3.4.85 \(\int \frac {(a+b \log (c (d+e x)^n)) (f+g \log (c (d+e x)^n))}{x^4} \, dx\) [385]

Optimal. Leaf size=234 \[ -\frac {b e^2 g n^2}{3 d^2 x}-\frac {b e^3 g n^2 \log (x)}{d^3}+\frac {b e^3 g n^2 \log (d+e x)}{3 d^3}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 x^3}-\frac {e n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac {e^2 n (d+e x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}+\frac {e^3 n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{3 d^3}-\frac {2 b e^3 g n^2 \text {Li}_2\left (\frac {d}{d+e x}\right )}{3 d^3} \]

[Out]

-1/3*b*e^2*g*n^2/d^2/x-b*e^3*g*n^2*ln(x)/d^3+1/3*b*e^3*g*n^2*ln(e*x+d)/d^3-1/3*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(c
*(e*x+d)^n))/x^3-1/6*e*n*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))/d/x^2+1/3*e^2*n*(e*x+d)*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n
))/d^3/x+1/3*e^3*n*(b*f+a*g+2*b*g*ln(c*(e*x+d)^n))*ln(1-d/(e*x+d))/d^3-2/3*b*e^3*g*n^2*polylog(2,d/(e*x+d))/d^
3

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Rubi [A]
time = 0.34, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2483, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \begin {gather*} -\frac {2 b e^3 g n^2 \text {PolyLog}\left (2,\frac {d}{d+e x}\right )}{3 d^3}+\frac {e^3 n \log \left (1-\frac {d}{d+e x}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{3 d^3}+\frac {e^2 n (d+e x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{3 d^3 x}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right )}{3 x^3}-\frac {e n \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )}{6 d x^2}-\frac {b e^3 g n^2 \log (x)}{d^3}+\frac {b e^3 g n^2 \log (d+e x)}{3 d^3}-\frac {b e^2 g n^2}{3 d^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/x^4,x]

[Out]

-1/3*(b*e^2*g*n^2)/(d^2*x) - (b*e^3*g*n^2*Log[x])/d^3 + (b*e^3*g*n^2*Log[d + e*x])/(3*d^3) - ((a + b*Log[c*(d
+ e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/(3*x^3) - (e*n*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(6*d*x^2) + (e^2
*n*(d + e*x)*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n]))/(3*d^3*x) + (e^3*n*(b*f + a*g + 2*b*g*Log[c*(d + e*x)^n])
*Log[1 - d/(d + e*x)])/(3*d^3) - (2*b*e^3*g*n^2*PolyLog[2, d/(d + e*x)])/(3*d^3)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2351

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

Rule 2356

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

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2438

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

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 2483

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 x^3}+\frac {1}{3} (b e n) \int \frac {f+g \log \left (c (d+e x)^n\right )}{x^3 (d+e x)} \, dx+\frac {1}{3} (e g n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (d+e x)} \, dx\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 x^3}+\frac {1}{3} (b n) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e x\right )+\frac {1}{3} (g n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e x\right )\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 x^3}+\frac {(b n) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e x\right )}{3 d}-\frac {(b e n) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{3 d}+\frac {(g n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+e x\right )}{3 d}-\frac {(e g n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{3 d}\\ &=-\frac {e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 d x^2}-\frac {b e n \left (f+g \log \left (c (d+e x)^n\right )\right )}{6 d x^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 x^3}-\frac {(b e n) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{3 d^2}+\frac {\left (b e^2 n\right ) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e x\right )}{3 d^2}-\frac {(e g n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{3 d^2}+\frac {\left (e^2 g n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e x\right )}{3 d^2}+2 \frac {\left (b e g n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{6 d}\\ &=-\frac {e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac {e^2 g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}-\frac {b e n \left (f+g \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac {b e^2 n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 x^3}+\frac {\left (b e^2 n\right ) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{3 d^3}-\frac {\left (b e^3 n\right ) \text {Subst}\left (\int \frac {f+g \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{3 d^3}+\frac {\left (e^2 g n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{3 d^3}-\frac {\left (e^3 g n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{3 d^3}+2 \frac {\left (b e g n^2\right ) \text {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+e x\right )}{6 d}-2 \frac {\left (b e^2 g n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{3 d^3}\\ &=-\frac {2 b e^3 g n^2 \log (x)}{3 d^3}+2 \left (-\frac {b e^2 g n^2}{6 d^2 x}-\frac {b e^3 g n^2 \log (x)}{6 d^3}+\frac {b e^3 g n^2 \log (d+e x)}{6 d^3}\right )-\frac {e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac {e^2 g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}+\frac {e^3 g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3}-\frac {e^3 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{6 b d^3}-\frac {b e n \left (f+g \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac {b e^2 n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}+\frac {b e^3 n \log \left (-\frac {e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 d^3}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 x^3}-\frac {b e^3 \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{6 d^3 g}-2 \frac {\left (b e^3 g n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{3 d^3}\\ &=-\frac {2 b e^3 g n^2 \log (x)}{3 d^3}+2 \left (-\frac {b e^2 g n^2}{6 d^2 x}-\frac {b e^3 g n^2 \log (x)}{6 d^3}+\frac {b e^3 g n^2 \log (d+e x)}{6 d^3}\right )-\frac {e g n \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac {e^2 g n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}+\frac {e^3 g n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 d^3}-\frac {e^3 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{6 b d^3}-\frac {b e n \left (f+g \log \left (c (d+e x)^n\right )\right )}{6 d x^2}+\frac {b e^2 n (d+e x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 d^3 x}+\frac {b e^3 n \log \left (-\frac {e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 d^3}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{3 x^3}-\frac {b e^3 \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{6 d^3 g}+\frac {2 b e^3 g n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{3 d^3}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 351, normalized size = 1.50 \begin {gather*} -\frac {a f}{3 x^3}-\frac {b e^2 g n^2}{3 d^2 x}-\frac {b e^3 g n^2 \log (x)}{d^3}+\frac {b e^3 g n^2 \log (d+e x)}{d^3}+\frac {1}{3} b e f n \left (-\frac {1}{2 d x^2}+\frac {e}{d^2 x}+\frac {e^2 \log (x)}{d^3}-\frac {e^2 \log (d+e x)}{d^3}\right )+\frac {1}{3} a e g n \left (-\frac {1}{2 d x^2}+\frac {e}{d^2 x}+\frac {e^2 \log (x)}{d^3}-\frac {e^2 \log (d+e x)}{d^3}\right )-\frac {b f \log \left (c (d+e x)^n\right )}{3 x^3}-\frac {a g \log \left (c (d+e x)^n\right )}{3 x^3}-\frac {b e g n \log \left (c (d+e x)^n\right )}{3 d x^2}+\frac {2 b e^2 g n \log \left (c (d+e x)^n\right )}{3 d^2 x}+\frac {2 b e^3 g n \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )}{3 d^3}-\frac {b e^3 g \log ^2\left (c (d+e x)^n\right )}{3 d^3}-\frac {b g \log ^2\left (c (d+e x)^n\right )}{3 x^3}+\frac {2 b e^3 g n^2 \text {Li}_2\left (\frac {d+e x}{d}\right )}{3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*(d + e*x)^n])*(f + g*Log[c*(d + e*x)^n]))/x^4,x]

