3.4.65 \(\int \frac {(f+g x^{3 n})^2 \log (c (d+e x^n)^p)}{x} \, dx\) [365]

Optimal. Leaf size=327 \[ -\frac {2 d^2 f g p x^n}{3 e^2 n}+\frac {d^5 g^2 p x^n}{6 e^5 n}+\frac {d f g p x^{2 n}}{3 e n}-\frac {d^4 g^2 p x^{2 n}}{12 e^4 n}-\frac {2 f g p x^{3 n}}{9 n}+\frac {d^3 g^2 p x^{3 n}}{18 e^3 n}-\frac {d^2 g^2 p x^{4 n}}{24 e^2 n}+\frac {d g^2 p x^{5 n}}{30 e n}-\frac {g^2 p x^{6 n}}{36 n}+\frac {2 d^3 f g p \log \left (d+e x^n\right )}{3 e^3 n}-\frac {d^6 g^2 p \log \left (d+e x^n\right )}{6 e^6 n}+\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n} \]

[Out]

-2/3*d^2*f*g*p*x^n/e^2/n+1/6*d^5*g^2*p*x^n/e^5/n+1/3*d*f*g*p*x^(2*n)/e/n-1/12*d^4*g^2*p*x^(2*n)/e^4/n-2/9*f*g*
p*x^(3*n)/n+1/18*d^3*g^2*p*x^(3*n)/e^3/n-1/24*d^2*g^2*p*x^(4*n)/e^2/n+1/30*d*g^2*p*x^(5*n)/e/n-1/36*g^2*p*x^(6
*n)/n+2/3*d^3*f*g*p*ln(d+e*x^n)/e^3/n-1/6*d^6*g^2*p*ln(d+e*x^n)/e^6/n+2/3*f*g*x^(3*n)*ln(c*(d+e*x^n)^p)/n+1/6*
g^2*x^(6*n)*ln(c*(d+e*x^n)^p)/n+f^2*ln(-e*x^n/d)*ln(c*(d+e*x^n)^p)/n+f^2*p*polylog(2,1+e*x^n/d)/n

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Rubi [A]
time = 0.21, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2525, 272, 45, 2463, 2441, 2352, 2442} \begin {gather*} \frac {f^2 p \text {PolyLog}\left (2,\frac {e x^n}{d}+1\right )}{n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 n}-\frac {d^6 g^2 p \log \left (d+e x^n\right )}{6 e^6 n}+\frac {d^5 g^2 p x^n}{6 e^5 n}-\frac {d^4 g^2 p x^{2 n}}{12 e^4 n}+\frac {2 d^3 f g p \log \left (d+e x^n\right )}{3 e^3 n}+\frac {d^3 g^2 p x^{3 n}}{18 e^3 n}-\frac {2 d^2 f g p x^n}{3 e^2 n}-\frac {d^2 g^2 p x^{4 n}}{24 e^2 n}+\frac {d f g p x^{2 n}}{3 e n}+\frac {d g^2 p x^{5 n}}{30 e n}-\frac {2 f g p x^{3 n}}{9 n}-\frac {g^2 p x^{6 n}}{36 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^2*f*g*p*x^n)/(3*e^2*n) + (d^5*g^2*p*x^n)/(6*e^5*n) + (d*f*g*p*x^(2*n))/(3*e*n) - (d^4*g^2*p*x^(2*n))/(12
*e^4*n) - (2*f*g*p*x^(3*n))/(9*n) + (d^3*g^2*p*x^(3*n))/(18*e^3*n) - (d^2*g^2*p*x^(4*n))/(24*e^2*n) + (d*g^2*p
*x^(5*n))/(30*e*n) - (g^2*p*x^(6*n))/(36*n) + (2*d^3*f*g*p*Log[d + e*x^n])/(3*e^3*n) - (d^6*g^2*p*Log[d + e*x^
n])/(6*e^6*n) + (2*f*g*x^(3*n)*Log[c*(d + e*x^n)^p])/(3*n) + (g^2*x^(6*n)*Log[c*(d + e*x^n)^p])/(6*n) + (f^2*L
og[-((e*x^n)/d)]*Log[c*(d + e*x^n)^p])/n + (f^2*p*PolyLog[2, 1 + (e*x^n)/d])/n

