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

Optimal. Leaf size=62 \[ \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac{b n}{2 e^2 (d+e x)}-\frac{b n \log (d+e x)}{2 d e^2} \]

[Out]

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

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Rubi [A]  time = 0.0500943, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2335, 43} \[ \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 d (d+e x)^2}-\frac{b n}{2 e^2 (d+e x)}-\frac{b n \log (d+e x)}{2 d e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2335

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

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])

Rubi steps

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

Mathematica [A]  time = 0.11525, size = 75, normalized size = 1.21 \[ \frac{b n \log (x)-\frac{a d (d+2 e x)+b d (d+2 e x) \log \left (c x^n\right )+b d n (d+e x)+b n (d+e x)^2 \log (d+e x)}{(d+e x)^2}}{2 d e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.11, size = 349, normalized size = 5.6 \begin{align*} -{\frac{b \left ( 2\,ex+d \right ) \ln \left ({x}^{n} \right ) }{2\, \left ( ex+d \right ) ^{2}{e}^{2}}}-{\frac{i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,i\pi \,bdex \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\,i\pi \,bdex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,i\pi \,bdex{\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}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +2\,i\pi \,bdex{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( ex+d \right ) b{e}^{2}n{x}^{2}-2\,\ln \left ( -x \right ) b{e}^{2}n{x}^{2}+4\,\ln \left ( ex+d \right ) bdenx-4\,\ln \left ( -x \right ) bdenx+4\,\ln \left ( c \right ) bdex+2\,\ln \left ( ex+d \right ) b{d}^{2}n-2\,\ln \left ( -x \right ) b{d}^{2}n+2\,bdenx+2\,\ln \left ( c \right ) b{d}^{2}+4\,adex+2\,b{d}^{2}n+2\,a{d}^{2}}{4\,d{e}^{2} \left ( ex+d \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.11261, size = 154, normalized size = 2.48 \begin{align*} -\frac{1}{2} \, b n{\left (\frac{1}{e^{3} x + d e^{2}} + \frac{\log \left (e x + d\right )}{d e^{2}} - \frac{\log \left (x\right )}{d e^{2}}\right )} - \frac{{\left (2 \, e x + d\right )} b \log \left (c x^{n}\right )}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} - \frac{{\left (2 \, e x + d\right )} a}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.06569, size = 251, normalized size = 4.05 \begin{align*} \frac{b e^{2} n x^{2} \log \left (x\right ) - b d^{2} n - a d^{2} -{\left (b d e n + 2 \, a d e\right )} x -{\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 (2 \, b d e x + b d^{2}\right )} \log \left (c\right )}{2 \,{\left (d e^{4} x^{2} + 2 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 23.424, size = 425, normalized size = 6.85 \begin{align*} \begin{cases} \tilde{\infty } \left (- \frac{a}{x} - \frac{b n \log{\left (x \right )}}{x} - \frac{b n}{x} - \frac{b \log{\left (c \right )}}{x}\right ) & \text{for}\: d = 0 \wedge e = 0 \\\frac{\frac{a x^{2}}{2} + \frac{b n x^{2} \log{\left (x \right )}}{2} - \frac{b n x^{2}}{4} + \frac{b x^{2} \log{\left (c \right )}}{2}}{d^{3}} & \text{for}\: e = 0 \\\frac{- \frac{a}{x} - \frac{b n \log{\left (x \right )}}{x} - \frac{b n}{x} - \frac{b \log{\left (c \right )}}{x}}{e^{3}} & \text{for}\: d = 0 \\\frac{a e^{2} x^{2}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} - \frac{b d^{2} n \log{\left (\frac{d}{e} + x \right )}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} - \frac{2 b d e n x \log{\left (\frac{d}{e} + x \right )}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} + \frac{b d e n x}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} + \frac{b e^{2} n x^{2} \log{\left (x \right )}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} - \frac{b e^{2} n x^{2} \log{\left (\frac{d}{e} + x \right )}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} + \frac{b e^{2} n x^{2}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} + \frac{b e^{2} x^{2} \log{\left (c \right )}}{2 d^{3} e^{2} + 4 d^{2} e^{3} x + 2 d e^{4} x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.26169, size = 165, normalized size = 2.66 \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 ) + b d n x e + b d^{2} n \log \left (x e + d\right ) + 2 \, b d x e \log \left (c\right ) + b d^{2} n + 2 \, a d x e + b d^{2} \log \left (c\right ) + a d^{2}}{2 \,{\left (d x^{2} e^{4} + 2 \, d^{2} x e^{3} + d^{3} e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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