[Out]

-1/3*(a*f)/x^3 - (b*e^2*g*n^2)/(3*d^2*x) - (b*e^3*g*n^2*Log[x])/d^3 + (b*e^3*g*n^2*Log[d + e*x])/d^3 + (b*e*f*
n*(-1/2*1/(d*x^2) + e/(d^2*x) + (e^2*Log[x])/d^3 - (e^2*Log[d + e*x])/d^3))/3 + (a*e*g*n*(-1/2*1/(d*x^2) + e/(
d^2*x) + (e^2*Log[x])/d^3 - (e^2*Log[d + e*x])/d^3))/3 - (b*f*Log[c*(d + e*x)^n])/(3*x^3) - (a*g*Log[c*(d + e*
x)^n])/(3*x^3) - (b*e*g*n*Log[c*(d + e*x)^n])/(3*d*x^2) + (2*b*e^2*g*n*Log[c*(d + e*x)^n])/(3*d^2*x) + (2*b*e^
3*g*n*Log[-((e*x)/d)]*Log[c*(d + e*x)^n])/(3*d^3) - (b*e^3*g*Log[c*(d + e*x)^n]^2)/(3*d^3) - (b*g*Log[c*(d + e
*x)^n]^2)/(3*x^3) + (2*b*e^3*g*n^2*PolyLog[2, (d + e*x)/d])/(3*d^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.32, size = 1437, normalized size = 6.14

