\(\int (f+g x^3)^2 \log (c (d+e x^2)^p) \, dx\) [289]

Optimal result

Integrand size = 22, antiderivative size = 231 \[ \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-2 f^2 p x+\frac {2 d^3 g^2 p x}{7 e^3}+\frac {d f g p x^2}{2 e}-\frac {2 d^2 g^2 p x^3}{21 e^2}-\frac {1}{4} f g p x^4+\frac {2 d g^2 p x^5}{35 e}-\frac {2}{49} g^2 p x^7+\frac {2 \sqrt {d} f^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {2 d^{7/2} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}-\frac {d^2 f g p \log \left (d+e x^2\right )}{2 e^2}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right ) \]

[Out]

-2*f^2*p*x+2/7*d^3*g^2*p*x/e^3+1/2*d*f*g*p*x^2/e-2/21*d^2*g^2*p*x^3/e^2-1/4*f*g*p*x^4+2/35*d*g^2*p*x^5/e-2/49*
g^2*p*x^7-2/7*d^(7/2)*g^2*p*arctan(x*e^(1/2)/d^(1/2))/e^(7/2)-1/2*d^2*f*g*p*ln(e*x^2+d)/e^2+f^2*x*ln(c*(e*x^2+
d)^p)+1/2*f*g*x^4*ln(c*(e*x^2+d)^p)+1/7*g^2*x^7*ln(c*(e*x^2+d)^p)+2*f^2*p*arctan(x*e^(1/2)/d^(1/2))*d^(1/2)/e^
(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2521, 2498, 327, 211, 2504, 2442, 45, 2505, 308} \[ \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {2 d^{7/2} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {2 \sqrt {d} f^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d^3 g^2 p x}{7 e^3}-\frac {d^2 f g p \log \left (d+e x^2\right )}{2 e^2}-\frac {2 d^2 g^2 p x^3}{21 e^2}+\frac {d f g p x^2}{2 e}+\frac {2 d g^2 p x^5}{35 e}-2 f^2 p x-\frac {1}{4} f g p x^4-\frac {2}{49} g^2 p x^7 \]

[In]

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

[Out]

-2*f^2*p*x + (2*d^3*g^2*p*x)/(7*e^3) + (d*f*g*p*x^2)/(2*e) - (2*d^2*g^2*p*x^3)/(21*e^2) - (f*g*p*x^4)/4 + (2*d
*g^2*p*x^5)/(35*e) - (2*g^2*p*x^7)/49 + (2*Sqrt[d]*f^2*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] - (2*d^(7/2)*g^2
*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) - (d^2*f*g*p*Log[d + e*x^2])/(2*e^2) + f^2*x*Log[c*(d + e*x^2)^p]
+ (f*g*x^4*Log[c*(d + e*x^2)^p])/2 + (g^2*x^7*Log[c*(d + e*x^2)^p])/7

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 2442

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

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 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 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]))

Rubi steps \begin{align*} \text {integral}& = \int \left (f^2 \log \left (c \left (d+e x^2\right )^p\right )+2 f g x^3 \log \left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx \\ & = f^2 \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^3 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx \\ & = f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+(f g) \text {Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )-\left (2 e f^2 p\right ) \int \frac {x^2}{d+e x^2} \, dx-\frac {1}{7} \left (2 e g^2 p\right ) \int \frac {x^8}{d+e x^2} \, dx \\ & = -2 f^2 p x+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\left (2 d f^2 p\right ) \int \frac {1}{d+e x^2} \, dx-\frac {1}{2} (e f g p) \text {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^2\right )-\frac {1}{7} \left (2 e g^2 p\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 \\ & = -2 f^2 p x+\frac {2 d^3 g^2 p x}{7 e^3}-\frac {2 d^2 g^2 p x^3}{21 e^2}+\frac {2 d g^2 p x^5}{35 e}-\frac {2}{49} g^2 p x^7+\frac {2 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{2} (e f g p) \text {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,x^2\right )-\frac {\left (2 d^4 g^2 p\right ) \int \frac {1}{d+e x^2} \, dx}{7 e^3} \\ & = -2 f^2 p x+\frac {2 d^3 g^2 p x}{7 e^3}+\frac {d f g p x^2}{2 e}-\frac {2 d^2 g^2 p x^3}{21 e^2}-\frac {1}{4} f g p x^4+\frac {2 d g^2 p x^5}{35 e}-\frac {2}{49} g^2 p x^7+\frac {2 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {2 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}-\frac {d^2 f g p \log \left (d+e x^2\right )}{2 e^2}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.77 \[ \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {p x \left (840 d^3 g^2-280 d^2 e g^2 x^2+42 d e^2 g x \left (35 f+4 g x^3\right )-15 e^3 \left (392 f^2+49 f g x^3+8 g^2 x^6\right )\right )}{2940 e^3}-\frac {2 \sqrt {d} \left (-7 e^3 f^2+d^3 g^2\right ) p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}-\frac {d^2 f g p \log \left (d+e x^2\right )}{2 e^2}+\frac {1}{14} x \left (14 f^2+7 f g x^3+2 g^2 x^6\right ) \log \left (c \left (d+e x^2\right )^p\right ) \]