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 2441

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

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 2463

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

Rule 2525

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rubi steps

\begin {align*} \int \frac {\left (f+g x^{3 n}\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (f+g x^3\right )^2 \log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {f^2 \log \left (c (d+e x)^p\right )}{x}+2 f g x^2 \log \left (c (d+e x)^p\right )+g^2 x^5 \log \left (c (d+e x)^p\right )\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {f^2 \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}+\frac {(2 f g) \text {Subst}\left (\int x^2 \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}+\frac {g^2 \text {Subst}\left (\int x^5 \log \left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}-\frac {\left (e f^2 p\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}-\frac {(2 e f g p) \text {Subst}\left (\int \frac {x^3}{d+e x} \, dx,x,x^n\right )}{3 n}-\frac {\left (e g^2 p\right ) \text {Subst}\left (\int \frac {x^6}{d+e x} \, dx,x,x^n\right )}{6 n}\\ &=\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}-\frac {(2 e f g p) \text {Subst}\left (\int \left (\frac {d^2}{e^3}-\frac {d x}{e^2}+\frac {x^2}{e}-\frac {d^3}{e^3 (d+e x)}\right ) \, dx,x,x^n\right )}{3 n}-\frac {\left (e g^2 p\right ) \text {Subst}\left (\int \left (-\frac {d^5}{e^6}+\frac {d^4 x}{e^5}-\frac {d^3 x^2}{e^4}+\frac {d^2 x^3}{e^3}-\frac {d x^4}{e^2}+\frac {x^5}{e}+\frac {d^6}{e^6 (d+e x)}\right ) \, dx,x,x^n\right )}{6 n}\\ &=-\frac {2 d^2 f g p x^n}{3 e^2 n}+\frac {d^5 g^2 p x^n}{6 e^5 n}+\frac {d f g p x^{2 n}}{3 e n}-\frac {d^4 g^2 p x^{2 n}}{12 e^4 n}-\frac {2 f g p x^{3 n}}{9 n}+\frac {d^3 g^2 p x^{3 n}}{18 e^3 n}-\frac {d^2 g^2 p x^{4 n}}{24 e^2 n}+\frac {d g^2 p x^{5 n}}{30 e n}-\frac {g^2 p x^{6 n}}{36 n}+\frac {2 d^3 f g p \log \left (d+e x^n\right )}{3 e^3 n}-\frac {d^6 g^2 p \log \left (d+e x^n\right )}{6 e^6 n}+\frac {2 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )}{3 n}+\frac {g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )}{6 n}+\frac {f^2 \log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac {f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{n}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 341, normalized size = 1.04 \begin {gather*} \frac {-240 d^2 e^4 f g p x^n+60 d^5 e g^2 p x^n+120 d e^5 f g p x^{2 n}-30 d^4 e^2 g^2 p x^{2 n}-80 e^6 f g p x^{3 n}+20 d^3 e^3 g^2 p x^{3 n}-15 d^2 e^4 g^2 p x^{4 n}+12 d e^5 g^2 p x^{5 n}-10 e^6 g^2 p x^{6 n}+240 d^3 e^3 f g p \log \left (d-d x^n\right )-60 d^6 g^2 p \log \left (d-d x^n\right )+360 e^6 f^2 p \log \left (-\frac {e x^n}{d}\right ) \log \left (d+e x^n\right )+240 e^6 f g x^{3 n} \log \left (c \left (d+e x^n\right )^p\right )+60 e^6 g^2 x^{6 n} \log \left (c \left (d+e x^n\right )^p\right )-60 n \log (x) \left (d^3 g \left (-4 e^3 f+d^3 g\right ) p+6 e^6 f^2 p \log \left (d+e x^n\right )-6 e^6 f^2 \log \left (c \left (d+e x^n\right )^p\right )\right )+360 e^6 f^2 p \text {Li}_2\left (1+\frac {e x^n}{d}\right )}{360 e^6 n} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-240*d^2*e^4*f*g*p*x^n + 60*d^5*e*g^2*p*x^n + 120*d*e^5*f*g*p*x^(2*n) - 30*d^4*e^2*g^2*p*x^(2*n) - 80*e^6*f*g
*p*x^(3*n) + 20*d^3*e^3*g^2*p*x^(3*n) - 15*d^2*e^4*g^2*p*x^(4*n) + 12*d*e^5*g^2*p*x^(5*n) - 10*e^6*g^2*p*x^(6*
n) + 240*d^3*e^3*f*g*p*Log[d - d*x^n] - 60*d^6*g^2*p*Log[d - d*x^n] + 360*e^6*f^2*p*Log[-((e*x^n)/d)]*Log[d +
e*x^n] + 240*e^6*f*g*x^(3*n)*Log[c*(d + e*x^n)^p] + 60*e^6*g^2*x^(6*n)*Log[c*(d + e*x^n)^p] - 60*n*Log[x]*(d^3
*g*(-4*e^3*f + d^3*g)*p + 6*e^6*f^2*p*Log[d + e*x^n] - 6*e^6*f^2*Log[c*(d + e*x^n)^p]) + 360*e^6*f^2*p*PolyLog
[2, 1 + (e*x^n)/d])/(360*e^6*n)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.34, size = 795, normalized size = 2.43

