3.54 \(\int \frac{x^4 (a+b \log (c x^n))}{(d+e x)^4} \, dx\)

Optimal. Leaf size=183 \[ -\frac{4 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^5}-\frac{x^3 \left (4 a+4 b \log \left (c x^n\right )+b n\right )}{6 e^2 (d+e x)^2}-\frac{x^2 \left (12 a+12 b \log \left (c x^n\right )+7 b n\right )}{6 e^3 (d+e x)}-\frac{d \log \left (\frac{e x}{d}+1\right ) \left (12 a+12 b \log \left (c x^n\right )+13 b n\right )}{3 e^5}-\frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}+\frac{x (12 a+13 b n)}{3 e^4}+\frac{4 b x \log \left (c x^n\right )}{e^4}-\frac{4 b n x}{e^4} \]

[Out]

(-4*b*n*x)/e^4 + ((12*a + 13*b*n)*x)/(3*e^4) + (4*b*x*Log[c*x^n])/e^4 - (x^4*(a + b*Log[c*x^n]))/(3*e*(d + e*x
)^3) - (x^3*(4*a + b*n + 4*b*Log[c*x^n]))/(6*e^2*(d + e*x)^2) - (x^2*(12*a + 7*b*n + 12*b*Log[c*x^n]))/(6*e^3*
(d + e*x)) - (d*(12*a + 13*b*n + 12*b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*e^5) - (4*b*d*n*PolyLog[2, -((e*x)/d)])
/e^5

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Rubi [A]  time = 0.281067, antiderivative size = 211, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {43, 2351, 2295, 2319, 44, 2314, 31, 2317, 2391} \[ -\frac{4 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^5}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)}-\frac{4 d \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^5}+\frac{a x}{e^4}+\frac{b x \log \left (c x^n\right )}{e^4}+\frac{b d^3 n}{6 e^5 (d+e x)^2}-\frac{5 b d^2 n}{3 e^5 (d+e x)}-\frac{5 b d n \log (x)}{3 e^5}-\frac{13 b d n \log (d+e x)}{3 e^5}-\frac{b n x}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x)/e^4 - (b*n*x)/e^4 + (b*d^3*n)/(6*e^5*(d + e*x)^2) - (5*b*d^2*n)/(3*e^5*(d + e*x)) - (5*b*d*n*Log[x])/(3*
e^5) + (b*x*Log[c*x^n])/e^4 - (d^4*(a + b*Log[c*x^n]))/(3*e^5*(d + e*x)^3) + (2*d^3*(a + b*Log[c*x^n]))/(e^5*(
d + e*x)^2) + (6*d*x*(a + b*Log[c*x^n]))/(e^4*(d + e*x)) - (13*b*d*n*Log[d + e*x])/(3*e^5) - (4*d*(a + b*Log[c
*x^n])*Log[1 + (e*x)/d])/e^5 - (4*b*d*n*PolyLog[2, -((e*x)/d)])/e^5

Rule 43

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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2295

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

Rule 2319

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 44

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] && L
tQ[m + n + 2, 0])

Rule 2314

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 31

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

Rule 2317

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

Rule 2391

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]

