Integrand size = 24, antiderivative size = 24 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=-48 f^2 p^3 x+\frac {351136 d^3 g^2 p^3 x}{25725 e^3}+\frac {6 d f g p^3 x^2}{e}-\frac {55456 d^2 g^2 p^3 x^3}{77175 e^2}+\frac {5232 d g^2 p^3 x^5}{42875 e}-\frac {48 g^2 p^3 x^7}{2401}-\frac {3 f g p^3 \left (d+e x^2\right )^2}{8 e^2}+\frac {48 \sqrt {d} f^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {351136 d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{25725 e^{7/2}}-\frac {24 i \sqrt {d} f^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {1408 i d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{245 e^{7/2}}-\frac {48 \sqrt {d} f^2 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 {2816 d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{245 e^{7/2}}+24 f^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {1408 d^3 g^2 p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{245 e^3}+\frac {568 d^2 g^2 p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )}{735 e^2}-\frac {288 d g^2 p^2 x^5 \log \left (c \left (d+e x^2\right )^p\right )}{1225 e}+\frac {24}{343} g^2 p^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {6 d f g p^2 \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {3 f g p^2 \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac {24 \sqrt {d} f^2 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 {1408 d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{245 e^{7/2}}-6 f^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {6 d^3 g^2 p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {2 d^2 g^2 p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {6 d g^2 p x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {6}{49} g^2 p x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3 d f g p \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {3 f g p \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+f^2 x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac {24 i \sqrt {d} f^2 p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {1408 i d^{7/2} g^2 p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{245 e^{7/2}}+6 d f^2 p \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )-\frac {6 d^4 g^2 p \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{7 e^3} \]
24*f^2*p^2*x*ln(c*(e*x^2+d)^p)+24/343*g^2*p^2*x^7*ln(c*(e*x^2+d)^p)-6*f^2* p*x*ln(c*(e*x^2+d)^p)^2-6/49*g^2*p*x^7*ln(c*(e*x^2+d)^p)^2+6*d*f^2*p*Unint egrable(ln(c*(e*x^2+d)^p)^2/(e*x^2+d),x)-48*f^2*p^3*x-48/2401*g^2*p^3*x^7+ 1/7*g^2*x^7*ln(c*(e*x^2+d)^p)^3-351136/25725*d^(7/2)*g^2*p^3*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)^3/e^2+48*f^2*p^ 3*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/e^(1/2)-6/7*d^4*g^2*p*Unintegrable(ln( c*(e*x^2+d)^p)^2/(e*x^2+d),x)/e^3+351136/25725*d^3*g^2*p^3*x/e^3-55456/771 75*d^2*g^2*p^3*x^3/e^2+5232/42875*d*g^2*p^3*x^5/e-3/8*f*g*p^3*(e*x^2+d)^2/ e^2+6*d*f*g*p^3*x^2/e-1408/245*d^3*g^2*p^2*x*ln(c*(e*x^2+d)^p)/e^3+568/735 *d^2*g^2*p^2*x^3*ln(c*(e*x^2+d)^p)/e^2-288/1225*d*g^2*p^2*x^5*ln(c*(e*x^2+ d)^p)/e+3/4*f*g*p^2*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)/e^2+1408/245*d^(7/2)*g^2 *p^2*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)/e^(7/2)+6/7*d^3*g^2*p*x*l n(c*(e*x^2+d)^p)^2/e^3-2/7*d^2*g^2*p*x^3*ln(c*(e*x^2+d)^p)^2/e^2+6/35*d*g^ 2*p*x^5*ln(c*(e*x^2+d)^p)^2/e-3/4*f*g*p*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)^2/e^ 2+2816/245*d^(7/2)*g^2*p^3*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2) +I*x*e^(1/2)))/e^(7/2)-24*f^2*p^2*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d) ^p)*d^(1/2)/e^(1/2)-48*f^2*p^3*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)-24*I*f^2*p^3*arctan(x*e^(1/2)/d^(1/2))^ 2*d^(1/2)/e^(1/2)-24*I*f^2*p^3*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)) )*d^(1/2)/e^(1/2)-d*f*g*(e*x^2+d)*ln(c*(e*x^2+d)^p)^3/e^2+f^2*x*ln(c*(e...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2385\) vs. \(2(1126)=2252\).