[In]

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

[Out]

(p*x*(840*d^3*g^2 - 280*d^2*e*g^2*x^2 + 42*d*e^2*g*x*(35*f + 4*g*x^3) - 15*e^3*(392*f^2 + 49*f*g*x^3 + 8*g^2*x
^6)))/(2940*e^3) - (2*Sqrt[d]*(-7*e^3*f^2 + d^3*g^2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(7*e^(7/2)) - (d^2*f*g*p*L
og[d + e*x^2])/(2*e^2) + (x*(14*f^2 + 7*f*g*x^3 + 2*g^2*x^6)*Log[c*(d + e*x^2)^p])/14

Maple [A] (verified)

Time = 2.72 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.84

method	result	size
parts	\(\frac {g^{2} x^{7} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{7}+\frac {f g \,x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2}+f^{2} x \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )-\frac {p e \left (-\frac {-\frac {2}{7} e^{3} g^{2} x^{7}+\frac {2}{5} d \,e^{2} g^{2} x^{5}-\frac {7}{4} e^{3} f g \,x^{4}-\frac {2}{3} d^{2} e \,g^{2} x^{3}+\frac {7}{2} d f g \,x^{2} e^{2}+2 x \,d^{3} g^{2}-14 x \,e^{3} f^{2}}{e^{4}}+\frac {d \left (\frac {7 d e f g \ln \left (e \,x^{2}+d \right )}{2}+\frac {\left (2 d^{3} g^{2}-14 e^{3} f^{2}\right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{\sqrt {d e}}\right )}{e^{4}}\right )}{7}\)	\(195\)
risch	\(x \ln \left (c \right ) f^{2}+\frac {i \pi \,g^{2} x^{7} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{14}-\frac {i \pi f g \,x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{4}-\frac {p \ln \left (-d^{4} g^{2}+7 d \,e^{3} f^{2}-\sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}\, x \right ) d^{2} f g}{2 e^{2}}-\frac {p \ln \left (-d^{4} g^{2}+7 d \,e^{3} f^{2}+\sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}\, x \right ) d^{2} f g}{2 e^{2}}+\frac {\ln \left (c \right ) g^{2} x^{7}}{7}+\frac {i x \pi \,f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\frac {i x \pi \,f^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,g^{2} x^{7} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{14}-\frac {i x \pi \,f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}-\frac {p \ln \left (-d^{4} g^{2}+7 d \,e^{3} f^{2}+\sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}\, x \right ) \sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}}{7 e^{4}}+\frac {p \ln \left (-d^{4} g^{2}+7 d \,e^{3} f^{2}-\sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}\, x \right ) \sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}}{7 e^{4}}+\frac {\ln \left (c \right ) f g \,x^{4}}{2}-\frac {i \pi f g \,x^{4} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{4}-\frac {i \pi \,g^{2} x^{7} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{14}+\frac {i \pi f g \,x^{4} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{4}+\frac {i \pi f g \,x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{4}-\frac {i x \pi \,f^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}+\frac {i \pi \,g^{2} x^{7} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{14}+\left (\frac {1}{7} g^{2} x^{7}+\frac {1}{2} f g \,x^{4}+f^{2} x \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )-\frac {2 g^{2} p \,x^{7}}{49}-2 f^{2} p x +\frac {2 d^{3} g^{2} p x}{7 e^{3}}-\frac {2 d^{2} g^{2} p \,x^{3}}{21 e^{2}}+\frac {2 d \,g^{2} p \,x^{5}}{35 e}+\frac {d f g p \,x^{2}}{2 e}-\frac {f g p \,x^{4}}{4}\)	\(869\)