Rubi steps

\begin{align*} \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{e^4}+\frac{d^4 \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)^4}-\frac{4 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)^3}+\frac{6 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)^2}-\frac{4 d \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)}\right ) \, dx\\ &=\frac{\int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^4}-\frac{(4 d) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^4}+\frac{\left (6 d^2\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^4}-\frac{\left (4 d^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^4}+\frac{d^4 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^4}\\ &=\frac{a x}{e^4}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)}-\frac{4 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^5}+\frac{b \int \log \left (c x^n\right ) \, dx}{e^4}+\frac{(4 b d n) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^5}-\frac{\left (2 b d^3 n\right ) \int \frac{1}{x (d+e x)^2} \, dx}{e^5}+\frac{\left (b d^4 n\right ) \int \frac{1}{x (d+e x)^3} \, dx}{3 e^5}-\frac{(6 b d n) \int \frac{1}{d+e x} \, dx}{e^4}\\ &=\frac{a x}{e^4}-\frac{b n x}{e^4}+\frac{b x \log \left (c x^n\right )}{e^4}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)}-\frac{6 b d n \log (d+e x)}{e^5}-\frac{4 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^5}-\frac{4 b d n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^5}-\frac{\left (2 b d^3 n\right ) \int \left (\frac{1}{d^2 x}-\frac{e}{d (d+e x)^2}-\frac{e}{d^2 (d+e x)}\right ) \, dx}{e^5}+\frac{\left (b d^4 n\right ) \int \left (\frac{1}{d^3 x}-\frac{e}{d (d+e x)^3}-\frac{e}{d^2 (d+e x)^2}-\frac{e}{d^3 (d+e x)}\right ) \, dx}{3 e^5}\\ &=\frac{a x}{e^4}-\frac{b n x}{e^4}+\frac{b d^3 n}{6 e^5 (d+e x)^2}-\frac{5 b d^2 n}{3 e^5 (d+e x)}-\frac{5 b d n \log (x)}{3 e^5}+\frac{b x \log \left (c x^n\right )}{e^4}-\frac{d^4 \left (a+b \log \left (c x^n\right )\right )}{3 e^5 (d+e x)^3}+\frac{2 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^2}+\frac{6 d x \left (a+b \log \left (c x^n\right )\right )}{e^4 (d+e x)}-\frac{13 b d n \log (d+e x)}{3 e^5}-\frac{4 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^5}-\frac{4 b d n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^5}\\ \end{align*}

Mathematica [A]  time = 0.244707, size = 207, normalized size = 1.13 \[ \frac{-24 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right )-\frac{2 d^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}+\frac{12 d^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac{36 d^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}-24 d \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+6 a e x+6 b e x \log \left (c x^n\right )+b d n \left (\frac{d (3 d+2 e x)}{(d+e x)^2}-2 \log (d+e x)+2 \log (x)\right )+36 b d n (\log (x)-\log (d+e x))-12 b d n \left (\frac{d}{d+e x}-\log (d+e x)+\log (x)\right )-6 b e n x}{6 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a*e*x - 6*b*e*n*x + 6*b*e*x*Log[c*x^n] - (2*d^4*(a + b*Log[c*x^n]))/(d + e*x)^3 + (12*d^3*(a + b*Log[c*x^n]
))/(d + e*x)^2 - (36*d^2*(a + b*Log[c*x^n]))/(d + e*x) + b*d*n*((d*(3*d + 2*e*x))/(d + e*x)^2 + 2*Log[x] - 2*L
og[d + e*x]) + 36*b*d*n*(Log[x] - Log[d + e*x]) - 12*b*d*n*(d/(d + e*x) + Log[x] - Log[d + e*x]) - 24*d*(a + b
*Log[c*x^n])*Log[1 + (e*x)/d] - 24*b*d*n*PolyLog[2, -((e*x)/d)])/(6*e^5)

