Integrand size = 22, antiderivative size = 22 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=-48 f p^3 x+\frac {208 d g p^3 x}{9 e}-\frac {16}{27} g p^3 x^3+\frac {48 \sqrt {d} f p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {208 d^{3/2} g p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {24 i \sqrt {d} f p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {32 i d^{3/2} g p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-\frac {48 \sqrt {d} f p^3 \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 {64 d^{3/2} g p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac {24 i \sqrt {d} f p^3 \operatorname {PolyLog}\left (2,-\frac {\sqrt {d}-i \sqrt {e} x}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {32 i d^{3/2} g p^3 \operatorname {PolyLog}\left (2,-\frac {\sqrt {d}-i \sqrt {e} x}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}-\frac {2 d (-3 e f+d g) p \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{e} \] Output:
-48*f*p^3*x+208/9*d*g*p^3*x/e-16/27*g*p^3*x^3+48*d^(1/2)*f*p^3*arctan(e^(1 /2)*x/d^(1/2))/e^(1/2)-208/9*d^(3/2)*g*p^3*arctan(e^(1/2)*x/d^(1/2))/e^(3/ 2)-24*I*d^(1/2)*f*p^3*arctan(e^(1/2)*x/d^(1/2))^2/e^(1/2)-24*I*d^(1/2)*f*p ^3*polylog(2,-(d^(1/2)-I*e^(1/2)*x)/(d^(1/2)+I*e^(1/2)*x))/e^(1/2)-48*d^(1 /2)*f*p^3*arctan(e^(1/2)*x/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*e^(1/2)*x))/e^ (1/2)+64/3*d^(3/2)*g*p^3*arctan(e^(1/2)*x/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I *e^(1/2)*x))/e^(3/2)+24*f*p^2*x*ln(c*(e*x^2+d)^p)-32/3*d*g*p^2*x*ln(c*(e*x ^2+d)^p)/e+8/9*g*p^2*x^3*ln(c*(e*x^2+d)^p)-24*d^(1/2)*f*p^2*arctan(e^(1/2) *x/d^(1/2))*ln(c*(e*x^2+d)^p)/e^(1/2)+32/3*d^(3/2)*g*p^2*arctan(e^(1/2)*x/ d^(1/2))*ln(c*(e*x^2+d)^p)/e^(3/2)-6*f*p*x*ln(c*(e*x^2+d)^p)^2+2*d*g*p*x*l n(c*(e*x^2+d)^p)^2/e-2/3*g*p*x^3*ln(c*(e*x^2+d)^p)^2+f*x*ln(c*(e*x^2+d)^p) ^3+1/3*g*x^3*ln(c*(e*x^2+d)^p)^3+32/3*I*d^(3/2)*g*p^3*polylog(2,-(d^(1/2)- I*e^(1/2)*x)/(d^(1/2)+I*e^(1/2)*x))/e^(3/2)+32/3*I*d^(3/2)*g*p^3*arctan(e^ (1/2)*x/d^(1/2))^2/e^(3/2)-2*d*(d*g-3*e*f)*p*Defer(Int)(ln(c*(e*x^2+d)^p)^ 2/(e*x^2+d),x)/e
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1772\) vs. \(2(683)=1366\).
Time = 12.98 (sec) , antiderivative size = 1772, normalized size of antiderivative = 80.55 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx =\text {Too large to display} \] Input:
Integrate[(f + g*x^2)*Log[c*(d + e*x^2)^p]^3,x]
Output:
(2*d*g*p*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/e + (6*Sqrt[d]* f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p ])^2)/Sqrt[e] - (2*d^(3/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-(p*Log[d + e* x^2]) + Log[c*(d + e*x^2)^p])^2)/e^(3/2) + 3*f*p*x*Log[d + e*x^2]*(-(p*Log [d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + g*p*x^3*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + f*x*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-6*p - p*Log[d + e*x^2] + Log[c*(d + e*x^2)^p]) + (g*x^3* (-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-2*p - p*Log[d + e*x^2] + Log[c*(d + e*x^2)^p]))/3 + 3*f*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2 )^p])*(x*Log[d + e*x^2]^2 - (4*((-I)*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[e]*x*(-2 + Log[d + e*x^2]) - Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*( -2 + 2*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + Log[d + e*x^2]) - I*Sqrt [d]*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)]))/Sqrt[ e]) + 3*g*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*((x^3*Log[d + e *x^2]^2)/3 - (4*((9*I)*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + 3*d^(3/2)*A rcTan[(Sqrt[e]*x)/Sqrt[d]]*(-8 + 6*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x) ] + 3*Log[d + e*x^2]) + Sqrt[e]*x*(24*d - 2*e*x^2 + (-9*d + 3*e*x^2)*Log[d + e*x^2]) + (9*I)*d^(3/2)*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d ] + Sqrt[e]*x)]))/(27*e^(3/2))) + (g*p^3*(416*Sqrt[-d]*d^(3/2)*Sqrt[d + e* x^2]*Sqrt[1 - d/(d + e*x^2)]*ArcSin[Sqrt[d]/Sqrt[d + e*x^2]] + 36*Sqrt[...