[In]

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

[Out]

1/7*g^2*x^7*ln(c*(e*x^2+d)^p)+1/2*f*g*x^4*ln(c*(e*x^2+d)^p)+f^2*x*ln(c*(e*x^2+d)^p)-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*ln(e*x^2+d)+(2*d^3*g^2-14*e^3*f^2)/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.97 \[ \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\left [-\frac {120 \, e^{3} g^{2} p x^{7} - 168 \, d e^{2} g^{2} p x^{5} + 735 \, e^{3} f g p x^{4} + 280 \, d^{2} e g^{2} p x^{3} - 1470 \, d e^{2} f g p x^{2} + 420 \, {\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p \sqrt {-\frac {d}{e}} \log \left (\frac {e x^{2} - 2 \, e x \sqrt {-\frac {d}{e}} - d}{e x^{2} + d}\right ) + 840 \, {\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p x - 210 \, {\left (2 \, e^{3} g^{2} p x^{7} + 7 \, e^{3} f g p x^{4} + 14 \, e^{3} f^{2} p x - 7 \, d^{2} e f g p\right )} \log \left (e x^{2} + d\right ) - 210 \, {\left (2 \, e^{3} g^{2} x^{7} + 7 \, e^{3} f g x^{4} + 14 \, e^{3} f^{2} x\right )} \log \left (c\right )}{2940 \, e^{3}}, -\frac {120 \, e^{3} g^{2} p x^{7} - 168 \, d e^{2} g^{2} p x^{5} + 735 \, e^{3} f g p x^{4} + 280 \, d^{2} e g^{2} p x^{3} - 1470 \, d e^{2} f g p x^{2} - 840 \, {\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p \sqrt {\frac {d}{e}} \arctan \left (\frac {e x \sqrt {\frac {d}{e}}}{d}\right ) + 840 \, {\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p x - 210 \, {\left (2 \, e^{3} g^{2} p x^{7} + 7 \, e^{3} f g p x^{4} + 14 \, e^{3} f^{2} p x - 7 \, d^{2} e f g p\right )} \log \left (e x^{2} + d\right ) - 210 \, {\left (2 \, e^{3} g^{2} x^{7} + 7 \, e^{3} f g x^{4} + 14 \, e^{3} f^{2} x\right )} \log \left (c\right )}{2940 \, e^{3}}\right ] \]

[In]

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

[Out]

[-1/2940*(120*e^3*g^2*p*x^7 - 168*d*e^2*g^2*p*x^5 + 735*e^3*f*g*p*x^4 + 280*d^2*e*g^2*p*x^3 - 1470*d*e^2*f*g*p
*x^2 + 420*(7*e^3*f^2 - d^3*g^2)*p*sqrt(-d/e)*log((e*x^2 - 2*e*x*sqrt(-d/e) - d)/(e*x^2 + d)) + 840*(7*e^3*f^2
- d^3*g^2)*p*x - 210*(2*e^3*g^2*p*x^7 + 7*e^3*f*g*p*x^4 + 14*e^3*f^2*p*x - 7*d^2*e*f*g*p)*log(e*x^2 + d) - 21
0*(2*e^3*g^2*x^7 + 7*e^3*f*g*x^4 + 14*e^3*f^2*x)*log(c))/e^3, -1/2940*(120*e^3*g^2*p*x^7 - 168*d*e^2*g^2*p*x^5
+ 735*e^3*f*g*p*x^4 + 280*d^2*e*g^2*p*x^3 - 1470*d*e^2*f*g*p*x^2 - 840*(7*e^3*f^2 - d^3*g^2)*p*sqrt(d/e)*arct
an(e*x*sqrt(d/e)/d) + 840*(7*e^3*f^2 - d^3*g^2)*p*x - 210*(2*e^3*g^2*p*x^7 + 7*e^3*f*g*p*x^4 + 14*e^3*f^2*p*x
- 7*d^2*e*f*g*p)*log(e*x^2 + d) - 210*(2*e^3*g^2*x^7 + 7*e^3*f*g*x^4 + 14*e^3*f^2*x)*log(c))/e^3]

Sympy [A] (verification not implemented)

