3.3.74 \(\int (f+g x^2) \log ^2(c (d+e x^2)^p) \, dx\) [274]

Optimal. Leaf size=548 \[ 8 f p^2 x-\frac {32 d g p^2 x}{9 e}+\frac {8}{27} g p^2 x^3-\frac {8 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}+\frac {4 i \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {8 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {8 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4}{9} g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {4 d^{3/2} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {4 i \sqrt {d} f p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {4 i d^{3/2} g p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}} \]

[Out]

8*f*p^2*x-32/9*d*g*p^2*x/e+8/27*g*p^2*x^3+32/9*d^(3/2)*g*p^2*arctan(x*e^(1/2)/d^(1/2))/e^(3/2)-4/3*I*d^(3/2)*g
*p^2*arctan(x*e^(1/2)/d^(1/2))^2/e^(3/2)-4*f*p*x*ln(c*(e*x^2+d)^p)+4/3*d*g*p*x*ln(c*(e*x^2+d)^p)/e-4/9*g*p*x^3
*ln(c*(e*x^2+d)^p)-4/3*d^(3/2)*g*p*arctan(x*e^(1/2)/d^(1/2))*ln(c*(e*x^2+d)^p)/e^(3/2)+f*x*ln(c*(e*x^2+d)^p)^2
+1/3*g*x^3*ln(c*(e*x^2+d)^p)^2-8/3*d^(3/2)*g*p^2*arctan(x*e^(1/2)/d^(1/2))*ln(2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))
/e^(3/2)+4*I*f*p^2*arctan(x*e^(1/2)/d^(1/2))^2*d^(1/2)/e^(1/2)-8*f*p^2*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/e^(1/
2)+4*I*f*p^2*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))*d^(1/2)/e^(1/2)+4*f*p*arctan(x*e^(1/2)/d^(1/2))*ln(c
*(e*x^2+d)^p)*d^(1/2)/e^(1/2)+8*f*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)-4/3*I*d^(3/2)*g*p^2*polylog(2,1-2*d^(1/2)/(d^(1/2)+I*x*e^(1/2)))/e^(3/2)

________________________________________________________________________________________

Rubi [A]
time = 0.46, antiderivative size = 548, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {2521, 2500, 2526, 2498, 327, 211, 2520, 12, 5040, 4964, 2449, 2352, 2507, 2505, 308} \begin {gather*} -\frac {4 i d^{3/2} g p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+\frac {4 i \sqrt {d} f p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {4 d^{3/2} g p \text {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 {4 \sqrt {d} f p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {4 i d^{3/2} g p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {32 d^{3/2} g p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {8 d^{3/2} g p^2 \text {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 {4 i \sqrt {d} f p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 \sqrt {d} f p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {8 \sqrt {d} f p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {4}{9} g p x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x}{9 e}+8 f p^2 x+\frac {8}{27} g p^2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