method	result	size
risch	\(\text {Expression too large to display}\)	\(1437\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))*(f+g*ln(c*(e*x+d)^n))/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*b*e^2*g*n^2/d^2/x+1/3*I*e^3*n/d^3*ln(e*x+d)*Pi*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/6*I*
e*n/d/x^2*Pi*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/3*I*e^3*n/d^3*ln(e*x+d)*Pi*b*g*csgn(I*c)*cs
gn(I*c*(e*x+d)^n)^2-1/3*I*e^2*n/d^2/x*Pi*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/3/x^3*b*g*ln((e
*x+d)^n)^2-2/3*b*g*e^3*n^2/d^3*dilog((e*x+d)/d)+1/3*b*g*e^3*n^2/d^3*ln(e*x+d)^2-1/3*I*ln((e*x+d)^n)/x^3*Pi*b*g
*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/3*I*ln((e*x+d)^n)/x^3*Pi*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-2/3*b*
g*e^3*n^2/d^3*ln(x)*ln((e*x+d)/d)-1/3*e*n/d/x^2*ln(c)*b*g+2/3*e^2*n/d^2/x*ln(c)*b*g-2/3*e^3*n/d^3*ln(e*x+d)*ln
(c)*b*g+2/3*e^3*n/d^3*ln(x)*ln(c)*b*g-1/3*I*e^3*n/d^3*ln(x)*Pi*b*g*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d
)^n)-b*e^3*g*n^2*ln(x)/d^3+b*e^3*g*n^2*ln(e*x+d)/d^3+1/3*I*ln((e*x+d)^n)/x^3*Pi*b*g*csgn(I*c)*csgn(I*(e*x+d)^n
)*csgn(I*c*(e*x+d)^n)+1/3*I*e^2*n/d^2/x*Pi*b*g*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/6*I*e*n/d/x^2*Pi*b*g*csgn(I*c
*(e*x+d)^n)^3-1/3*I*e^2*n/d^2/x*Pi*b*g*csgn(I*c*(e*x+d)^n)^3-2/3*b*g*e^3*n*ln((e*x+d)^n)/d^3*ln(e*x+d)-1/3*b*g
*e*n*ln((e*x+d)^n)/d/x^2+2/3*b*g*e^3*n*ln((e*x+d)^n)/d^3*ln(x)-1/3*ln((e*x+d)^n)/x^3*a*g-1/3*ln((e*x+d)^n)/x^3
*b*f-1/3*I*e^3*n/d^3*ln(e*x+d)*Pi*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/3*I*e^3*n/d^3*ln(x)*Pi*b*g*csg
n(I*c*(e*x+d)^n)^3+1/3*I*e^3*n/d^3*ln(e*x+d)*Pi*b*g*csgn(I*c*(e*x+d)^n)^3+1/3*I*e^3*n/d^3*ln(x)*Pi*b*g*csgn(I*
c)*csgn(I*c*(e*x+d)^n)^2+1/3*I*e^3*n/d^3*ln(x)*Pi*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-2/3*ln((e*x+d)^n
)/x^3*ln(c)*b*g-1/6*e*n/d/x^2*a*g-1/6*e*n/d/x^2*b*f+1/3*e^2*n/d^2/x*a*g+1/3*e^2*n/d^2/x*b*f-1/3*e^3*n/d^3*ln(e
*x+d)*a*g-1/3*e^3*n/d^3*ln(e*x+d)*b*f+1/3*e^3*n/d^3*ln(x)*a*g+1/3*e^3*n/d^3*ln(x)*b*f-1/6*I*e*n/d/x^2*Pi*b*g*c
sgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/3*I*e^2*n/d^2/x*Pi*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+2/3*b*g*e^2*n*
ln((e*x+d)^n)/d^2/x-1/12*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e
*x+d)^n)^2+I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*a)*(-I*g*Pi
*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*g*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*g*Pi*csgn(I*(e*x+d)^
n)*csgn(I*c*(e*x+d)^n)^2-I*g*Pi*csgn(I*c*(e*x+d)^n)^3+2*g*ln(c)+2*f)/x^3+1/3*I*ln((e*x+d)^n)/x^3*Pi*b*g*csgn(I
*c*(e*x+d)^n)^3-1/6*I*e*n/d/x^2*Pi*b*g*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2

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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((a+b*log(c*(e*x+d)^n))*(f+g*log(c*(e*x+d)^n))/x^4,x, algorithm="maxima")

[Out]

-1/6*b*f*n*(2*e^2*log(x*e + d)/d^3 - 2*e^2*log(x)/d^3 - (2*x*e - d)/(d^2*x^2))*e - 1/6*a*g*n*(2*e^2*log(x*e +
d)/d^3 - 2*e^2*log(x)/d^3 - (2*x*e - d)/(d^2*x^2))*e - 1/3*b*g*(log((x*e + d)^n)^2/x^3 - 3*integrate(1/3*(3*x*
e*log(c)^2 + 3*d*log(c)^2 + 2*((n + 3*log(c))*x*e + 3*d*log(c))*log((x*e + d)^n))/(x^5*e + d*x^4), x)) - 1/3*b
*f*log((x*e + d)^n*c)/x^3 - 1/3*a*g*log((x*e + d)^n*c)/x^3 - 1/3*a*f/x^3

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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((a+b*log(c*(e*x+d)^n))*(f+g*log(c*(e*x+d)^n))/x^4,x, algorithm="fricas")

[Out]

integral((b*g*log((x*e + d)^n*c)^2 + a*f + (b*f + a*g)*log((x*e + d)^n*c))/x^4, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g \log {\left (c \left (d + e x\right )^{n} \right )}\right )}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))*(f+g*ln(c*(e*x+d)**n))/x**4,x)

[Out]

Integral((a + b*log(c*(d + e*x)**n))*(f + g*log(c*(d + e*x)**n))/x**4, 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((a+b*log(c*(e*x+d)^n))*(f+g*log(c*(e*x+d)^n))/x^4,x, algorithm="giac")

[Out]

integrate((b*log((x*e + d)^n*c) + a)*(g*log((x*e + d)^n*c) + f)/x^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*log(c*(d + e*x)^n))*(f + g*log(c*(d + e*x)^n)))/x^4,x)

[Out]

int(((a + b*log(c*(d + e*x)^n))*(f + g*log(c*(d + e*x)^n)))/x^4, x)

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