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

Optimal. Leaf size=110 \[ \frac{b e^2 n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d^3}-\frac{e^2 \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{2 d x^2}+\frac{b e n}{d^2 x}-\frac{b n}{4 d x^2} \]

[Out]

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

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Rubi [A]  time = 0.170924, antiderivative size = 135, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {44, 2351, 2304, 2301, 2317, 2391} \[ -\frac{b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^3}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac{e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{2 d x^2}+\frac{b e n}{d^2 x}-\frac{b n}{4 d x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, 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{a+b \log \left (c x^n\right )}{x^3 (d+e x)} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{d x^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x^2}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{d}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{d^2}+\frac{e^2 \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^3}-\frac{e^3 \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^3}\\ &=-\frac{b n}{4 d x^2}+\frac{b e n}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{2 d x^2}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^3}+\frac{\left (b e^2 n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^3}\\ &=-\frac{b n}{4 d x^2}+\frac{b e n}{d^2 x}-\frac{a+b \log \left (c x^n\right )}{2 d x^2}+\frac{e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^3 n}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^3}-\frac{b e^2 n \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.197236, size = 124, normalized size = 1.13 \[ -\frac{4 b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}+4 e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{4 d e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{2 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+\frac{b d^2 n}{x^2}-\frac{4 b d e n}{x}}{4 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.158, size = 689, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 123.158, size = 246, normalized size = 2.24 \begin{align*} - \frac{a}{2 d x^{2}} + \frac{a e}{d^{2} x} - \frac{a e^{3} \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right )}{d^{3}} + \frac{a e^{2} \log{\left (x \right )}}{d^{3}} - \frac{b n}{4 d x^{2}} - \frac{b \log{\left (c x^{n} \right )}}{2 d x^{2}} + \frac{b e n}{d^{2} x} + \frac{b e \log{\left (c x^{n} \right )}}{d^{2} x} + \frac{b e^{3} n \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\begin{cases} \log{\left (d \right )} \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (d \right )} \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\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 (d \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (d \right )} - \operatorname{Li}_{2}\left (\frac{e x e^{i \pi }}{d}\right ) & \text{otherwise} \end{cases}}{e} & \text{otherwise} \end{cases}\right )}{d^{3}} - \frac{b e^{3} \left (\begin{cases} \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (d + e x \right )}}{e} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )}}{d^{3}} - \frac{b e^{2} n \log{\left (x \right )}^{2}}{2 d^{3}} + \frac{b e^{2} \log{\left (x \right )} \log{\left (c x^{n} \right )}}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(2*d*x**2) + a*e/(d**2*x) - a*e**3*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/d**3 + a*e**2*log(x)/
d**3 - b*n/(4*d*x**2) - b*log(c*x**n)/(2*d*x**2) + b*e*n/(d**2*x) + b*e*log(c*x**n)/(d**2*x) + b*e**3*n*Piecew
ise((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) + mei
jerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x*exp_polar(I*pi)/d), True))/e, True))/d**3 - b*e**3
*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))*log(c*x**n)/d**3 - b*e**2*n*log(x)**2/(2*d**3) + b*e**2*lo
g(x)*log(c*x**n)/d**3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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