\(\int (f+g x^3)^3 \log ^2(c (d+e x^2)^p) \, dx\) [293]

Optimal result

Integrand size = 24, antiderivative size = 1221 \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}-\frac {3 d f^2 g p^2 x^2}{e}+\frac {d^4 g^3 p^2 x^2}{e^4}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {24}{343} f g^2 p^2 x^7+\frac {3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac {d^3 g^3 p^2 \left (d+e x^2\right )^2}{2 e^5}+\frac {2 d^2 g^3 p^2 \left (d+e x^2\right )^3}{9 e^5}-\frac {d g^3 p^2 \left (d+e x^2\right )^4}{16 e^5}+\frac {g^3 p^2 \left (d+e x^2\right )^5}{125 e^5}-\frac {8 \sqrt {d} f^3 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} f g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{245 e^{7/2}}+\frac {4 i \sqrt {d} f^3 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {8 \sqrt {d} f^3 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 {24 d^{7/2} f 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}}-\frac {d^5 g^3 p^2 \log ^2\left (d+e x^2\right )}{10 e^5}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {d^4 g^3 p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^5}-\frac {3 f^2 g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {d^3 g^3 p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^5}-\frac {2 d^2 g^3 p \left (d+e x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 e^5}+\frac {d g^3 p \left (d+e x^2\right )^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^5}-\frac {g^3 p \left (d+e x^2\right )^5 \log \left (c \left (d+e x^2\right )^p\right )}{25 e^5}+\frac {4 \sqrt {d} f^3 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {12 d^{7/2} f 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}}+\frac {d^5 g^3 p \log \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^5}+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {3 f^2 g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {4 i \sqrt {d} f^3 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{7 e^{7/2}} \]

[Out]

8*f^3*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)-d^4*g^3*p*(e*x^2+d)*ln
(c*(e*x^2+d)^p)/e^5+d^3*g^3*p*(e*x^2+d)^2*ln(c*(e*x^2+d)^p)/e^5+d^4*g^3*p^2*x^2/e^4-1/10*d^5*g^3*p^2*ln(e*x^2+
d)^2/e^5-12/49*f*g^2*p*x^7*ln(c*(e*x^2+d)^p)-1/25*g^3*p*(e*x^2+d)^5*ln(c*(e*x^2+d)^p)/e^5+3/4*f^2*g*(e*x^2+d)^
2*ln(c*(e*x^2+d)^p)^2/e^2-8*f^3*p^2*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/e^(1/2)+4*I*f^3*p^2*arctan(x*e^(1/2)/d^(
1/2))^2*d^(1/2)/e^(1/2)+4*I*f^3*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^3*p^2*x+1
/10*g^3*x^10*ln(c*(e*x^2+d)^p)^2-1408/245*d^3*f*g^2*p^2*x/e^3-3*d*f^2*g*p^2*x^2/e+568/735*d^2*f*g^2*p^2*x^3/e^
2-288/1225*d*f*g^2*p^2*x^5/e+1408/245*d^(7/2)*f*g^2*p^2*arctan(x*e^(1/2)/d^(1/2))/e^(7/2)-3/4*f^2*g*p*(e*x^2+d
)^2*ln(c*(e*x^2+d)^p)/e^2-2/3*d^2*g^3*p*(e*x^2+d)^3*ln(c*(e*x^2+d)^p)/e^5+1/4*d*g^3*p*(e*x^2+d)^4*ln(c*(e*x^2+
d)^p)/e^5+1/5*d^5*g^3*p*ln(e*x^2+d)*ln(c*(e*x^2+d)^p)/e^5-3/2*d*f^2*g*(e*x^2+d)*ln(c*(e*x^2+d)^p)^2/e^2+4*f^3*
p*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)*d^(1/2)/e^(1/2)+24/343*f*g^2*p^2*x^7+1/125*g^3*p^2*(e*x^2+d)^5/e
^5-4*f^3*p*x*ln(c*(e*x^2+d)^p)+3/7*f*g^2*x^7*ln(c*(e*x^2+d)^p)^2-4/7*d^2*f*g^2*p*x^3*ln(c*(e*x^2+d)^p)/e^2+12/
35*d*f*g^2*p*x^5*ln(c*(e*x^2+d)^p)/e+3*d*f^2*g*p*(e*x^2+d)*ln(c*(e*x^2+d)^p)/e^2-12/7*d^(7/2)*f*g^2*p*arctan(x
*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)/e^(7/2)-24/7*d^(7/2)*f*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)+12/7*d^3*f*g^2*p*x*ln(c*(e*x^2+d)^p)/e^3-12/7*I*d^(7/2)*f*g^2*p^2*arctan(x*e^(1/2)/
d^(1/2))^2/e^(7/2)-12/7*I*d^(7/2)*f*g^2*p^2*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))/e^(7/2)+3/8*f^2*g*p^2
*(e*x^2+d)^2/e^2-1/2*d^3*g^3*p^2*(e*x^2+d)^2/e^5+2/9*d^2*g^3*p^2*(e*x^2+d)^3/e^5-1/16*d*g^3*p^2*(e*x^2+d)^4/e^
5+f^3*x*ln(c*(e*x^2+d)^p)^2