Time = 8.70 (sec) , antiderivative size = 2385, normalized size of antiderivative = 99.38 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Result too large to show} \]
(f*g*p^3*(d + e*x^2)*(-8*d*(-6 + 6*Log[d + e*x^2] - 3*Log[d + e*x^2]^2 + L og[d + e*x^2]^3) + (d + e*x^2)*(-3 + 6*Log[d + e*x^2] - 6*Log[d + e*x^2]^2 + 4*Log[d + e*x^2]^3)))/(8*e^2) + 6*f*g*p^2*((x^4*Log[d + e*x^2]^2)/4 - e *((3*d*x^2)/(4*e^2) - x^4/(8*e) - (3*d^2*Log[d + e*x^2])/(4*e^3) - (d*x^2* Log[d + e*x^2])/(2*e^2) + (x^4*Log[d + e*x^2])/(4*e) + (d^2*Log[d + e*x^2] ^2)/(4*e^3)))*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p]) + (3*d*f*g*p*x^ 2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(2*e) - (2*d^2*g^2*p*x^3 *(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(7*e^2) + (6*d*g^2*p*x^5* (-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(35*e) - (3*d^2*f*g*p*Log[ d + e*x^2]*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2)/(2*e^2) + (3*p* x*(14*f^2 + 7*f*g*x^3 + 2*g^2*x^6)*Log[d + e*x^2]*(-(p*Log[d + e*x^2]) + L og[c*(d + e*x^2)^p])^2)/14 + (f*g*x^4*(-(p*Log[d + e*x^2]) + Log[c*(d + e* x^2)^p])^2*(-3*p + 2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/4 + (g ^2*x^7*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2*(-6*p + 7*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])))/49 + (x*(-(p*Log[d + e*x^2]) + Log[c* (d + e*x^2)^p])^2*(-42*e^3*f^2*p + 6*d^3*g^2*p + 7*e^3*f^2*(-(p*Log[d + e* x^2]) + Log[c*(d + e*x^2)^p])))/(7*e^3) - (6*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*( -7*d*e^3*f^2*p*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2 + d^4*g^2*p* (-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])^2))/(7*Sqrt[d]*e^(7/2)) + 3*f ^2*p^2*(-(p*Log[d + e*x^2]) + Log[c*(d + e*x^2)^p])*(x*Log[d + e*x^2]^2...
Not integrable
Time = 2.38 (sec) , antiderivative size = 24, 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.
| \(\displaystyle \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx\) |
| \(\Big \downarrow \) 2921 |
| \(\displaystyle \int \left (f^2 \log ^3\left (c \left (d+e x^2\right )^p\right )+2 f g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \log ^3\left (c \left (d+e x^2\right )^p\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {48 g^2 p^3 x^7}{2401}+\frac {1}{7} g^2 \log ^3\left (c \left (e x^2+d\right )^p\right ) x^7-\frac {6}{49} g^2 p \log ^2\left (c \left (e x^2+d\right )^p\right ) x^7+\frac {24}{343} g^2 p^2 \log \left (c \left (e x^2+d\right )^p\right ) x^7+\frac {5232 d g^2 p^3 x^5}{42875 e}+\frac {6 d g^2 p \log ^2\left (c \left (e x^2+d\right )^p\right ) x^5}{35 e}-\frac {288 d g^2 p^2 \log \left (c \left (e x^2+d\right )^p\right ) x^5}{1225 e}-\frac {55456 d^2 g^2 p^3 x^3}{77175 e^2}-\frac {2 d^2 g^2 p \log ^2\left (c \left (e x^2+d\right )^p\right ) x^3}{7 e^2}+\frac {568 d^2 g^2 p^2 \log \left (c \left (e x^2+d\right )^p\right ) x^3}{735 e^2}+\frac {6 d f g p^3 x^2}{e}-48 f^2 p^3 x+\frac {351136 d^3 g^2 p^3 x}{25725 e^3}+f^2 \log ^3\left (c \left (e x^2+d\right )^p\right ) x-6 f^2 p \log ^2\left (c \left (e x^2+d\right )^p\right ) x+\frac {6 d^3 g^2 p \log ^2\left (c \left (e x^2+d\right )^p\right ) x}{7 e^3}+24 f^2 p^2 \log \left (c \left (e x^2+d\right )^p\right ) x-\frac {1408 d^3 g^2 p^2 \log \left (c \left (e x^2+d\right )^p\right ) x}{245 e^3}+\frac {f g \left (e x^2+d\right )^2 \log ^3\left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac {d f g \left (e x^2+d\right ) \log ^3\left (c \left (e x^2+d\right )^p\right )}{e^2}-\frac {3 f g p^3 \left (e x^2+d\right )^2}{8 e^2}-\frac {24 i \sqrt {d} f^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {1408 i d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{245 e^{7/2}}-\frac {3 f g p \left (e x^2+d\right )^2 \log ^2\left (c \left (e x^2+d\right )^p\right )}{4 e^2}+\frac {3 d f g p \left (e x^2+d\right ) \log ^2\left (c \left (e x^2+d\right )^p\right )}{e^2}+\frac {48 \sqrt {d} f^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {351136 d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{25725 e^{7/2}}-\frac {48 \sqrt {d} f^2 p^3 \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 {2816 d^{7/2} g^2 p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{245 e^{7/2}}+\frac {3 f g p^2 \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{4 e^2}-\frac {6 d f g p^2 \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{e^2}-\frac {24 \sqrt {d} f^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{245 e^{7/2}}-\frac {24 i \sqrt {d} f^2 p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 i d^{7/2} g^2 p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{245 e^{7/2}}+6 d f^2 p \int \frac {\log ^2\left (c \left (e x^2+d\right )^p\right )}{e x^2+d}dx-\frac {6 d^4 g^2 p \int \frac {\log ^2\left (c \left (e x^2+d\right )^p\right )}{e x^2+d}dx}{7 e^3}\) |
3.3.98.3.1 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]))
Not integrable
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
\[\int \left (g \,x^{3}+f \right )^{2} {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}^{3}d x\]
Not integrable
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]
Not integrable
Time = 32.81 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f + g x^{3}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{3}\, dx \]
Exception generated. \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Exception raised: ValueError} \]
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
Not integrable
Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]
Not integrable
Time = 1.57 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \left (f+g x^3\right )^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^3\,{\left (g\,x^3+f\right )}^2 \,d x \]