3.329 \(\int \frac{x^3 (a+b \log (c x^n))}{d+\frac{e}{x}} \, dx\)

Optimal. Leaf size=185 \[ \frac{b e^4 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^5}+\frac{e^4 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac{e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac{e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}-\frac{a e^3 x}{d^4}-\frac{b e^3 x \log \left (c x^n\right )}{d^4}-\frac{b e^2 n x^2}{4 d^3}+\frac{b e^3 n x}{d^4}+\frac{b e n x^3}{9 d^2}-\frac{b n x^4}{16 d} \]

[Out]

-((a*e^3*x)/d^4) + (b*e^3*n*x)/d^4 - (b*e^2*n*x^2)/(4*d^3) + (b*e*n*x^3)/(9*d^2) - (b*n*x^4)/(16*d) - (b*e^3*x
*Log[c*x^n])/d^4 + (e^2*x^2*(a + b*Log[c*x^n]))/(2*d^3) - (e*x^3*(a + b*Log[c*x^n]))/(3*d^2) + (x^4*(a + b*Log
[c*x^n]))/(4*d) + (e^4*(a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d^5 + (b*e^4*n*PolyLog[2, -((d*x)/e)])/d^5

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Rubi [A]  time = 0.199915, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {263, 43, 2351, 2295, 2304, 2317, 2391} \[ \frac{b e^4 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^5}+\frac{e^4 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac{e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac{e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}-\frac{a e^3 x}{d^4}-\frac{b e^3 x \log \left (c x^n\right )}{d^4}-\frac{b e^2 n x^2}{4 d^3}+\frac{b e^3 n x}{d^4}+\frac{b e n x^3}{9 d^2}-\frac{b n x^4}{16 d} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*Log[c*x^n]))/(d + e/x),x]

[Out]

-((a*e^3*x)/d^4) + (b*e^3*n*x)/d^4 - (b*e^2*n*x^2)/(4*d^3) + (b*e*n*x^3)/(9*d^2) - (b*n*x^4)/(16*d) - (b*e^3*x
*Log[c*x^n])/d^4 + (e^2*x^2*(a + b*Log[c*x^n]))/(2*d^3) - (e*x^3*(a + b*Log[c*x^n]))/(3*d^2) + (x^4*(a + b*Log
[c*x^n]))/(4*d) + (e^4*(a + b*Log[c*x^n])*Log[1 + (d*x)/e])/d^5 + (b*e^4*n*PolyLog[2, -((d*x)/e)])/d^5

Rule 263

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

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 2304

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

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^3 \left (a+b \log \left (c x^n\right )\right )}{d+\frac{e}{x}} \, dx &=\int \left (-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac{e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{e x^2 \left (a+b \log \left (c x^n\right )\right )}{d^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{d}+\frac{e^4 \left (a+b \log \left (c x^n\right )\right )}{d^4 (e+d x)}\right ) \, dx\\ &=\frac{\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx}{d}-\frac{e \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^2}+\frac{e^2 \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^3}-\frac{e^3 \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{d^4}+\frac{e^4 \int \frac{a+b \log \left (c x^n\right )}{e+d x} \, dx}{d^4}\\ &=-\frac{a e^3 x}{d^4}-\frac{b e^2 n x^2}{4 d^3}+\frac{b e n x^3}{9 d^2}-\frac{b n x^4}{16 d}+\frac{e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac{e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}+\frac{e^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x}{e}\right )}{d^5}-\frac{\left (b e^3\right ) \int \log \left (c x^n\right ) \, dx}{d^4}-\frac{\left (b e^4 n\right ) \int \frac{\log \left (1+\frac{d x}{e}\right )}{x} \, dx}{d^5}\\ &=-\frac{a e^3 x}{d^4}+\frac{b e^3 n x}{d^4}-\frac{b e^2 n x^2}{4 d^3}+\frac{b e n x^3}{9 d^2}-\frac{b n x^4}{16 d}-\frac{b e^3 x \log \left (c x^n\right )}{d^4}+\frac{e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac{e x^3 \left (a+b \log \left (c x^n\right )\right )}{3 d^2}+\frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d}+\frac{e^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x}{e}\right )}{d^5}+\frac{b e^4 n \text{Li}_2\left (-\frac{d x}{e}\right )}{d^5}\\ \end{align*}

Mathematica [A]  time = 0.101681, size = 171, normalized size = 0.92 \[ \frac{144 b e^4 n \text{PolyLog}\left (2,-\frac{d x}{e}\right )+72 d^2 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-48 d^3 e x^3 \left (a+b \log \left (c x^n\right )\right )+36 d^4 x^4 \left (a+b \log \left (c x^n\right )\right )+144 e^4 \log \left (\frac{d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )-144 a d e^3 x-144 b d e^3 x \log \left (c x^n\right )-36 b d^2 e^2 n x^2+16 b d^3 e n x^3-9 b d^4 n x^4+144 b d e^3 n x}{144 d^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e/x),x]

[Out]

(-144*a*d*e^3*x + 144*b*d*e^3*n*x - 36*b*d^2*e^2*n*x^2 + 16*b*d^3*e*n*x^3 - 9*b*d^4*n*x^4 - 144*b*d*e^3*x*Log[
c*x^n] + 72*d^2*e^2*x^2*(a + b*Log[c*x^n]) - 48*d^3*e*x^3*(a + b*Log[c*x^n]) + 36*d^4*x^4*(a + b*Log[c*x^n]) +
144*e^4*(a + b*Log[c*x^n])*Log[1 + (d*x)/e] + 144*b*e^4*n*PolyLog[2, -((d*x)/e)])/(144*d^5)

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Maple [C]  time = 0.183, size = 867, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*ln(c*x^n))/(d+e/x),x)