method	result	size
risch	\(\frac {d^{5} g^{2} p \,x^{n}}{6 e^{5} n}-\frac {d^{4} g^{2} p \,x^{2 n}}{12 e^{4} n}-\frac {g^{2} p \,x^{6 n}}{36 n}+\frac {2 d^{3} f g p \ln \left (d +e \,x^{n}\right )}{3 e^{3} n}-\frac {d^{6} g^{2} p \ln \left (d +e \,x^{n}\right )}{6 e^{6} n}+\frac {d^{3} g^{2} p \,x^{3 n}}{18 e^{3} n}-\frac {d^{2} g^{2} p \,x^{4 n}}{24 e^{2} n}+\frac {d \,g^{2} p \,x^{5 n}}{30 e n}-\frac {2 d^{2} f g p \,x^{n}}{3 e^{2} n}+\frac {d f g p \,x^{2 n}}{3 e n}-\frac {2 f g p \,x^{3 n}}{9 n}+\frac {2 \ln \left (c \right ) f g \,x^{3 n}}{3 n}+\frac {\left (g^{2} x^{6 n}+4 f g \,x^{3 n}+6 f^{2} \ln \left (x \right ) n \right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{6 n}+\frac {\ln \left (c \right ) g^{2} x^{6 n}}{6 n}-\frac {p \,f^{2} \dilog \left (\frac {d +e \,x^{n}}{d}\right )}{n}-p \,f^{2} \ln \left (x \right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )+\frac {\ln \left (c \right ) f^{2} \ln \left (x^{n}\right )}{n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) g^{2} x^{6 n}}{12 n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} g^{2} x^{6 n}}{12 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} f g \,x^{3 n}}{3 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) f^{2} \ln \left (x^{n}\right )}{2 n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} f^{2} \ln \left (x^{n}\right )}{2 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) f g \,x^{3 n}}{3 n}+\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) f g \,x^{3 n}}{3 n}+\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} f g \,x^{3 n}}{3 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) g^{2} x^{6 n}}{12 n}-\frac {i \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right ) f^{2} \ln \left (x^{n}\right )}{2 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} g^{2} x^{6 n}}{12 n}-\frac {i \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3} f^{2} \ln \left (x^{n}\right )}{2 n}\)	\(795\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*d^5*g^2*p*x^n/e^5/n+2/3*d^3*f*g*p*ln(d+e*x^n)/e^3/n-1/6*d^6*g^2*p*ln(d+e*x^n)/e^6/n-2/3*d^2*f*g*p*x^n/e^2/
n+1/6*(g^2*(x^n)^6+4*f*g*(x^n)^3+6*f^2*ln(x)*n)/n*ln((d+e*x^n)^p)-1/36*p/n*g^2*(x^n)^6-p/n*f^2*dilog((d+e*x^n)
/d)-p*f^2*ln(x)*ln((d+e*x^n)/d)+1/6/n*ln(c)*g^2*(x^n)^6+1/n*ln(c)*f^2*ln(x^n)+2/3/n*ln(c)*f*g*(x^n)^3-2/9*p/n*
f*g*(x^n)^3+1/12*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)*g^2*(x^n)^6+1/2*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(
I*c)*f^2*ln(x^n)+1/3*p/e/n*f*g*d*(x^n)^2+1/12*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2*g^2*(x^n)^6+1
/2*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2*f^2*ln(x^n)-1/3*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^3*f*g*(x^n)
^3-1/3*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn(I*c)*f*g*(x^n)^3-1/2*I/n*Pi*csgn(I*(d+e*x^n)^p)*c
sgn(I*c*(d+e*x^n)^p)*csgn(I*c)*f^2*ln(x^n)+1/3*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^2*csgn(I*c)*f*g*(x^n)^3+1/3*I/n*Pi
*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)^2*f*g*(x^n)^3-1/12*I/n*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)
*csgn(I*c)*g^2*(x^n)^6-1/2*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^3*f^2*ln(x^n)+1/30*p/e/n*g^2*(x^n)^5*d-1/24*p/e^2/n*g^
2*(x^n)^4*d^2+1/18*p/e^3/n*g^2*d^3*(x^n)^3-1/12*p/e^4/n*g^2*(x^n)^2*d^4-1/12*I/n*Pi*csgn(I*c*(d+e*x^n)^p)^3*g^
2*(x^n)^6

