Integrand size = 24, antiderivative size = 835 \[ \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}-\frac {2 d f g p^2 x^2}{e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {8}{343} g^2 p^2 x^7+\frac {f g p^2 \left (d+e x^2\right )^2}{4 e^2}-\frac {8 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\frac {4 i \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {8 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {8 d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{7 e^{7/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d f g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {f g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {4 \sqrt {d} f^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {4 d^{7/2} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {4 i \sqrt {d} f^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{7 e^{7/2}} \]
-d*f*g*(e*x^2+d)*ln(c*(e*x^2+d)^p)^2/e^2-4/49*g^2*p*x^7*ln(c*(e*x^2+d)^p)- 4*f^2*p*x*ln(c*(e*x^2+d)^p)+8/343*g^2*p^2*x^7+1/7*g^2*x^7*ln(c*(e*x^2+d)^p )^2+8*f^2*p^2*x-1408/735*d^3*g^2*p^2*x/e^3+568/2205*d^2*g^2*p^2*x^3/e^2-96 /1225*d*g^2*p^2*x^5/e+1/4*f*g*p^2*(e*x^2+d)^2/e^2+1408/735*d^(7/2)*g^2*p^2 *arctan(x*e^(1/2)/d^(1/2))/e^(7/2)+1/2*f*g*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)^2 /e^2+4*f^2*p*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)*d^(1/2)/e^(1/2)+8 *f^2*p^2*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))*d^( 1/2)/e^(1/2)-8/7*d^(7/2)*g^2*p^2*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d ^(1/2)+I*x*e^(1/2)))/e^(7/2)-4/7*I*d^(7/2)*g^2*p^2*arctan(x*e^(1/2)/d^(1/2 ))^2/e^(7/2)-4/7*I*d^(7/2)*g^2*p^2*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x*e^(1 /2)))/e^(7/2)-2*d*f*g*p^2*x^2/e+4/7*d^3*g^2*p*x*ln(c*(e*x^2+d)^p)/e^3-4/21 *d^2*g^2*p*x^3*ln(c*(e*x^2+d)^p)/e^2+4/35*d*g^2*p*x^5*ln(c*(e*x^2+d)^p)/e- 1/2*f*g*p*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)/e^2-4/7*d^(7/2)*g^2*p*arctan(x*e^( 1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)/e^(7/2)+f^2*x*ln(c*(e*x^2+d)^p)^2+2*d*f*g* p*(e*x^2+d)*ln(c*(e*x^2+d)^p)/e^2+4*I*f^2*p^2*arctan(x*e^(1/2)/d^(1/2))^2* d^(1/2)/e^(1/2)+4*I*f^2*p^2*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))*d ^(1/2)/e^(1/2)-8*f^2*p^2*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/e^(1/2)
Time = 0.34 (sec) , antiderivative size = 475, normalized size of antiderivative = 0.57 \[ \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {-176400 i \sqrt {d} \left (-7 e^3 f^2+d^3 g^2\right ) p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2-1680 \sqrt {d} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (2 \left (735 e^3 f^2-176 d^3 g^2\right ) p-210 \left (7 e^3 f^2-d^3 g^2\right ) p \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )-105 \left (7 e^3 f^2-d^3 g^2\right ) \log \left (c \left (d+e x^2\right )^p\right )\right )+\sqrt {e} \left (p^2 x \left (-591360 d^3 g^2+79520 d^2 e g^2 x^2-378 d e^2 g x \left (1225 f+64 g x^3\right )+225 e^3 \left (10976 f^2+343 f g x^3+32 g^2 x^6\right )\right )+154350 d^2 e f g p^2 \log \left (d+e x^2\right )-210 p \left (-840 d^3 g^2 x+70 d^2 e g \left (-21 f+4 g x^3\right )-42 d e^2 g x^2 \left (35 f+4 g x^3\right )+15 e^3 x \left (392 f^2+49 f g x^3+8 g^2 x^6\right )\right ) \log \left (c \left (d+e x^2\right )^p\right )+22050 \left (-7 d^2 e f g+e^3 x \left (14 f^2+7 f g x^3+2 g^2 x^6\right )\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )\right )-176400 i \sqrt {d} \left (-7 e^3 f^2+d^3 g^2\right ) p^2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )}{308700 e^{7/2}} \]