Time = 120.70 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.90 \[ \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\begin {cases} \left (f^{2} x + \frac {f g x^{4}}{2} + \frac {g^{2} x^{7}}{7}\right ) \log {\left (0^{p} c \right )} & \text {for}\: d = 0 \wedge e = 0 \\\left (f^{2} x + \frac {f g x^{4}}{2} + \frac {g^{2} x^{7}}{7}\right ) \log {\left (c d^{p} \right )} & \text {for}\: e = 0 \\- 2 f^{2} p x + f^{2} x \log {\left (c \left (e x^{2}\right )^{p} \right )} - \frac {f g p x^{4}}{4} + \frac {f g x^{4} \log {\left (c \left (e x^{2}\right )^{p} \right )}}{2} - \frac {2 g^{2} p x^{7}}{49} + \frac {g^{2} x^{7} \log {\left (c \left (e x^{2}\right )^{p} \right )}}{7} & \text {for}\: d = 0 \\- \frac {2 d^{4} g^{2} p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{7 e^{4} \sqrt {- \frac {d}{e}}} + \frac {d^{4} g^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{7 e^{4} \sqrt {- \frac {d}{e}}} + \frac {2 d^{3} g^{2} p x}{7 e^{3}} - \frac {d^{2} f g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2 e^{2}} - \frac {2 d^{2} g^{2} p x^{3}}{21 e^{2}} + \frac {2 d f^{2} p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{e \sqrt {- \frac {d}{e}}} - \frac {d f^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{e \sqrt {- \frac {d}{e}}} + \frac {d f g p x^{2}}{2 e} + \frac {2 d g^{2} p x^{5}}{35 e} - 2 f^{2} p x + f^{2} x \log {\left (c \left (d + e x^{2}\right )^{p} \right )} - \frac {f g p x^{4}}{4} + \frac {f g x^{4} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2} - \frac {2 g^{2} p x^{7}}{49} + \frac {g^{2} x^{7} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{7} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((f**2*x + f*g*x**4/2 + g**2*x**7/7)*log(0**p*c), Eq(d, 0) & Eq(e, 0)), ((f**2*x + f*g*x**4/2 + g**2
*x**7/7)*log(c*d**p), Eq(e, 0)), (-2*f**2*p*x + f**2*x*log(c*(e*x**2)**p) - f*g*p*x**4/4 + f*g*x**4*log(c*(e*x
**2)**p)/2 - 2*g**2*p*x**7/49 + g**2*x**7*log(c*(e*x**2)**p)/7, Eq(d, 0)), (-2*d**4*g**2*p*log(x - sqrt(-d/e))
/(7*e**4*sqrt(-d/e)) + d**4*g**2*log(c*(d + e*x**2)**p)/(7*e**4*sqrt(-d/e)) + 2*d**3*g**2*p*x/(7*e**3) - d**2*
f*g*log(c*(d + e*x**2)**p)/(2*e**2) - 2*d**2*g**2*p*x**3/(21*e**2) + 2*d*f**2*p*log(x - sqrt(-d/e))/(e*sqrt(-d
/e)) - d*f**2*log(c*(d + e*x**2)**p)/(e*sqrt(-d/e)) + d*f*g*p*x**2/(2*e) + 2*d*g**2*p*x**5/(35*e) - 2*f**2*p*x
+ f**2*x*log(c*(d + e*x**2)**p) - f*g*p*x**4/4 + f*g*x**4*log(c*(d + e*x**2)**p)/2 - 2*g**2*p*x**7/49 + g**2*
x**7*log(c*(d + e*x**2)**p)/7, True))

Maxima [F(-2)]

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

[In]

integrate((g*x^3+f)^2*log(c*(e*x^2+d)^p),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 [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.88 \[ \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {2 \, d g^{2} p x^{5}}{35 \, e} - \frac {1}{49} \, {\left (2 \, g^{2} p - 7 \, g^{2} \log \left (c\right )\right )} x^{7} - \frac {2 \, d^{2} g^{2} p x^{3}}{21 \, e^{2}} + \frac {d f g p x^{2}}{2 \, e} - \frac {1}{4} \, {\left (f g p - 2 \, f g \log \left (c\right )\right )} x^{4} - \frac {d^{2} f g p \log \left (e x^{2} + d\right )}{2 \, e^{2}} + \frac {1}{14} \, {\left (2 \, g^{2} p x^{7} + 7 \, f g p x^{4} + 14 \, f^{2} p x\right )} \log \left (e x^{2} + d\right ) - \frac {{\left (14 \, e^{3} f^{2} p - 2 \, d^{3} g^{2} p - 7 \, e^{3} f^{2} \log \left (c\right )\right )} x}{7 \, e^{3}} + \frac {2 \, {\left (7 \, d e^{3} f^{2} p - d^{4} g^{2} p\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{7 \, \sqrt {d e} e^{3}} \]