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 1221, normalized size of antiderivative = 1.00, number of steps used = 55, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.208, Rules used = {2521, 2500, 2526, 2498, 327, 211, 2520, 12, 5040, 4964, 2449, 2352, 2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341, 2507, 2505, 308, 2445, 2458, 45, 2372, 14, 2338} \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {1}{10} g^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^{10}+\frac {24}{343} f g^2 p^2 x^7+\frac {3}{7} f g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^7-\frac {12}{49} f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^7-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {12 d f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5}{35 e}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac {4 d^2 f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{7 e^2}+\frac {d^4 g^3 p^2 x^2}{e^4}-\frac {3 d f^2 g p^2 x^2}{e}+8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}+f^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x-4 f^3 p \log \left (c \left (e x^2+d\right )^p\right ) x+\frac {12 d^3 f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{7 e^3}+\frac {g^3 p^2 \left (e x^2+d\right )^5}{125 e^5}-\frac {d g^3 p^2 \left (e x^2+d\right )^4}{16 e^5}+\frac {2 d^2 g^3 p^2 \left (e x^2+d\right )^3}{9 e^5}-\frac {d^3 g^3 p^2 \left (e x^2+d\right )^2}{2 e^5}+\frac {3 f^2 g p^2 \left (e x^2+d\right )^2}{8 e^2}+\frac {4 i \sqrt {d} f^3 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}-\frac {d^5 g^3 p^2 \log ^2\left (e x^2+d\right )}{10 e^5}+\frac {3 f^2 g \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^2 g \left (e x^2+d\right ) \log ^2\left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac {8 \sqrt {d} f^3 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} f g^2 p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{245 e^{7/2}}+\frac {8 \sqrt {d} f^3 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 {24 d^{7/2} f 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 {g^3 p \left (e x^2+d\right )^5 \log \left (c \left (e x^2+d\right )^p\right )}{25 e^5}+\frac {d g^3 p \left (e x^2+d\right )^4 \log \left (c \left (e x^2+d\right )^p\right )}{4 e^5}-\frac {2 d^2 g^3 p \left (e x^2+d\right )^3 \log \left (c \left (e x^2+d\right )^p\right )}{3 e^5}+\frac {d^3 g^3 p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{e^5}-\frac {3 f^2 g p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{4 e^2}-\frac {d^4 g^3 p \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{e^5}+\frac {3 d f^2 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^3 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{\sqrt {e}}-\frac {12 d^{7/2} f 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 {d^5 g^3 p \log \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{5 e^5}+\frac {4 i \sqrt {d} f^3 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{7 e^{7/2}} \]

[In]

Int[(f + g*x^3)^3*Log[c*(d + e*x^2)^p]^2,x]

[Out]

