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

Optimal. Leaf size=95 \[ -\frac{a+b \log \left (c x^n\right )}{3 e (d+e x)^3}+\frac{b n}{3 d^2 e (d+e x)}+\frac{b n \log (x)}{3 d^3 e}-\frac{b n \log (d+e x)}{3 d^3 e}+\frac{b n}{6 d e (d+e x)^2} \]

[Out]

(b*n)/(6*d*e*(d + e*x)^2) + (b*n)/(3*d^2*e*(d + e*x)) + (b*n*Log[x])/(3*d^3*e) - (a + b*Log[c*x^n])/(3*e*(d +
e*x)^3) - (b*n*Log[d + e*x])/(3*d^3*e)

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Rubi [A]  time = 0.0411906, antiderivative size = 95, 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} \[ -\frac{a+b \log \left (c x^n\right )}{3 e (d+e x)^3}+\frac{b n}{3 d^2 e (d+e x)}+\frac{b n \log (x)}{3 d^3 e}-\frac{b n \log (d+e x)}{3 d^3 e}+\frac{b n}{6 d e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(d + e*x)^4,x]

[Out]

(b*n)/(6*d*e*(d + e*x)^2) + (b*n)/(3*d^2*e*(d + e*x)) + (b*n*Log[x])/(3*d^3*e) - (a + b*Log[c*x^n])/(3*e*(d +
e*x)^3) - (b*n*Log[d + e*x])/(3*d^3*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)^4} \, dx &=-\frac{a+b \log \left (c x^n\right )}{3 e (d+e x)^3}+\frac{(b n) \int \frac{1}{x (d+e x)^3} \, dx}{3 e}\\ &=-\frac{a+b \log \left (c x^n\right )}{3 e (d+e x)^3}+\frac{(b n) \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}\\ &=\frac{b n}{6 d e (d+e x)^2}+\frac{b n}{3 d^2 e (d+e x)}+\frac{b n \log (x)}{3 d^3 e}-\frac{a+b \log \left (c x^n\right )}{3 e (d+e x)^3}-\frac{b n \log (d+e x)}{3 d^3 e}\\ \end{align*}

Mathematica [A]  time = 0.0740458, size = 66, normalized size = 0.69 \[ \frac{\frac{b n \left (\frac{d (3 d+2 e x)}{(d+e x)^2}-2 \log (d+e x)+2 \log (x)\right )}{2 d^3}-\frac{a+b \log \left (c x^n\right )}{(d+e x)^3}}{3 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.101, size = 284, normalized size = 3. \begin{align*} -{\frac{b\ln \left ({x}^{n} \right ) }{3\, \left ( ex+d \right ) ^{3}e}}-{\frac{-i\pi \,b{d}^{3}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +i\pi \,b{d}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +i\pi \,b{d}^{3}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,b{d}^{3} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,\ln \left ( ex+d \right ) b{e}^{3}n{x}^{3}-2\,\ln \left ( -x \right ) b{e}^{3}n{x}^{3}+6\,\ln \left ( ex+d \right ) bd{e}^{2}n{x}^{2}-6\,\ln \left ( -x \right ) bd{e}^{2}n{x}^{2}+6\,\ln \left ( ex+d \right ) b{d}^{2}enx-6\,\ln \left ( -x \right ) b{d}^{2}enx-2\,bd{e}^{2}n{x}^{2}+2\,\ln \left ( ex+d \right ) b{d}^{3}n-2\,\ln \left ( -x \right ) b{d}^{3}n-5\,b{d}^{2}enx+2\,\ln \left ( c \right ) b{d}^{3}-3\,b{d}^{3}n+2\,a{d}^{3}}{6\,{d}^{3}e \left ( ex+d \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b/e/(e*x+d)^3*ln(x^n)-1/6*(-I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+I*Pi*b*d^3*csgn(I*c*x^n)^2*csg
n(I*c)+I*Pi*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*d^3*csgn(I*c*x^n)^3+2*ln(e*x+d)*b*e^3*n*x^3-2*ln(-x)*b*e^
3*n*x^3+6*ln(e*x+d)*b*d*e^2*n*x^2-6*ln(-x)*b*d*e^2*n*x^2+6*ln(e*x+d)*b*d^2*e*n*x-6*ln(-x)*b*d^2*e*n*x-2*b*d*e^
2*n*x^2+2*ln(e*x+d)*b*d^3*n-2*ln(-x)*b*d^3*n-5*b*d^2*e*n*x+2*ln(c)*b*d^3-3*b*d^3*n+2*a*d^3)/d^3/e/(e*x+d)^3

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Maxima [A]  time = 1.17066, size = 194, normalized size = 2.04 \begin{align*} \frac{1}{6} \, b n{\left (\frac{2 \, e x + 3 \, d}{d^{2} e^{3} x^{2} + 2 \, d^{3} e^{2} x + d^{4} e} - \frac{2 \, \log \left (e x + d\right )}{d^{3} e} + \frac{2 \, \log \left (x\right )}{d^{3} e}\right )} - \frac{b \log \left (c x^{n}\right )}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac{a}{3 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} 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)^4,x, algorithm="maxima")

[Out]

1/6*b*n*((2*e*x + 3*d)/(d^2*e^3*x^2 + 2*d^3*e^2*x + d^4*e) - 2*log(e*x + d)/(d^3*e) + 2*log(x)/(d^3*e)) - 1/3*
b*log(c*x^n)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 1/3*a/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*
e)

