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

Optimal. Leaf size=76 \[ -\frac{a+b \log \left (c x^n\right )}{2 e (d+e x)^2}+\frac{b n \log (x)}{2 d^2 e}-\frac{b n \log (d+e x)}{2 d^2 e}+\frac{b n}{2 d e (d+e x)} \]

[Out]

(b*n)/(2*d*e*(d + e*x)) + (b*n*Log[x])/(2*d^2*e) - (a + b*Log[c*x^n])/(2*e*(d + e*x)^2) - (b*n*Log[d + e*x])/(
2*d^2*e)

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Rubi [A]  time = 0.0340487, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2319, 44} \[ -\frac{a+b \log \left (c x^n\right )}{2 e (d+e x)^2}+\frac{b n \log (x)}{2 d^2 e}-\frac{b n \log (d+e x)}{2 d^2 e}+\frac{b n}{2 d e (d+e x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(d + e*x)^3,x]

[Out]

(b*n)/(2*d*e*(d + e*x)) + (b*n*Log[x])/(2*d^2*e) - (a + b*Log[c*x^n])/(2*e*(d + e*x)^2) - (b*n*Log[d + e*x])/(
2*d^2*e)

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])

Rubi steps

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

Mathematica [A]  time = 0.053675, size = 53, normalized size = 0.7 \[ \frac{\frac{b n \left (\frac{d}{d+e x}-\log (d+e x)+\log (x)\right )}{d^2}-\frac{a+b \log \left (c x^n\right )}{(d+e x)^2}}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a + b*Log[c*x^n])/(d + e*x)^2) + (b*n*(d/(d + e*x) + Log[x] - Log[d + e*x]))/d^2)/(2*e)

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Maple [C]  time = 0.101, size = 235, normalized size = 3.1 \begin{align*} -{\frac{b\ln \left ({x}^{n} \right ) }{2\, \left ( ex+d \right ) ^{2}e}}-{\frac{i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,\ln \left ( -x \right ) b{e}^{2}n{x}^{2}+2\,\ln \left ( ex+d \right ) b{e}^{2}n{x}^{2}-4\,\ln \left ( -x \right ) bdenx+4\,\ln \left ( ex+d \right ) bdenx-2\,\ln \left ( -x \right ) b{d}^{2}n+2\,\ln \left ( ex+d \right ) b{d}^{2}n-2\,bdenx+2\,\ln \left ( c \right ) b{d}^{2}-2\,b{d}^{2}n+2\,a{d}^{2}}{4\,e{d}^{2} \left ( ex+d \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08569, size = 134, normalized size = 1.76 \begin{align*} \frac{1}{2} \, b n{\left (\frac{1}{d e^{2} x + d^{2} e} - \frac{\log \left (e x + d\right )}{d^{2} e} + \frac{\log \left (x\right )}{d^{2} e}\right )} - \frac{b \log \left (c x^{n}\right )}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac{a}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.0672, size = 238, normalized size = 3.13 \begin{align*} \frac{b d e n x + b d^{2} n - b d^{2} \log \left (c\right ) - a d^{2} -{\left (b e^{2} n x^{2} + 2 \, b d e n x + b d^{2} n\right )} \log \left (e x + d\right ) +{\left (b e^{2} n x^{2} + 2 \, b d e n x\right )} \log \left (x\right )}{2 \,{\left (d^{2} e^{3} x^{2} + 2 \, d^{3} e^{2} x + d^{4} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 5.93654, size = 559, normalized size = 7.36 \begin{align*} \begin{cases} \tilde{\infty } \left (- \frac{a}{2 x^{2}} - \frac{b n \log{\left (x \right )}}{2 x^{2}} - \frac{b n}{4 x^{2}} - \frac{b \log{\left (c \right )}}{2 x^{2}}\right ) & \text{for}\: d = 0 \wedge e = 0 \\\frac{- \frac{a}{2 x^{2}} - \frac{b n \log{\left (x \right )}}{2 x^{2}} - \frac{b n}{4 x^{2}} - \frac{b \log{\left (c \right )}}{2 x^{2}}}{e^{3}} & \text{for}\: d = 0 \\\frac{a x + b n x \log{\left (x \right )} - b n x + b x \log{\left (c \right )}}{d^{3}} & \text{for}\: e = 0 \\\frac{2 a d e x}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} + \frac{a e^{2} x^{2}}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} - \frac{b d^{2} n \log{\left (\frac{d}{e} + x \right )}}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} + \frac{2 b d e n x \log{\left (x \right )}}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} - \frac{2 b d e n x \log{\left (\frac{d}{e} + x \right )}}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} - \frac{b d e n x}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} + \frac{2 b d e x \log{\left (c \right )}}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} + \frac{b e^{2} n x^{2} \log{\left (x \right )}}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} - \frac{b e^{2} n x^{2} \log{\left (\frac{d}{e} + x \right )}}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} - \frac{b e^{2} n x^{2}}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} + \frac{b e^{2} x^{2} \log{\left (c \right )}}{2 d^{4} e + 4 d^{3} e^{2} x + 2 d^{2} e^{3} x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(-a/(2*x**2) - b*n*log(x)/(2*x**2) - b*n/(4*x**2) - b*log(c)/(2*x**2)), Eq(d, 0) & Eq(e, 0)), (
(-a/(2*x**2) - b*n*log(x)/(2*x**2) - b*n/(4*x**2) - b*log(c)/(2*x**2))/e**3, Eq(d, 0)), ((a*x + b*n*x*log(x) -
b*n*x + b*x*log(c))/d**3, Eq(e, 0)), (2*a*d*e*x/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2) + a*e**2*x**2/(
2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2) - b*d**2*n*log(d/e + x)/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x
**2) + 2*b*d*e*n*x*log(x)/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2) - 2*b*d*e*n*x*log(d/e + x)/(2*d**4*e +
4*d**3*e**2*x + 2*d**2*e**3*x**2) - b*d*e*n*x/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2) + 2*b*d*e*x*log(c
)/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2) + b*e**2*n*x**2*log(x)/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3
*x**2) - b*e**2*n*x**2*log(d/e + x)/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2) - b*e**2*n*x**2/(2*d**4*e +
4*d**3*e**2*x + 2*d**2*e**3*x**2) + b*e**2*x**2*log(c)/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2), True))

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Giac [A]  time = 1.26493, size = 162, normalized size = 2.13 \begin{align*} -\frac{b n x^{2} e^{2} \log \left (x e + d\right ) + 2 \, b d n x e \log \left (x e + d\right ) - b n x^{2} e^{2} \log \left (x\right ) - 2 \, b d n x e \log \left (x\right ) - b d n x e + b d^{2} n \log \left (x e + d\right ) - b d^{2} n + b d^{2} \log \left (c\right ) + a d^{2}}{2 \,{\left (d^{2} x^{2} e^{3} + 2 \, d^{3} x e^{2} + d^{4} e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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