8*f^3*p^2*x - (1408*d^3*f*g^2*p^2*x)/(245*e^3) - (3*d*f^2*g*p^2*x^2)/e + (d^4*g^3*p^2*x^2)/e^4 + (568*d^2*f*g^
2*p^2*x^3)/(735*e^2) - (288*d*f*g^2*p^2*x^5)/(1225*e) + (24*f*g^2*p^2*x^7)/343 + (3*f^2*g*p^2*(d + e*x^2)^2)/(
8*e^2) - (d^3*g^3*p^2*(d + e*x^2)^2)/(2*e^5) + (2*d^2*g^3*p^2*(d + e*x^2)^3)/(9*e^5) - (d*g^3*p^2*(d + e*x^2)^
4)/(16*e^5) + (g^3*p^2*(d + e*x^2)^5)/(125*e^5) - (8*Sqrt[d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (1
408*d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(245*e^(7/2)) + ((4*I)*Sqrt[d]*f^3*p^2*ArcTan[(Sqrt[e]*x)/S
qrt[d]]^2)/Sqrt[e] - (((12*I)/7)*d^(7/2)*f*g^2*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(7/2) + (8*Sqrt[d]*f^3*p^2
*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (24*d^(7/2)*f*g^2*p^2*ArcTan[
(Sqrt[e]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(7*e^(7/2)) - (d^5*g^3*p^2*Log[d + e*x^2]^2)/(1
0*e^5) - 4*f^3*p*x*Log[c*(d + e*x^2)^p] + (12*d^3*f*g^2*p*x*Log[c*(d + e*x^2)^p])/(7*e^3) - (4*d^2*f*g^2*p*x^3
*Log[c*(d + e*x^2)^p])/(7*e^2) + (12*d*f*g^2*p*x^5*Log[c*(d + e*x^2)^p])/(35*e) - (12*f*g^2*p*x^7*Log[c*(d + e
*x^2)^p])/49 + (3*d*f^2*g*p*(d + e*x^2)*Log[c*(d + e*x^2)^p])/e^2 - (d^4*g^3*p*(d + e*x^2)*Log[c*(d + e*x^2)^p
])/e^5 - (3*f^2*g*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p])/(4*e^2) + (d^3*g^3*p*(d + e*x^2)^2*Log[c*(d + e*x^2)^p
])/e^5 - (2*d^2*g^3*p*(d + e*x^2)^3*Log[c*(d + e*x^2)^p])/(3*e^5) + (d*g^3*p*(d + e*x^2)^4*Log[c*(d + e*x^2)^p
])/(4*e^5) - (g^3*p*(d + e*x^2)^5*Log[c*(d + e*x^2)^p])/(25*e^5) + (4*Sqrt[d]*f^3*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]
]*Log[c*(d + e*x^2)^p])/Sqrt[e] - (12*d^(7/2)*f*g^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(7*e^(
7/2)) + (d^5*g^3*p*Log[d + e*x^2]*Log[c*(d + e*x^2)^p])/(5*e^5) + f^3*x*Log[c*(d + e*x^2)^p]^2 + (3*f*g^2*x^7*
Log[c*(d + e*x^2)^p]^2)/7 + (g^3*x^10*Log[c*(d + e*x^2)^p]^2)/10 - (3*d*f^2*g*(d + e*x^2)*Log[c*(d + e*x^2)^p]
^2)/(2*e^2) + (3*f^2*g*(d + e*x^2)^2*Log[c*(d + e*x^2)^p]^2)/(4*e^2) + ((4*I)*Sqrt[d]*f^3*p^2*PolyLog[2, 1 - (
2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (((12*I)/7)*d^(7/2)*f*g^2*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[
d] + I*Sqrt[e]*x)])/e^(7/2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2500

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x^
n)^p])^q, x] - Dist[b*e*n*p*q, Int[x^n*((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a,
b, c, d, e, n, p}, x] && IGtQ[q, 0] && (EqQ[q, 1] || IntegerQ[n])

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 2507

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)
^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])^q/(f*(m + 1))), x] - Dist[b*e*n*p*(q/(f^n*(m + 1))), Int[(f*x)^(m + n)*
((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1]
&& IntegerQ[n] && NeQ[m, -1]

Rule 2520

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[u*(x^(n - 1)/(d + e*x^n)
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 2521