[In]

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

[Out]

2/35*d*g^2*p*x^5/e - 1/49*(2*g^2*p - 7*g^2*log(c))*x^7 - 2/21*d^2*g^2*p*x^3/e^2 + 1/2*d*f*g*p*x^2/e - 1/4*(f*g
*p - 2*f*g*log(c))*x^4 - 1/2*d^2*f*g*p*log(e*x^2 + d)/e^2 + 1/14*(2*g^2*p*x^7 + 7*f*g*p*x^4 + 14*f^2*p*x)*log(
e*x^2 + d) - 1/7*(14*e^3*f^2*p - 2*d^3*g^2*p - 7*e^3*f^2*log(c))*x/e^3 + 2/7*(7*d*e^3*f^2*p - d^4*g^2*p)*arcta
n(e*x/sqrt(d*e))/(sqrt(d*e)*e^3)

Mupad [B] (verification not implemented)

Time = 4.06 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.37 \[ \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {g^2\,x^7\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{7}-2\,f^2\,p\,x-\frac {2\,g^2\,p\,x^7}{49}+f^2\,x\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )+\frac {f\,g\,x^4\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{2}-\frac {f\,g\,p\,x^4}{4}+\frac {2\,d\,g^2\,p\,x^5}{35\,e}+\frac {2\,d^3\,g^2\,p\,x}{7\,e^3}-\frac {2\,\sqrt {d}\,f^2\,p\,\mathrm {atan}\left (\frac {7\,\sqrt {d}\,e^{7/2}\,f^2\,p\,x}{d^4\,g^2\,p-7\,d\,e^3\,f^2\,p}-\frac {d^{7/2}\,\sqrt {e}\,g^2\,p\,x}{d^4\,g^2\,p-7\,d\,e^3\,f^2\,p}\right )}{\sqrt {e}}+\frac {2\,d^{7/2}\,g^2\,p\,\mathrm {atan}\left (\frac {7\,\sqrt {d}\,e^{7/2}\,f^2\,p\,x}{d^4\,g^2\,p-7\,d\,e^3\,f^2\,p}-\frac {d^{7/2}\,\sqrt {e}\,g^2\,p\,x}{d^4\,g^2\,p-7\,d\,e^3\,f^2\,p}\right )}{7\,e^{7/2}}-\frac {2\,d^2\,g^2\,p\,x^3}{21\,e^2}+\frac {d\,f\,g\,p\,x^2}{2\,e}-\frac {d^2\,f\,g\,p\,\ln \left (e\,x^2+d\right )}{2\,e^2} \]

[In]

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

[Out]

(g^2*x^7*log(c*(d + e*x^2)^p))/7 - 2*f^2*p*x - (2*g^2*p*x^7)/49 + f^2*x*log(c*(d + e*x^2)^p) + (f*g*x^4*log(c*
(d + e*x^2)^p))/2 - (f*g*p*x^4)/4 + (2*d*g^2*p*x^5)/(35*e) + (2*d^3*g^2*p*x)/(7*e^3) - (2*d^(1/2)*f^2*p*atan((
7*d^(1/2)*e^(7/2)*f^2*p*x)/(d^4*g^2*p - 7*d*e^3*f^2*p) - (d^(7/2)*e^(1/2)*g^2*p*x)/(d^4*g^2*p - 7*d*e^3*f^2*p)
))/e^(1/2) + (2*d^(7/2)*g^2*p*atan((7*d^(1/2)*e^(7/2)*f^2*p*x)/(d^4*g^2*p - 7*d*e^3*f^2*p) - (d^(7/2)*e^(1/2)*
g^2*p*x)/(d^4*g^2*p - 7*d*e^3*f^2*p)))/(7*e^(7/2)) - (2*d^2*g^2*p*x^3)/(21*e^2) + (d*f*g*p*x^2)/(2*e) - (d^2*f
*g*p*log(d + e*x^2))/(2*e^2)