} g^{2} \log \left (c\right )}{2 \, e^{3}} \]
1/2*(e*x^2 + d)^2*f*g*p*log(e*x^2 + d)/e^2 + 1/6*(e*x^2 + d)^3*g^2*p*log(e *x^2 + d)/e^3 - 1/2*(e*x^2 + d)^2*d*g^2*p*log(e*x^2 + d)/e^3 - 1/4*(e*x^2 + d)^2*f*g*p/e^2 - 1/18*(e*x^2 + d)^3*g^2*p/e^3 + 1/4*(e*x^2 + d)^2*d*g^2* p/e^3 + 1/2*(e*x^2 + d)^2*f*g*log(c)/e^2 + 1/6*(e*x^2 + d)^3*g^2*log(c)/e^ 3 - 1/2*(e*x^2 + d)^2*d*g^2*log(c)/e^3 - 1/2*((e*x^2 - (e*x^2 + d)*log(e*x ^2 + d) + d)*e^2*f^2*p - 2*(e*x^2 - (e*x^2 + d)*log(e*x^2 + d) + d)*d*e*f* g*p + (e*x^2 - (e*x^2 + d)*log(e*x^2 + d) + d)*d^2*g^2*p - (e*x^2 + d)*e^2 *f^2*log(c) + 2*(e*x^2 + d)*d*e*f*g*log(c) - (e*x^2 + d)*d^2*g^2*log(c))/e ^3
Time = 1.64 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15 \[ \int x \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2\,x^2}{2}+\frac {f\,g\,x^4}{2}+\frac {g^2\,x^6}{6}\right )-x^2\,\left (\frac {f^2\,p}{2}-\frac {d\,\left (f\,g\,p-\frac {d\,g^2\,p}{3\,e}\right )}{2\,e}\right )-x^4\,\left (\frac {f\,g\,p}{4}-\frac {d\,g^2\,p}{12\,e}\right )+\frac {\ln \left (e\,x^2+d\right )\,\left (p\,d^3\,g^2-3\,p\,d^2\,e\,f\,g+3\,p\,d\,e^2\,f^2\right )}{6\,e^3}-\frac {g^2\,p\,x^6}{18} \]