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]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x
^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (f^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+3 f^2 g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+3 f g^2 x^6 \log ^2\left (c \left (d+e x^2\right )^p\right )+g^3 x^9 \log ^2\left (c \left (d+e x^2\right )^p\right )\right ) \, dx \\ & = f^3 \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\left (3 f^2 g\right ) \int x^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\left (3 f g^2\right ) \int x^6 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+g^3 \int x^9 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx \\ & = f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} \left (3 f^2 g\right ) \text {Subst}\left (\int x \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )+\frac {1}{2} g^3 \text {Subst}\left (\int x^4 \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )-\left (4 e f^3 p\right ) \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{7} \left (12 e f g^2 p\right ) \int \frac {x^8 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx \\ & = f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} \left (3 f^2 g\right ) \text {Subst}\left (\int \left (-\frac {d \log ^2\left (c (d+e x)^p\right )}{e}+\frac {(d+e x) \log ^2\left (c (d+e x)^p\right )}{e}\right ) \, dx,x,x^2\right )-\left (4 e f^3 p\right ) \int \left (\frac {\log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac {d \log \left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx-\frac {1}{7} \left (12 e f g^2 p\right ) \int \left (-\frac {d^3 \log \left (c \left (d+e x^2\right )^p\right )}{e^4}+\frac {d^2 x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac {d x^4 \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^6 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {d^4 \log \left (c \left (d+e x^2\right )^p\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx-\frac {1}{5} \left (e g^3 p\right ) \text {Subst}\left (\int \frac {x^5 \log \left (c (d+e x)^p\right )}{d+e x} \, dx,x,x^2\right ) \\ & = f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {\left (3 f^2 g\right ) \text {Subst}\left (\int (d+e x) \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 e}-\frac {\left (3 d f^2 g\right ) \text {Subst}\left (\int \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 e}-\left (4 f^3 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\left (4 d f^3 p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{7} \left (12 f g^2 p\right ) \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\frac {\left (12 d^3 f g^2 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}-\frac {\left (12 d^4 f g^2 p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}-\frac {\left (12 d^2 f g^2 p\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^2}+\frac {\left (12 d f g^2 p\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e}-\frac {1}{5} \left (g^3 p\right ) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^5 \log \left (c x^p\right )}{x} \, dx,x,d+e x^2\right ) \\ & = -4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^4 g^3 p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^5}+\frac {d^3 g^3 p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^5}-\frac {2 d^2 g^3 p \left (d+e x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 e^5}+\frac {d g^3 p \left (d+e x^2\right )^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^5}-\frac {g^3 p \left (d+e x^2\right )^5 \log \left (c \left (d+e x^2\right )^p\right )}{25 e^5}+\frac {4 \sqrt {d} f^3 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+\frac {d^5 g^3 p \log \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^5}+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {\left (3 f^2 g\right ) \text {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}-\frac {\left (3 d f^2 g\right ) \text {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}+\left (8 e f^3 p^2\right ) \int \frac {x^2}{d+e x^2} \, dx-\left (8 d e f^3 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx-\frac {1}{35} \left (24 d f g^2 p^2\right ) \int \frac {x^6}{d+e x^2} \, dx-\frac {\left (24 d^3 f g^2 p^2\right ) \int \frac {x^2}{d+e x^2} \, dx}{7 e^2}+\frac {\left (24 d^4 f g^2 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx}{7 e^2}+\frac {\left (8 d^2 f g^2 p^2\right ) \int \frac {x^4}{d+e x^2} \, dx}{7 e}+\frac {1}{49} \left (24 e f g^2 p^2\right ) \int \frac {x^8}{d+e x^2} \, dx+\frac {1}{5} \left (g^3 p^2\right ) \text {Subst}\left (\int \frac {300 d^4 x-300 d^3 x^2+200 d^2 x^3-75 d x^4+12 x^5-60 d^5 \log (x)}{60 e^5 x} \, dx,x,d+e x^2\right ) \\ & = 8 f^3 p^2 x-\frac {24 d^3 f g^2 p^2 x}{7 e^3}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^4 g^3 p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^5}+\frac {d^3 g^3 p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^5}-\frac {2 d^2 g^3 p \left (d+e x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 e^5}+\frac {d g^3 p \left (d+e x^2\right )^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^5}-\frac {g^3 p \left (d+e x^2\right )^5 \log \left (c \left (d+e x^2\right )^p\right )}{25 e^5}+\frac {4 \sqrt {d} f^3 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+\frac {d^5 g^3 p \log \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^5}+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {3 f^2 g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac {\left (3 f^2 g p\right ) \text {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}+\frac {\left (3 d f^2 g p\right ) \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\left (8 d f^3 p^2\right ) \int \frac {1}{d+e x^2} \, dx-\left (8 \sqrt {d} \sqrt {e} f^3 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx-\frac {1}{35} \left (24 d f g^2 p^2\right ) \int \left (\frac {d^2}{e^3}-\frac {d x^2}{e^2}+\frac {x^4}{e}-\frac {d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx+\frac {\left (24 d^4 f g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{7 e^3}+\frac {\left (24 d^{7/2} f g^2 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx}{7 e^{5/2}}+\frac {\left (8 d^2 f g^2 p^2\right ) \int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx}{7 e}+\frac {1}{49} \left (24 e f g^2 p^2\right ) \int \left (-\frac {d^3}{e^4}+\frac {d^2 x^2}{e^3}-\frac {d x^4}{e^2}+\frac {x^6}{e}+\frac {d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx+\frac {\left (g^3 p^2\right ) \text {Subst}\left (\int \frac {300 d^4 x-300 d^3 x^2+200 d^2 x^3-75 d x^4+12 x^5-60 d^5 \log (x)}{x} \, dx,x,d+e x^2\right )}{300 e^5} \\ & = 8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}-\frac {3 d f^2 g p^2 x^2}{e}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {24}{343} f g^2 p^2 x^7+\frac {3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac {8 \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {24 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {4 i \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {d^4 g^3 p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^5}-\frac {3 f^2 g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {d^3 g^3 p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^5}-\frac {2 d^2 g^3 p \left (d+e x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 e^5}+\frac {d g^3 p \left (d+e x^2\right )^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^5}-\frac {g^3 p \left (d+e x^2\right )^5 \log \left (c \left (d+e x^2\right )^p\right )}{25 e^5}+\frac {4 \sqrt {d} f^3 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+\frac {d^5 g^3 p \log \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^5}+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {3 f^2 g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\left (8 f^3 p^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx-\frac {\left (24 d^3 f g^2 p^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx}{7 e^3}+\frac {\left (24 d^4 f g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{49 e^3}+\frac {\left (24 d^4 f g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{35 e^3}+\frac {\left (8 d^4 f g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{7 e^3}+\frac {\left (g^3 p^2\right ) \text {Subst}\left (\int \left (300 d^4-300 d^3 x+200 d^2 x^2-75 d x^3+12 x^4-\frac {60 d^5 \log (x)}{x}\right ) \, dx,x,d+e x^2\right )}{300 e^5} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 780, normalized size of antiderivative = 0.64 \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{4} f^2 g x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3 f^2 g \left (e p^2 x^2 \left (-6 d+e x^2\right )+2 d^2 p^2 \log \left (d+e x^2\right )+2 p \left (2 d^2+2 d e x^2-e^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )-2 d^2 \log ^2\left (c \left (d+e x^2\right )^p\right )\right )}{8 e^2}+\frac {g^3 \left (e p^2 x^2 \left (8220 d^4-2310 d^3 e x^2+940 d^2 e^2 x^4-405 d e^3 x^6+144 e^4 x^8\right )-4620 d^5 p^2 \log \left (d+e x^2\right )-60 p \left (60 d^5+60 d^4 e x^2-30 d^3 e^2 x^4+20 d^2 e^3 x^6-15 d e^4 x^8+12 e^5 x^{10}\right ) \log \left (c \left (d+e x^2\right )^p\right )+1800 d^5 \log ^2\left (c \left (d+e x^2\right )^p\right )\right )}{18000 e^5}+\frac {4 f^3 p \left (i \sqrt {d} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2+\sqrt {e} x \left (2 p-\log \left (c \left (d+e x^2\right )^p\right )\right )+\sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-2 p+2 p \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )+i \sqrt {d} p \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )\right )}{\sqrt {e}}+\frac {4 f g^2 p \left (-11025 i d^{7/2} p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2-105 d^{7/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-352 p+210 p \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+105 \log \left (c \left (d+e x^2\right )^p\right )\right )+\sqrt {e} x \left (2 p \left (-18480 d^3+2485 d^2 e x^2-756 d e^2 x^4+225 e^3 x^6\right )+105 \left (105 d^3-35 d^2 e x^2+21 d e^2 x^4-15 e^3 x^6\right ) \log \left (c \left (d+e x^2\right )^p\right )\right )-11025 i d^{7/2} p \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )\right )}{25725 e^{7/2}} \]