Not integrable
Time = 2.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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.
| \(\displaystyle \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2921 |
| \(\displaystyle \int \left (f \log ^3\left (c \left (d+e x^2\right )^p\right )+g x^2 \log ^3\left (c \left (d+e x^2\right )^p\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 d^2 g p \int \frac {\log ^2\left (c \left (e x^2+d\right )^p\right )}{e x^2+d}dx}{e}+6 d f p \int \frac {\log ^2\left (c \left (e x^2+d\right )^p\right )}{e x^2+d}dx+\frac {32 d^{3/2} g p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-\frac {24 \sqrt {d} f p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {32 i d^{3/2} g p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-\frac {208 d^{3/2} g p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}+\frac {64 d^{3/2} g p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}-\frac {24 i \sqrt {d} f p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {48 \sqrt {d} f p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {48 \sqrt {d} f p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {32 i d^{3/2} g p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{3 e^{3/2}}-\frac {24 i \sqrt {d} f p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}+\frac {208 d g p^3 x}{9 e}-48 f p^3 x-\frac {16}{27} g p^3 x^3\) |
Input:
Int[(f + g*x^2)*Log[c*(d + e*x^2)^p]^3,x]
Output:
$Aborted
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]))
Not integrable
Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \left (g \,x^{2}+f \right ) {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}^{3}d x\]
Input:
int((g*x^2+f)*ln(c*(e*x^2+d)^p)^3,x)
Output:
int((g*x^2+f)*ln(c*(e*x^2+d)^p)^3,x)
Not integrable
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)^3,x, algorithm="fricas")
Output:
integral((g*x^2 + f)*log((e*x^2 + d)^p*c)^3, x)
Not integrable
Time = 10.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f + g x^{2}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{3}\, dx \] Input:
integrate((g*x**2+f)*ln(c*(e*x**2+d)**p)**3,x)
Output:
Integral((f + g*x**2)*log(c*(d + e*x**2)**p)**3, x)
Exception generated. \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)^3,x, algorithm="maxima")
Output:
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
Not integrable
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \] Input:
integrate((g*x^2+f)*log(c*(e*x^2+d)^p)^3,x, algorithm="giac")
Output:
integrate((g*x^2 + f)*log((e*x^2 + d)^p*c)^3, x)
Not integrable
Time = 25.74 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^3\,\left (g\,x^2+f\right ) \,d x \] Input:
int(log(c*(d + e*x^2)^p)^3*(f + g*x^2),x)
Output:
int(log(c*(d + e*x^2)^p)^3*(f + g*x^2), x)
Not integrable
Time = 0.16 (sec) , antiderivative size = 397, normalized size of antiderivative = 18.05 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {-624 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) d g \,p^{3}+1296 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) e f \,p^{3}-54 \left (\int \frac {{\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}^{2}}{e \,x^{2}+d}d x \right ) d^{2} e g p +162 \left (\int \frac {{\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}^{2}}{e \,x^{2}+d}d x \right ) d \,e^{2} f p +288 \left (\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}{e \,x^{2}+d}d x \right ) d^{2} e g \,p^{2}-648 \left (\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}{e \,x^{2}+d}d x \right ) d \,e^{2} f \,p^{2}+27 {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}^{3} e^{2} f x +9 {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}^{3} e^{2} g \,x^{3}+54 {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}^{2} d e g p x -162 {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}^{2} e^{2} f p x -18 {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}^{2} e^{2} g p \,x^{3}-288 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d e g \,p^{2} x +648 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{2} f \,p^{2} x +24 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{2} g \,p^{2} x^{3}+624 d e g \,p^{3} x -1296 e^{2} f \,p^{3} x -16 e^{2} g \,p^{3} x^{3}}{27 e^{2}} \] Input:
int((g*x^2+f)*log(c*(e*x^2+d)^p)^3,x)
Output:
( - 624*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*d*g*p**3 + 1296*sqrt (e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*e*f*p**3 - 54*int(log((d + e*x** 2)**p*c)**2/(d + e*x**2),x)*d**2*e*g*p + 162*int(log((d + e*x**2)**p*c)**2 /(d + e*x**2),x)*d*e**2*f*p + 288*int(log((d + e*x**2)**p*c)/(d + e*x**2), x)*d**2*e*g*p**2 - 648*int(log((d + e*x**2)**p*c)/(d + e*x**2),x)*d*e**2*f *p**2 + 27*log((d + e*x**2)**p*c)**3*e**2*f*x + 9*log((d + e*x**2)**p*c)** 3*e**2*g*x**3 + 54*log((d + e*x**2)**p*c)**2*d*e*g*p*x - 162*log((d + e*x* *2)**p*c)**2*e**2*f*p*x - 18*log((d + e*x**2)**p*c)**2*e**2*g*p*x**3 - 288 *log((d + e*x**2)**p*c)*d*e*g*p**2*x + 648*log((d + e*x**2)**p*c)*e**2*f*p **2*x + 24*log((d + e*x**2)**p*c)*e**2*g*p**2*x**3 + 624*d*e*g*p**3*x - 12 96*e**2*f*p**3*x - 16*e**2*g*p**3*x**3)/(27*e**2)