8*f*p^2*x - (32*d*g*p^2*x)/(9*e) + (8*g*p^2*x^3)/27 - (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] +
(32*d^(3/2)*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(9*e^(3/2)) + ((4*I)*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^
2)/Sqrt[e] - (((4*I)/3)*d^(3/2)*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2)/e^(3/2) + (8*Sqrt[d]*f*p^2*ArcTan[(Sqrt[e
]*x)/Sqrt[d]]*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/Sqrt[e] - (8*d^(3/2)*g*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]
*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/(3*e^(3/2)) - 4*f*p*x*Log[c*(d + e*x^2)^p] + (4*d*g*p*x*Log[c*(d +
e*x^2)^p])/(3*e) - (4*g*p*x^3*Log[c*(d + e*x^2)^p])/9 + (4*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d +
e*x^2)^p])/Sqrt[e] - (4*d^(3/2)*g*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*Log[c*(d + e*x^2)^p])/(3*e^(3/2)) + f*x*Log[c*
(d + e*x^2)^p]^2 + (g*x^3*Log[c*(d + e*x^2)^p]^2)/3 + ((4*I)*Sqrt[d]*f*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d]
+ I*Sqrt[e]*x)])/Sqrt[e] - (((4*I)/3)*d^(3/2)*g*p^2*PolyLog[2, 1 - (2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)])/e^(3
/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 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 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 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 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 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*} \int \left (f+g x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f \log ^2\left (c \left (d+e x^2\right )^p\right )+g x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+g \int x^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )-(4 e f p) \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{3} (4 e g p) \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )-(4 e f p) \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}{3} (4 e g p) \int \left (-\frac {d \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )-(4 f p) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+(4 d f p) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{3} (4 g p) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\frac {(4 d g p) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{3 e}-\frac {\left (4 d^2 g p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{3 e}\\ &=-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4}{9} g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f 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 {4 d^{3/2} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\left (8 e f p^2\right ) \int \frac {x^2}{d+e x^2} \, dx-\left (8 d e f 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}{3} \left (8 d g p^2\right ) \int \frac {x^2}{d+e x^2} \, dx+\frac {1}{3} \left (8 d^2 g 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}{9} \left (8 e g p^2\right ) \int \frac {x^4}{d+e x^2} \, dx\\ &=8 f p^2 x-\frac {8 d g p^2 x}{3 e}-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4}{9} g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f 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 {4 d^{3/2} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (8 d f p^2\right ) \int \frac {1}{d+e x^2} \, dx-\left (8 \sqrt {d} \sqrt {e} f p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx+\frac {\left (8 d^2 g p^2\right ) \int \frac {1}{d+e x^2} \, dx}{3 e}+\frac {\left (8 d^{3/2} g p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx}{3 \sqrt {e}}+\frac {1}{9} \left (8 e g 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\\ &=8 f p^2 x-\frac {32 d g p^2 x}{9 e}+\frac {8}{27} g p^2 x^3-\frac {8 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {8 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}+\frac {4 i \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4}{9} g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f 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 {4 d^{3/2} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\left (8 f 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 (8 d g p^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx}{3 e}+\frac {\left (8 d^2 g p^2\right ) \int \frac {1}{d+e x^2} \, dx}{9 e}\\ &=8 f p^2 x-\frac {32 d g p^2 x}{9 e}+\frac {8}{27} g p^2 x^3-\frac {8 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}+\frac {4 i \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {8 \sqrt {d} f p^2 \tan ^{-1}\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^{3/2} g p^2 \tan ^{-1}\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}}-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4}{9} g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f 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 {4 d^{3/2} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (8 f p^2\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx+\frac {\left (8 d g p^2\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx}{3 e}\\ &=8 f p^2 x-\frac {32 d g p^2 x}{9 e}+\frac {8}{27} g p^2 x^3-\frac {8 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}+\frac {4 i \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {8 \sqrt {d} f p^2 \tan ^{-1}\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^{3/2} g p^2 \tan ^{-1}\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}}-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4}{9} g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f 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 {4 d^{3/2} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {\left (8 i \sqrt {d} f p^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{\sqrt {e}}-\frac {\left (8 i d^{3/2} g p^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{3 e^{3/2}}\\ &=8 f p^2 x-\frac {32 d g p^2 x}{9 e}+\frac {8}{27} g p^2 x^3-\frac {8 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}+\frac {4 i \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {8 \sqrt {d} f p^2 \tan ^{-1}\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^{3/2} g p^2 \tan ^{-1}\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}}-4 f p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4}{9} g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f 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 {4 d^{3/2} g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+f x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {4 i \sqrt {d} f p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {4 i d^{3/2} g p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 281, normalized size = 0.51 \begin {gather*} \frac {-36 i \sqrt {d} (-3 e f+d g) p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2-12 \sqrt {d} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (2 (9 e f-4 d g) p+6 (-3 e f+d g) p \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+(-9 e f+3 d g) \log \left (c \left (d+e x^2\right )^p\right )\right )+\sqrt {e} x \left (8 p^2 \left (27 e f-12 d g+e g x^2\right )-12 p \left (9 e f-3 d g+e g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )+9 e \left (3 f+g x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )\right )-36 i \sqrt {d} (-3 e f+d g) p^2 \text {Li}_2\left (\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )}{27 e^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-36*I)*Sqrt[d]*(-3*e*f + d*g)*p^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]]^2 - 12*Sqrt[d]*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(
2*(9*e*f - 4*d*g)*p + 6*(-3*e*f + d*g)*p*Log[(2*Sqrt[d])/(Sqrt[d] + I*Sqrt[e]*x)] + (-9*e*f + 3*d*g)*Log[c*(d
+ e*x^2)^p]) + Sqrt[e]*x*(8*p^2*(27*e*f - 12*d*g + e*g*x^2) - 12*p*(9*e*f - 3*d*g + e*g*x^2)*Log[c*(d + e*x^2)
^p] + 9*e*(3*f + g*x^2)*Log[c*(d + e*x^2)^p]^2) - (36*I)*Sqrt[d]*(-3*e*f + d*g)*p^2*PolyLog[2, (I*Sqrt[d] + Sq
rt[e]*x)/((-I)*Sqrt[d] + Sqrt[e]*x)])/(27*e^(3/2))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (g \,x^{2}+f \right ) \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(g*p^2*x^3 + 3*f*p^2*x)*log(x^2*e + d)^2 + integrate(1/3*(3*g*x^4*e*log(c)^2 + 3*d*f*log(c)^2 + 3*(d*g*log
(c)^2 + f*e*log(c)^2)*x^2 - 2*((2*g*p^2 - 3*g*p*log(c))*x^4*e - 3*d*f*p*log(c) - 3*(d*g*p*log(c) - (2*f*p^2 -
f*p*log(c))*e)*x^2)*log(x^2*e + d))/(x^2*e + d), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (f + g x^{2}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,\left (g\,x^2+f\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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