((-176400*I)*Sqrt[d]*(-7*e^3*f^2 + d^3*g^2)*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d] ]^2 - 1680*Sqrt[d]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(2*(735*e^3*f^2 - 176*d^3 *g^2)*p - 210*(7*e^3*f^2 - d^3*g^2)*p*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e] *x)] - 105*(7*e^3*f^2 - d^3*g^2)*Log[c*(d + e*x^2)^p]) + Sqrt[e]*(p^2*x*(- 591360*d^3*g^2 + 79520*d^2*e*g^2*x^2 - 378*d*e^2*g*x*(1225*f + 64*g*x^3) + 225*e^3*(10976*f^2 + 343*f*g*x^3 + 32*g^2*x^6)) + 154350*d^2*e*f*g*p^2*Lo g[d + e*x^2] - 210*p*(-840*d^3*g^2*x + 70*d^2*e*g*(-21*f + 4*g*x^3) - 42*d *e^2*g*x^2*(35*f + 4*g*x^3) + 15*e^3*x*(392*f^2 + 49*f*g*x^3 + 8*g^2*x^6)) *Log[c*(d + e*x^2)^p] + 22050*(-7*d^2*e*f*g + e^3*x*(14*f^2 + 7*f*g*x^3 + 2*g^2*x^6))*Log[c*(d + e*x^2)^p]^2) - (176400*I)*Sqrt[d]*(-7*e^3*f^2 + d^3 *g^2)*p^2*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)])/ (308700*e^(7/2))
Time = 1.28 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2921, 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.
| \(\displaystyle \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2921 |
| \(\displaystyle \int \left (f^2 \log ^2\left (c \left (d+e x^2\right )^p\right )+2 f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \log ^2\left (c \left (d+e x^2\right )^p\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8}{343} g^2 p^2 x^7+\frac {1}{7} g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^7-\frac {4}{49} g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^7-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {4 d g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5}{35 e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {4 d^2 g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{21 e^2}-\frac {2 d f g p^2 x^2}{e}+8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}+f^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x-4 f^2 p \log \left (c \left (e x^2+d\right )^p\right ) x+\frac {4 d^3 g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{7 e^3}+\frac {f g p^2 \left (e x^2+d\right )^2}{4 e^2}+\frac {4 i \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {f g \left (e x^2+d\right )^2 \log ^2\left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac {d f g \left (e x^2+d\right ) \log ^2\left (c \left (e x^2+d\right )^p\right )}{e^2}-\frac {8 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\frac {8 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {8 d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{7 e^{7/2}}-\frac {f g p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{2 e^2}+\frac {2 d f g p \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{e^2}+\frac {4 \sqrt {d} f^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{\sqrt {e}}-\frac {4 d^{7/2} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{7 e^{7/2}}+\frac {4 i \sqrt {d} f^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{7 e^{7/2}}\) |