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Maple [C]  time = 0.198, size = 969, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b*n/e^5*d*ln(e*x+d)*ln(-e*x/d)+2*b*ln(x^n)/e^5*d^3/(e*x+d)^2-1/3*b*ln(x^n)*d^4/e^5/(e*x+d)^3-6*b*ln(x^n)/e^5
*d^2/(e*x+d)-13/3*b*n/e^5*d*ln(e*x+d)-5/3*b*n/e^5*d^2/(e*x+d)+4*b*n/e^5*d*dilog(-e*x/d)+13/3*b*n/e^5*d*ln(e*x)
+1/6*b*n/e^5*d^3/(e*x+d)^2-1/2*I*b*Pi*csgn(I*c*x^n)^3/e^4*x+2*I*b*Pi*csgn(I*c*x^n)^3/e^5*d*ln(e*x+d)-1/3*b*ln(
c)*d^4/e^5/(e*x+d)^3-6*b*ln(c)/e^5*d^2/(e*x+d)-4*b*ln(c)/e^5*d*ln(e*x+d)+2*b*ln(c)/e^5*d^3/(e*x+d)^2+2*I*b*Pi*
csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^5*d*ln(e*x+d)+3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^5*d^2/(e*x+
d)-I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^5*d^3/(e*x+d)^2+1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)
*d^4/e^5/(e*x+d)^3-I*b*Pi*csgn(I*c*x^n)^3/e^5*d^3/(e*x+d)^2-2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^5*d*ln(e*x+
d)-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^5*d^2/(e*x+d)-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^4*x+I
*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^5*d^3/(e*x+d)^2-1/6*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*d^4/e^5/(e*x+d)^3-2*I*b
*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^5*d*ln(e*x+d)+a/e^4*x-3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^5*d^2/(e*x+d)+I*b*P
i*csgn(I*x^n)*csgn(I*c*x^n)^2/e^5*d^3/(e*x+d)^2-1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d^4/e^5/(e*x+d)^3+1/2*I
*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^4*x+3*I*b*Pi*csgn(I*c*x^n)^3/e^5*d^2/(e*x+d)-4*b*ln(x^n)/e^5*d*ln(e*x+d)+1
/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^4*x+1/6*I*b*Pi*csgn(I*c*x^n)^3*d^4/e^5/(e*x+d)^3+b*ln(x^n)/e^4*x-6*a/e^5
*d^2/(e*x+d)-4*a/e^5*d*ln(e*x+d)+b*ln(c)/e^4*x+2*a/e^5*d^3/(e*x+d)^2-1/3*a*d^4/e^5/(e*x+d)^3-b*n/e^5*d-b*n*x/e
^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{3} \, a{\left (\frac{18 \, d^{2} e^{2} x^{2} + 30 \, d^{3} e x + 13 \, d^{4}}{e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}} - \frac{3 \, x}{e^{4}} + \frac{12 \, d \log \left (e x + d\right )}{e^{5}}\right )} + b \int \frac{x^{4} \log \left (c\right ) + x^{4} \log \left (x^{n}\right )}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a*((18*d^2*e^2*x^2 + 30*d^3*e*x + 13*d^4)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) - 3*x/e^4 + 12*
d*log(e*x + d)/e^5) + b*integrate((x^4*log(c) + x^4*log(x^n))/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e
*x + d^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{4} \log \left (c x^{n}\right ) + a x^{4}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^4*log(c*x^n) + a*x^4)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x + d^4), x)

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Sympy [A]  time = 73.0398, size = 544, normalized size = 2.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**4*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/e**4 - 4*a*d**3*Piecewise((x/d**3, Eq(e, 0
)), (-1/(2*e*(d + e*x)**2), True))/e**4 + 6*a*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/e*
*4 - 4*a*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**4 + a*x/e**4 - b*d**4*n*Piecewise((x/d**4, Eq
(e, 0)), (-3*d/(6*d**4*e + 12*d**3*e**2*x + 6*d**2*e**3*x**2) - 2*e*x/(6*d**4*e + 12*d**3*e**2*x + 6*d**2*e**3
*x**2) - log(x)/(3*d**3*e) + log(d/e + x)/(3*d**3*e), True))/e**4 + b*d**4*Piecewise((x/d**4, Eq(e, 0)), (-1/(
3*e*(d + e*x)**3), True))*log(c*x**n)/e**4 + 4*b*d**3*n*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*d**2*e + 2*d*e**2
*x) - log(x)/(2*d**2*e) + log(d/e + x)/(2*d**2*e), True))/e**4 - 4*b*d**3*Piecewise((x/d**3, Eq(e, 0)), (-1/(2
*e*(d + e*x)**2), True))*log(c*x**n)/e**4 - 6*b*d**2*n*Piecewise((x/d**2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e
+ x)/(d*e), True))/e**4 + 6*b*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))*log(c*x**n)/e**4 +
4*b*d*n*Piecewise((x/d, Eq(e, 0)), (Piecewise((log(d)*log(x) - polylog(2, e*x*exp_polar(I*pi)/d), Abs(x) < 1)
, (-log(d)*log(1/x) - polylog(2, e*x*exp_polar(I*pi)/d), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()),
x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x*exp_polar(I*pi)/d), True))/e, True)
)/e**4 - 4*b*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))*log(c*x**n)/e**4 - b*n*x/e**4 + b*x*log(c*x*
*n)/e**4

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{4}}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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