3.1.58 \(\int \frac {a+b \log (c x^n)}{(d+e x)^4} \, dx\) [58]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, 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 = {2356, 46} \begin {gather*} -\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}+\frac {b n \log (x)}{3 d^3 e}-\frac {b n \log (d+e x)}{3 d^3 e}+\frac {b n}{3 d^2 e (d+e x)}+\frac {b n}{6 d e (d+e x)^2} \end {gather*}

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 46

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] && Lt
Q[m + n + 2, 0])

Rule 2356

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]))

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*}

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Mathematica [A]
time = 0.05, size = 66, normalized size = 0.69 \begin {gather*} \frac {-\frac {a+b \log \left (c x^n\right )}{(d+e x)^3}+\frac {b n \left (\frac {d (3 d+2 e x)}{(d+e x)^2}+2 \log (x)-2 \log (d+e x)\right )}{2 d^3}}{3 e} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.12, size = 284, normalized size = 2.99

method	result	size
risch	\(-\frac {b \ln \left (x^{n}\right )}{3 e \left (e x +d \right )^{3}}-\frac {-2 \ln \left (-x \right ) b \,e^{3} n \,x^{3}+2 \ln \left (e x +d \right ) b \,e^{3} n \,x^{3}+i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-i \pi b \,d^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-6 \ln \left (-x \right ) b d \,e^{2} n \,x^{2}+6 \ln \left (e x +d \right ) b d \,e^{2} n \,x^{2}-6 \ln \left (-x \right ) b \,d^{2} e n x +6 \ln \left (e x +d \right ) b \,d^{2} e n x -2 b d \,e^{2} n \,x^{2}-2 \ln \left (-x \right ) b \,d^{3} n +2 \ln \left (e x +d \right ) b \,d^{3} n -5 b \,d^{2} e n x +2 d^{3} b \ln \left (c \right )-3 b \,d^{3} n +2 a \,d^{3}}{6 d^{3} e \left (e x +d \right )^{3}}\)	\(284\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 139, normalized size = 1.46 \begin {gather*} \frac {1}{6} \, b n {\left (\frac {2 \, x e + 3 \, d}{d^{2} x^{2} e^{3} + 2 \, d^{3} x e^{2} + d^{4} e} - \frac {2 \, e^{\left (-1\right )} \log \left (x e + d\right )}{d^{3}} + \frac {2 \, e^{\left (-1\right )} \log \left (x\right )}{d^{3}}\right )} - \frac {b \log \left (c x^{n}\right )}{3 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} - \frac {a}{3 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}} \end {gather*}

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

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Fricas [A]
time = 0.37, size = 157, normalized size = 1.65 \begin {gather*} \frac {2 \, b d n x^{2} e^{2} + 5 \, b d^{2} n x e + 3 \, b d^{3} n - 2 \, b d^{3} \log \left (c\right ) - 2 \, a d^{3} - 2 \, {\left (b n x^{3} e^{3} + 3 \, b d n x^{2} e^{2} + 3 \, b d^{2} n x e + b d^{3} n\right )} \log \left (x e + d\right ) + 2 \, {\left (b n x^{3} e^{3} + 3 \, b d n x^{2} e^{2} + 3 \, b d^{2} n x e\right )} \log \left (x\right )}{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 {gather*}

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (83) = 166\).
time = 6.00, size = 700, normalized size = 7.37 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {a}{3 x^{3}} - \frac {b n}{9 x^{3}} - \frac {b \log {\left (c x^{n} \right )}}{3 x^{3}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {- \frac {a}{3 x^{3}} - \frac {b n}{9 x^{3}} - \frac {b \log {\left (c x^{n} \right )}}{3 x^{3}}}{e^{4}} & \text {for}\: d = 0 \\\frac {a x - b n x + b x \log {\left (c x^{n} \right )}}{d^{4}} & \text {for}\: e = 0 \\- \frac {2 a d^{3}}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} - \frac {2 b d^{3} n \log {\left (\frac {d}{e} + x \right )}}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} + \frac {3 b d^{3} n}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} - \frac {6 b d^{2} e n x \log {\left (\frac {d}{e} + x \right )}}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} + \frac {5 b d^{2} e n x}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} + \frac {6 b d^{2} e x \log {\left (c x^{n} \right )}}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} - \frac {6 b d e^{2} n x^{2} \log {\left (\frac {d}{e} + x \right )}}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} + \frac {2 b d e^{2} n x^{2}}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} + \frac {6 b d e^{2} x^{2} \log {\left (c x^{n} \right )}}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} - \frac {2 b e^{3} n x^{3} \log {\left (\frac {d}{e} + x \right )}}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} + \frac {2 b e^{3} x^{3} \log {\left (c x^{n} \right )}}{6 d^{6} e + 18 d^{5} e^{2} x + 18 d^{4} e^{3} x^{2} + 6 d^{3} e^{4} x^{3}} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (84) = 168\).
time = 3.80, size = 179, normalized size = 1.88 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 3.85, size = 127, normalized size = 1.34 \begin {gather*} \frac {\frac {3\,b\,n}{2}-a+\frac {b\,e^2\,n\,x^2}{d^2}+\frac {5\,b\,e\,n\,x}{2\,d}}{3\,d^3\,e+9\,d^2\,e^2\,x+9\,d\,e^3\,x^2+3\,e^4\,x^3}-\frac {b\,\ln \left (c\,x^n\right )}{3\,e\,\left (d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3\right )}-\frac {2\,b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{3\,d^3\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(d + e*x)^4,x)

[Out]

((3*b*n)/2 - a + (b*e^2*n*x^2)/d^2 + (5*b*e*n*x)/(2*d))/(3*d^3*e + 3*e^4*x^3 + 9*d^2*e^2*x + 9*d*e^3*x^2) - (b
*log(c*x^n))/(3*e*(d^3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x)) - (2*b*n*atanh((2*e*x)/d + 1))/(3*d^3*e)

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