[In]

Integrate[(f + g*x^3)^3*Log[c*(d + e*x^2)^p]^2,x]

[Out]

f^3*x*Log[c*(d + e*x^2)^p]^2 + (3*f^2*g*x^4*Log[c*(d + e*x^2)^p]^2)/4 + (3*f*g^2*x^7*Log[c*(d + e*x^2)^p]^2)/7
+ (g^3*x^10*Log[c*(d + e*x^2)^p]^2)/10 + (3*f^2*g*(e*p^2*x^2*(-6*d + e*x^2) + 2*d^2*p^2*Log[d + e*x^2] + 2*p*
(2*d^2 + 2*d*e*x^2 - e^2*x^4)*Log[c*(d + e*x^2)^p] - 2*d^2*Log[c*(d + e*x^2)^p]^2))/(8*e^2) + (g^3*(e*p^2*x^2*
(8220*d^4 - 2310*d^3*e*x^2 + 940*d^2*e^2*x^4 - 405*d*e^3*x^6 + 144*e^4*x^8) - 4620*d^5*p^2*Log[d + e*x^2] - 60
*p*(60*d^5 + 60*d^4*e*x^2 - 30*d^3*e^2*x^4 + 20*d^2*e^3*x^6 - 15*d*e^4*x^8 + 12*e^5*x^10)*Log[c*(d + e*x^2)^p]
+ 1800*d^5*Log[c*(d + e*x^2)^p]^2))/(18000*e^5) + (4*f^3*p*(I*Sqrt[d]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 + Sqrt[
e]*x*(2*p - Log[c*(d + e*x^2)^p]) + Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(-2*p + 2*p*Log[(2*Sqrt[d])/(Sqrt[d] +
I*Sqrt[e]*x)] + Log[c*(d + e*x^2)^p]) + I*Sqrt[d]*p*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + Sqrt[e
]*x)]))/Sqrt[e] + (4*f*g^2*p*((-11025*I)*d^(7/2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 - 105*d^(7/2)*ArcTan[(Sqrt[e]
*x)/Sqrt[d]]*(-352*p + 210*p*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + 105*Log[c*(d + e*x^2)^p]) + Sqrt[e]*x*
(2*p*(-18480*d^3 + 2485*d^2*e*x^2 - 756*d*e^2*x^4 + 225*e^3*x^6) + 105*(105*d^3 - 35*d^2*e*x^2 + 21*d*e^2*x^4
- 15*e^3*x^6)*Log[c*(d + e*x^2)^p]) - (11025*I)*d^(7/2)*p*PolyLog[2, (I*Sqrt[d] + Sqrt[e]*x)/((-I)*Sqrt[d] + S
qrt[e]*x)]))/(25725*e^(7/2))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 10.28 (sec) , antiderivative size = 1584, normalized size of antiderivative = 1.30

method	result	size
risch	\(\text {Expression too large to display}\)	\(1584\)

[In]

int((g*x^3+f)^3*ln(c*(e*x^2+d)^p)^2,x,method=_RETURNVERBOSE)

[Out]