8*f^2*p^2*x - (1408*d^3*g^2*p^2*x)/(735*e^3) - (2*d*f*g*p^2*x^2)/e + (568* d^2*g^2*p^2*x^3)/(2205*e^2) - (96*d*g^2*p^2*x^5)/(1225*e) + (8*g^2*p^2*x^7 )/343 + (f*g*p^2*(d + e*x^2)^2)/(4*e^2) - (8*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[ e]*x)/Sqrt[d]])/Sqrt[e] + (1408*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d] ])/(735*e^(7/2)) + ((4*I)*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/S qrt[e] - (((4*I)/7)*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(7/2) + (8*Sqrt[d]*f^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (8*d^(7/2)*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]] *Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(7*e^(7/2)) - 4*f^2*p*x*Log[c*( d + e*x^2)^p] + (4*d^3*g^2*p*x*Log[c*(d + e*x^2)^p])/(7*e^3) - (4*d^2*g^2* p*x^3*Log[c*(d + e*x^2)^p])/(21*e^2) + (4*d*g^2*p*x^5*Log[c*(d + e*x^2)^p] )/(35*e) - (4*g^2*p*x^7*Log[c*(d + e*x^2)^p])/49 + (2*d*f*g*p*(d + e*x^2)* Log[c*(d + e*x^2)^p])/e^2 - (f*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(2* e^2) + (4*Sqrt[d]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/ Sqrt[e] - (4*d^(7/2)*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p ])/(7*e^(7/2)) + f^2*x*Log[c*(d + e*x^2)^p]^2 + (g^2*x^7*Log[c*(d + e*x^2) ^p]^2)/7 - (d*f*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]^2)/e^2 + (f*g*(d + e*x^ 2)^2*Log[c*(d + e*x^2)^p]^2)/(2*e^2) + ((4*I)*Sqrt[d]*f^2*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (((4*I)/7)*d^(7/2)*g^2* p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/e^(7/2)
3.3.94.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> With[{t = ExpandIntegrand[(a + b*Log[ c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && Integ erQ[r] && IntegerQ[s] && (EqQ[q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.75 (sec) , antiderivative size = 1127, normalized size of antiderivative = 1.35
-4/49*p*g^2*x^7*ln((e*x^2+d)^p)-4/21*p/e^2*d^2*g^2*x^3*ln((e*x^2+d)^p)+4/7 *p/e^3*x*d^3*g^2*ln((e*x^2+d)^p)+p^2/e^2*d^2*f*g*ln(e*x^2+d)^2+3/2*p^2/e^2 *d^2*f*g*ln(e*x^2+d)+1408/735*p^2/e^3*g^2*d^4/(d*e)^(1/2)*arctan(x*e/(d*e) ^(1/2))-4*p*f^2*x*ln((e*x^2+d)^p)+1/4*(I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*( e*x^2+d)^p)^2-I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-I*P i*csgn(I*c*(e*x^2+d)^p)^3+I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+2*ln(c))^ 2*(1/7*g^2*x^7+1/2*f*g*x^4+f^2*x)+1/4*p^2*f*g*x^4-1/2*p*f*g*x^4*ln((e*x^2+ d)^p)-4*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^2*ln(e*x^2+d)+4*p*d/(d *e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^2*ln((e*x^2+d)^p)+4/35*p/e*d*g^2*x^5*l n((e*x^2+d)^p)-8*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^2+ln((e*x^2+d )^p)^2*x*f^2+1/2*ln((e*x^2+d)^p)^2*f*g*x^4+1/7*ln((e*x^2+d)^p)^2*g^2*x^7-2 /7*p^2*e*Sum(1/2*(ln(x-_alpha)*ln(e*x^2+d)-2*e*(1/4/_alpha/e*ln(x-_alpha)^ 2+1/2*_alpha/d*ln(x-_alpha)*ln(1/2*(x+_alpha)/_alpha)+1/2*_alpha/d*dilog(1 /2*(x+_alpha)/_alpha)))*d*(7*_alpha*d*e^2*f*g+2*d^3*g^2-14*e^3*f^2)/e^5/_a lpha,_alpha=RootOf(_Z^2*e+d))+(I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d) ^p)^2-I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)-I*Pi*csgn(I *c*(e*x^2+d)^p)^3+I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)+2*ln(c))*(1/7*ln( (e*x^2+d)^p)*g^2*x^7+1/2*ln((e*x^2+d)^p)*f*g*x^4+ln((e*x^2+d)^p)*x*f^2-1/7 *p*e*(1/e^4*(2/7*e^3*g^2*x^7-2/5*d*e^2*g^2*x^5+7/4*e^3*f*g*x^4+2/3*d^2*e*g ^2*x^3-7/2*d*f*g*x^2*e^2-2*x*d^3*g^2+14*x*e^3*f^2)+d/e^4*(7/2*d*e*f*g*l...
\[ \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]
\[ \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f + g x^{3}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \]
Exception generated. \[ \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]
Timed out. \[ \int \left (f+g x^3\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (g\,x^3+f\right )}^2 \,d x \]