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Fricas [A]  time = 1.08493, size = 356, normalized size = 3.75 \begin{align*} \frac{2 \, b d e^{2} n x^{2} + 5 \, b d^{2} e n x + 3 \, b d^{3} n - 2 \, b d^{3} \log \left (c\right ) - 2 \, a d^{3} - 2 \,{\left (b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2} + 3 \, b d^{2} e n x + b d^{3} n\right )} \log \left (e x + d\right ) + 2 \,{\left (b e^{3} n x^{3} + 3 \, b d e^{2} n x^{2} + 3 \, b d^{2} e n x\right )} \log \left (x\right )}{6 \,{\left (d^{3} e^{4} x^{3} + 3 \, d^{4} e^{3} x^{2} + 3 \, d^{5} e^{2} x + d^{6} 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)^4,x, algorithm="fricas")

[Out]

1/6*(2*b*d*e^2*n*x^2 + 5*b*d^2*e*n*x + 3*b*d^3*n - 2*b*d^3*log(c) - 2*a*d^3 - 2*(b*e^3*n*x^3 + 3*b*d*e^2*n*x^2
+ 3*b*d^2*e*n*x + b*d^3*n)*log(e*x + d) + 2*(b*e^3*n*x^3 + 3*b*d*e^2*n*x^2 + 3*b*d^2*e*n*x)*log(x))/(d^3*e^4*
x^3 + 3*d^4*e^3*x^2 + 3*d^5*e^2*x + d^6*e)

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Sympy [A]  time = 13.0407, size = 881, normalized size = 9.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(-a/(3*x**3) - b*n*log(x)/(3*x**3) - b*n/(9*x**3) - b*log(c)/(3*x**3)), Eq(d, 0) & Eq(e, 0)), (
(a*x + b*n*x*log(x) - b*n*x + b*x*log(c))/d**4, Eq(e, 0)), ((-a/(3*x**3) - b*n*log(x)/(3*x**3) - b*n/(9*x**3)
- b*log(c)/(3*x**3))/e**4, Eq(d, 0)), (-6*a*d**3/(18*d**6*e + 54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**
4*x**3) - 6*b*d**3*n*log(d/e + x)/(18*d**6*e + 54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**4*x**3) + 7*b*d
**3*n/(18*d**6*e + 54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**4*x**3) + 18*b*d**2*e*n*x*log(x)/(18*d**6*e
+ 54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**4*x**3) - 18*b*d**2*e*n*x*log(d/e + x)/(18*d**6*e + 54*d**5
*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**4*x**3) + 9*b*d**2*e*n*x/(18*d**6*e + 54*d**5*e**2*x + 54*d**4*e**3*x
**2 + 18*d**3*e**4*x**3) + 18*b*d**2*e*x*log(c)/(18*d**6*e + 54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**4
*x**3) + 18*b*d*e**2*n*x**2*log(x)/(18*d**6*e + 54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**4*x**3) - 18*b
*d*e**2*n*x**2*log(d/e + x)/(18*d**6*e + 54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**4*x**3) + 18*b*d*e**2
*x**2*log(c)/(18*d**6*e + 54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**4*x**3) + 6*b*e**3*n*x**3*log(x)/(18
*d**6*e + 54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**4*x**3) - 6*b*e**3*n*x**3*log(d/e + x)/(18*d**6*e +
54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d**3*e**4*x**3) - 2*b*e**3*n*x**3/(18*d**6*e + 54*d**5*e**2*x + 54*d**
4*e**3*x**2 + 18*d**3*e**4*x**3) + 6*b*e**3*x**3*log(c)/(18*d**6*e + 54*d**5*e**2*x + 54*d**4*e**3*x**2 + 18*d
**3*e**4*x**3), True))

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Giac [B]  time = 1.30056, size = 242, normalized size = 2.55 \begin{align*} -\frac{2 \, b n x^{3} e^{3} \log \left (x e + d\right ) + 6 \, b d n x^{2} e^{2} \log \left (x e + d\right ) + 6 \, b d^{2} n x e \log \left (x e + d\right ) - 2 \, b n x^{3} e^{3} \log \left (x\right ) - 6 \, b d n x^{2} e^{2} \log \left (x\right ) - 6 \, b d^{2} n x e \log \left (x\right ) - 2 \, b d n x^{2} e^{2} - 5 \, b d^{2} n x e + 2 \, b d^{3} n \log \left (x e + d\right ) - 3 \, b d^{3} n + 2 \, b d^{3} \log \left (c\right ) + 2 \, a d^{3}}{6 \,{\left (d^{3} x^{3} e^{4} + 3 \, d^{4} x^{2} e^{3} + 3 \, d^{5} x e^{2} + d^{6} 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)^4,x, algorithm="giac")

[Out]

-1/6*(2*b*n*x^3*e^3*log(x*e + d) + 6*b*d*n*x^2*e^2*log(x*e + d) + 6*b*d^2*n*x*e*log(x*e + d) - 2*b*n*x^3*e^3*l
og(x) - 6*b*d*n*x^2*e^2*log(x) - 6*b*d^2*n*x*e*log(x) - 2*b*d*n*x^2*e^2 - 5*b*d^2*n*x*e + 2*b*d^3*n*log(x*e +
d) - 3*b*d^3*n + 2*b*d^3*log(c) + 2*a*d^3)/(d^3*x^3*e^4 + 3*d^4*x^2*e^3 + 3*d^5*x*e^2 + d^6*e)