47/900*p^2/e^2*d^2*g^3*x^6-77/600*p^2/e^3*x^4*d^3*g^3+3/8*p^2*x^4*f^2*g-137/300*p^2/e^5*d^5*ln(e*x^2+d)*g^3-12
/49*p*f*g^2*x^7*ln((e*x^2+d)^p)-3/4*p*f^2*g*x^4*ln((e*x^2+d)^p)-8*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^
3-12/7*p/e^3*d^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f*g^2*ln((e*x^2+d)^p)+12/7*p^2/e^3*d^4/(d*e)^(1/2)*arctan
(x*e/(d*e)^(1/2))*f*g^2*ln(e*x^2+d)-9/400*p^2/e*d*g^3*x^8+1/10*ln((e*x^2+d)^p)^2*g^3*x^10+ln((e*x^2+d)^p)^2*x*
f^3+3/7*ln((e*x^2+d)^p)^2*g^2*f*x^7+3/4*ln((e*x^2+d)^p)^2*f^2*g*x^4+1/125*p^2*g^3*x^10-1/25*p*g^3*x^10*ln((e*x
^2+d)^p)-4*p*x*f^3*ln((e*x^2+d)^p)+12/35*p/e*d*f*g^2*x^5*ln((e*x^2+d)^p)-4/7*p/e^2*d^2*f*g^2*x^3*ln((e*x^2+d)^
p)+3/2*p/e*d*f^2*g*x^2*ln((e*x^2+d)^p)+12/7*p/e^3*x*d^3*f*g^2*ln((e*x^2+d)^p)-3/2*p/e^2*d^2*ln(e*x^2+d)*f^2*g*
ln((e*x^2+d)^p)+1408/245*p^2/e^3*f*g^2*d^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-1/35*p^2*e*Sum(-1/2*(ln(x-_alph
a)*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*(14*_alpha*d^4*g^3-105*_alpha*d*e^3*f^2*g-60*d^3*e*f*g^2+140*e^4*f^3)/e^6/
_alpha,_alpha=RootOf(_Z^2*e+d))-4*p^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*f^3*ln(e*x^2+d)+4*p*d/(d*e)^(1/2)*
arctan(x*e/(d*e)^(1/2))*f^3*ln((e*x^2+d)^p)+1/20*p/e*d*g^3*x^8*ln((e*x^2+d)^p)-1/15*p/e^2*d^2*g^3*x^6*ln((e*x^
2+d)^p)+1/10*p/e^3*d^3*g^3*x^4*ln((e*x^2+d)^p)-1/5*p/e^4*d^4*g^3*x^2*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*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))^2*(1/10*g^3*x^10+3/7*f*g^2*x^7+3/4*f^2*g*x^4+f^3*x)+1/5*p
/e^5*d^5*ln(e*x^2+d)*g^3*ln((e*x^2+d)^p)+3/2*p^2/e^2*d^2*ln(e*x^2+d)^2*f^2*g+9/4*p^2/e^2*d^2*ln(e*x^2+d)*f^2*g
+8*f^3*p^2*x+(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))*(1/10*ln((e*x^2+d)^p)*g
^3*x^10+3/7*ln((e*x^2+d)^p)*g^2*f*x^7+3/4*ln((e*x^2+d)^p)*f^2*g*x^4+ln((e*x^2+d)^p)*x*f^3-1/70*p*e*(1/e^5*(7/5
*e^4*g^3*x^10-7/4*d*e^3*g^3*x^8+60/7*e^4*f*g^2*x^7+7/3*d^2*e^2*g^3*x^6-12*d*e^3*f*g^2*x^5-7/2*d^3*e*g^3*x^4+10
5/4*e^4*f^2*g*x^4+20*d^2*e^2*f*g^2*x^3+7*d^4*g^3*x^2-105/2*d*f^2*g*x^2*e^3-60*x*d^3*f*g^2*e+140*x*e^4*f^3)-d/e
^5*(1/2*(14*d^4*g^3-105*d*e^3*f^2*g)/e*ln(e*x^2+d)+(-60*d^3*e*f*g^2+140*e^4*f^3)/(d*e)^(1/2)*arctan(x*e/(d*e)^
(1/2)))))-1408/245*d^3*f*g^2*p^2*x/e^3-9/4*d*f^2*g*p^2*x^2/e+568/735*d^2*f*g^2*p^2*x^3/e^2-288/1225*d*f*g^2*p^
2*x^5/e-1/5*d^5*g^3*p^2*ln(e*x^2+d)^2/e^5+24/343*f*g^2*p^2*x^7+137/300*d^4*g^3*p^2*x^2/e^4

Fricas [F]

\[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]

[In]

integrate((g*x^3+f)^3*log(c*(e*x^2+d)^p)^2,x, algorithm="fricas")

[Out]

integral((g^3*x^9 + 3*f*g^2*x^6 + 3*f^2*g*x^3 + f^3)*log((e*x^2 + d)^p*c)^2, x)

Sympy [F]

\[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f + g x^{3}\right )^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \]

[In]

integrate((g*x**3+f)**3*ln(c*(e*x**2+d)**p)**2,x)

[Out]

Integral((f + g*x**3)**3*log(c*(d + e*x**2)**p)**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((g*x^3+f)^3*log(c*(e*x^2+d)^p)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{3} + f\right )}^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2} \,d x } \]

[In]

integrate((g*x^3+f)^3*log(c*(e*x^2+d)^p)^2,x, algorithm="giac")

[Out]

integrate((g*x^3 + f)^3*log((e*x^2 + d)^p*c)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (g\,x^3+f\right )}^3 \,d x \]

[In]

int(log(c*(d + e*x^2)^p)^2*(f + g*x^3)^3,x)

[Out]

int(log(c*(d + e*x^2)^p)^2*(f + g*x^3)^3, x)