[Out]

-1/8*I*b*Pi*csgn(I*c*x^n)^3/d*x^4+1/6*I*b*Pi*csgn(I*c*x^n)^3/d^2*e*x^3-1/4*I*b*Pi*csgn(I*c*x^n)^3/d^3*x^2*e^2+
1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*e^4/d^5*ln(d*x+e)-1/2*I*b*Pi*csgn(I*c*x^n)^3*e^4/d^5*ln(d*x+e)-1/8*I*b*Pi
*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d*x^4+1/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^3*x^2*e^2-1/6*I*b*Pi*csgn(I*
x^n)*csgn(I*c*x^n)^2/d^2*e*x^3+1/4*b*ln(x^n)/d*x^4-b*n*e^4/d^5*ln(d*x+e)*ln(-d*x/e)+a*e^4/d^5*ln(d*x+e)-1/3*a/
d^2*e*x^3+1/2*a/d^3*x^2*e^2+1/4*b*ln(c)/d*x^4+205/144*b*n*e^4/d^5+1/8*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d*x^4-1
/3*b*ln(x^n)/d^2*e*x^3+1/2*b*ln(x^n)/d^3*x^2*e^2-b*ln(x^n)/d^4*x*e^3+b*ln(x^n)*e^4/d^5*ln(d*x+e)-1/4*I*b*Pi*cs
gn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^3*x^2*e^2+1/8*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d*x^4-1/2*I*b*Pi*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*c)*e^4/d^5*ln(d*x+e)+1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3*x^2*e^2-1/2*I*b*Pi*cs
gn(I*x^n)*csgn(I*c*x^n)^2/d^4*x*e^3+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e^4/d^5*ln(d*x+e)-1/6*I*b*Pi*csgn(I
*c*x^n)^2*csgn(I*c)/d^2*e*x^3+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^4*x*e^3+1/6*I*b*Pi*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)/d^2*e*x^3-1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^4*x*e^3+1/2*I*b*Pi*csgn(I*c*x^n)^3/d^
4*x*e^3-b*n*e^4/d^5*dilog(-d*x/e)-1/3*b*ln(c)/d^2*e*x^3+1/2*b*ln(c)/d^3*x^2*e^2-b*ln(c)/d^4*x*e^3+b*ln(c)*e^4/
d^5*ln(d*x+e)+1/4*a/d*x^4-a*e^3*x/d^4-1/4*b*e^2*n*x^2/d^3+1/9*b*e*n*x^3/d^2-1/16*b*n*x^4/d+b*e^3*n*x/d^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))/(d+e/x),x, algorithm="maxima")

[Out]

1/12*a*(12*e^4*log(d*x + e)/d^5 + (3*d^3*x^4 - 4*d^2*e*x^3 + 6*d*e^2*x^2 - 12*e^3*x)/d^4) + b*integrate((x^4*l
og(c) + x^4*log(x^n))/(d*x + e), 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}}{d x + e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))/(d+e/x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 135.23, size = 298, normalized size = 1.61 \begin{align*} \frac{a x^{4}}{4 d} - \frac{a e x^{3}}{3 d^{2}} + \frac{a e^{2} x^{2}}{2 d^{3}} + \frac{a e^{4} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right )}{d^{4}} - \frac{a e^{3} x}{d^{4}} - \frac{b n x^{4}}{16 d} + \frac{b x^{4} \log{\left (c x^{n} \right )}}{4 d} + \frac{b e n x^{3}}{9 d^{2}} - \frac{b e x^{3} \log{\left (c x^{n} \right )}}{3 d^{2}} - \frac{b e^{2} n x^{2}}{4 d^{3}} + \frac{b e^{2} x^{2} \log{\left (c x^{n} \right )}}{2 d^{3}} - \frac{b e^{4} n \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\begin{cases} \log{\left (e \right )} \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (e \right )} \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\right ) & \text{for}\: \frac{1}{\left |{x}\right |} < 1 \\-{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x} \right )} \log{\left (e \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (e \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\right ) & \text{otherwise} \end{cases}}{d} & \text{otherwise} \end{cases}\right )}{d^{4}} + \frac{b e^{4} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )}}{d^{4}} + \frac{b e^{3} n x}{d^{4}} - \frac{b e^{3} x \log{\left (c x^{n} \right )}}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*ln(c*x**n))/(d+e/x),x)

[Out]

a*x**4/(4*d) - a*e*x**3/(3*d**2) + a*e**2*x**2/(2*d**3) + a*e**4*Piecewise((x/e, Eq(d, 0)), (log(d*x + e)/d, T
rue))/d**4 - a*e**3*x/d**4 - b*n*x**4/(16*d) + b*x**4*log(c*x**n)/(4*d) + b*e*n*x**3/(9*d**2) - b*e*x**3*log(c
*x**n)/(3*d**2) - b*e**2*n*x**2/(4*d**3) + b*e**2*x**2*log(c*x**n)/(2*d**3) - b*e**4*n*Piecewise((x/e, Eq(d, 0
)), (Piecewise((log(e)*log(x) - polylog(2, d*x*exp_polar(I*pi)/e), Abs(x) < 1), (-log(e)*log(1/x) - polylog(2,
d*x*exp_polar(I*pi)/e), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(e) + meijerg(((1, 1), ())
, ((), (0, 0)), x)*log(e) - polylog(2, d*x*exp_polar(I*pi)/e), True))/d, True))/d**4 + b*e**4*Piecewise((x/e,
Eq(d, 0)), (log(d*x + e)/d, True))*log(c*x**n)/d**4 + b*e**3*n*x/d**4 - b*e**3*x*log(c*x**n)/d**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^{3}}{d + \frac{e}{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*x^n))/(d+e/x),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^3/(d + e/x), x)