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

[Out]

-1/360*(180*f^2*n^2*p*e^6*log(x)^2 + 15*d^2*g^2*p*e^(4*n*log(x) + 4) - 12*d*g^2*p*e^(5*n*log(x) + 5) - 20*(d^3
*g^2*p*e^3 - 4*(f*g*p - 3*f*g*log(c))*e^6)*x^(3*n) + 30*(d^4*g^2*p*e^2 - 4*d*f*g*p*e^5)*x^(2*n) - 60*(d^5*g^2*
p*e - 4*d^2*f*g*p*e^4)*x^n + 10*(g^2*p - 6*g^2*log(c))*e^(6*n*log(x) + 6) - 60*(6*f^2*n*e^6*log(x) + g^2*e^(6*
n*log(x) + 6) + 4*f*g*e^(3*n*log(x) + 6))*log((d + e^(n*log(x) + 1))^p) + 60*(d^6*g^2*n*p - 4*d^3*f*g*n*p*e^3
- 6*f^2*n*e^6*log(c))*log(x))*e^(-6)/n + integrate(1/6*(d^7*g^2*p - 4*d^4*f*g*p*e^3 + 6*d*f^2*n*p*e^6*log(x))/
(d*x*e^6 + x*e^(n*log(x) + 7)), x)

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Fricas [A]
time = 0.38, size = 273, normalized size = 0.83 \begin {gather*} -\frac {{\left (15 \, d^{2} g^{2} p x^{4 \, n} e^{4} + 360 \, f^{2} n p e^{6} \log \left (x\right ) \log \left (\frac {x^{n} e + d}{d}\right ) - 12 \, d g^{2} p x^{5 \, n} e^{5} - 360 \, f^{2} n e^{6} \log \left (c\right ) \log \left (x\right ) + 360 \, f^{2} p {\rm Li}_2\left (-\frac {x^{n} e + d}{d} + 1\right ) e^{6} + 10 \, {\left (g^{2} p e^{6} - 6 \, g^{2} e^{6} \log \left (c\right )\right )} x^{6 \, n} - 20 \, {\left (d^{3} g^{2} p e^{3} - 4 \, f g p e^{6} + 12 \, f g e^{6} \log \left (c\right )\right )} x^{3 \, n} + 30 \, {\left (d^{4} g^{2} p e^{2} - 4 \, d f g p e^{5}\right )} x^{2 \, n} - 60 \, {\left (d^{5} g^{2} p e - 4 \, d^{2} f g p e^{4}\right )} x^{n} + 60 \, {\left (d^{6} g^{2} p - 4 \, d^{3} f g p e^{3} - 6 \, f^{2} n p e^{6} \log \left (x\right ) - g^{2} p x^{6 \, n} e^{6} - 4 \, f g p x^{3 \, n} e^{6}\right )} \log \left (x^{n} e + d\right )\right )} e^{\left (-6\right )}}{360 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/360*(15*d^2*g^2*p*x^(4*n)*e^4 + 360*f^2*n*p*e^6*log(x)*log((x^n*e + d)/d) - 12*d*g^2*p*x^(5*n)*e^5 - 360*f^
2*n*e^6*log(c)*log(x) + 360*f^2*p*dilog(-(x^n*e + d)/d + 1)*e^6 + 10*(g^2*p*e^6 - 6*g^2*e^6*log(c))*x^(6*n) -
20*(d^3*g^2*p*e^3 - 4*f*g*p*e^6 + 12*f*g*e^6*log(c))*x^(3*n) + 30*(d^4*g^2*p*e^2 - 4*d*f*g*p*e^5)*x^(2*n) - 60
*(d^5*g^2*p*e - 4*d^2*f*g*p*e^4)*x^n + 60*(d^6*g^2*p - 4*d^3*f*g*p*e^3 - 6*f^2*n*p*e^6*log(x) - g^2*p*x^(6*n)*
e^6 - 4*f*g*p*x^(3*n)*e^6)*log(x^n*e + d))*e^(-6)/n

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+g\,x^{3\,n}\right )}^2}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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