3.83 \(\int \frac{(d+e x) (a+b \log (c x^n))^2}{x^5} \, dx\)

Optimal. Leaf size=109 \[ -\frac{b d n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{2 b e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{b^2 d n^2}{32 x^4}-\frac{2 b^2 e n^2}{27 x^3} \]

[Out]

-(b^2*d*n^2)/(32*x^4) - (2*b^2*e*n^2)/(27*x^3) - (b*d*n*(a + b*Log[c*x^n]))/(8*x^4) - (2*b*e*n*(a + b*Log[c*x^
n]))/(9*x^3) - (d*(a + b*Log[c*x^n])^2)/(4*x^4) - (e*(a + b*Log[c*x^n])^2)/(3*x^3)

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Rubi [A]  time = 0.135152, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2353, 2305, 2304} \[ -\frac{b d n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{2 b e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac{b^2 d n^2}{32 x^4}-\frac{2 b^2 e n^2}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(a + b*Log[c*x^n])^2)/x^5,x]

[Out]

-(b^2*d*n^2)/(32*x^4) - (2*b^2*e*n^2)/(27*x^3) - (b*d*n*(a + b*Log[c*x^n]))/(8*x^4) - (2*b*e*n*(a + b*Log[c*x^
n]))/(9*x^3) - (d*(a + b*Log[c*x^n])^2)/(4*x^4) - (e*(a + b*Log[c*x^n])^2)/(3*x^3)

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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
]

Rubi steps

\begin{align*} \int \frac{(d+e x) \left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx &=\int \left (\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{x^5}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{x^4}\right ) \, dx\\ &=d \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^5} \, dx+e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx\\ &=-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}+\frac{1}{2} (b d n) \int \frac{a+b \log \left (c x^n\right )}{x^5} \, dx+\frac{1}{3} (2 b e n) \int \frac{a+b \log \left (c x^n\right )}{x^4} \, dx\\ &=-\frac{b^2 d n^2}{32 x^4}-\frac{2 b^2 e n^2}{27 x^3}-\frac{b d n \left (a+b \log \left (c x^n\right )\right )}{8 x^4}-\frac{2 b e n \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{4 x^4}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0532247, size = 82, normalized size = 0.75 \[ -\frac{216 d \left (a+b \log \left (c x^n\right )\right )^2+27 b d n \left (4 a+4 b \log \left (c x^n\right )+b n\right )+288 e x \left (a+b \log \left (c x^n\right )\right )^2+64 b e n x \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{864 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(a + b*Log[c*x^n])^2)/x^5,x]

[Out]

-(216*d*(a + b*Log[c*x^n])^2 + 288*e*x*(a + b*Log[c*x^n])^2 + 64*b*e*n*x*(3*a + b*n + 3*b*Log[c*x^n]) + 27*b*d
*n*(4*a + b*n + 4*b*Log[c*x^n]))/(864*x^4)

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Maple [C]  time = 0.204, size = 1486, normalized size = 13.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(a+b*ln(c*x^n))^2/x^5,x)

[Out]

-1/12*b^2*(4*e*x+3*d)/x^4*ln(x^n)^2-1/72*(24*I*Pi*b^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2-24*I*Pi*b^2*e*x*csgn(I*x
^n)*csgn(I*c*x^n)*csgn(I*c)-24*I*Pi*b^2*e*x*csgn(I*c*x^n)^3+24*I*Pi*b^2*e*x*csgn(I*c*x^n)^2*csgn(I*c)+48*ln(c)
*b^2*e*x+16*b^2*e*n*x+48*a*b*e*x+18*I*Pi*b^2*d*csgn(I*x^n)*csgn(I*c*x^n)^2-18*I*Pi*b^2*d*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)-18*I*Pi*b^2*d*csgn(I*c*x^n)^3+18*I*Pi*b^2*d*csgn(I*c*x^n)^2*csgn(I*c)+36*ln(c)*b^2*d+9*b^2*d*n+
36*a*b*d)/x^4*ln(x^n)-1/864*(-54*Pi^2*b^2*d*csgn(I*x^n)^2*csgn(I*c*x^n)^4+108*Pi^2*b^2*d*csgn(I*x^n)*csgn(I*c*
x^n)^5+288*a^2*e*x+144*Pi^2*b^2*e*x*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-72*Pi^2*b^2*e*x*csgn(I*x^n)^2*csgn
(I*c*x^n)^2*csgn(I*c)^2-288*Pi^2*b^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+144*Pi^2*b^2*e*x*csgn(I*x^n)*cs
gn(I*c*x^n)^3*csgn(I*c)^2+288*I*ln(c)*Pi*b^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2+96*I*n*Pi*b^2*e*x*csgn(I*x^n)*csg
n(I*c*x^n)^2+96*I*n*Pi*b^2*e*x*csgn(I*c*x^n)^2*csgn(I*c)+288*I*Pi*a*b*e*x*csgn(I*x^n)*csgn(I*c*x^n)^2+108*a*b*
n*d-54*I*Pi*b^2*d*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-216*I*Pi*a*b*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-216
*I*ln(c)*Pi*b^2*d*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+216*a^2*d+108*Pi^2*b^2*d*csgn(I*c*x^n)^5*csgn(I*c)-288*I
*Pi*a*b*e*x*csgn(I*c*x^n)^3+27*b^2*d*n^2+64*b^2*e*n^2*x+288*I*ln(c)*Pi*b^2*e*x*csgn(I*c*x^n)^2*csgn(I*c)+288*I
*Pi*a*b*e*x*csgn(I*c*x^n)^2*csgn(I*c)-54*I*Pi*b^2*d*n*csgn(I*c*x^n)^3-216*I*Pi*a*b*d*csgn(I*c*x^n)^3-216*I*ln(
c)*Pi*b^2*d*csgn(I*c*x^n)^3+108*ln(c)*b^2*d*n+432*ln(c)*a*b*d-96*I*n*Pi*b^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn
(I*c)-288*I*ln(c)*Pi*b^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-288*I*Pi*a*b*e*x*csgn(I*x^n)*csgn(I*c*x^n)*cs
gn(I*c)+144*Pi^2*b^2*e*x*csgn(I*x^n)*csgn(I*c*x^n)^5+144*Pi^2*b^2*e*x*csgn(I*c*x^n)^5*csgn(I*c)-72*Pi^2*b^2*e*
x*csgn(I*c*x^n)^4*csgn(I*c)^2+288*ln(c)^2*b^2*e*x+54*I*Pi*b^2*d*n*csgn(I*x^n)*csgn(I*c*x^n)^2+54*I*Pi*b^2*d*n*
csgn(I*c*x^n)^2*csgn(I*c)+576*ln(c)*a*b*e*x+192*ln(c)*b^2*e*n*x+216*ln(c)^2*b^2*d-54*Pi^2*b^2*d*csgn(I*c*x^n)^
4*csgn(I*c)^2+216*I*ln(c)*Pi*b^2*d*csgn(I*x^n)*csgn(I*c*x^n)^2+216*I*ln(c)*Pi*b^2*d*csgn(I*c*x^n)^2*csgn(I*c)+
216*I*Pi*a*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2+216*I*Pi*a*b*d*csgn(I*c*x^n)^2*csgn(I*c)-72*Pi^2*b^2*e*x*csgn(I*c*x
^n)^6-72*Pi^2*b^2*e*x*csgn(I*x^n)^2*csgn(I*c*x^n)^4-54*Pi^2*b^2*d*csgn(I*c*x^n)^6+192*a*b*e*n*x+108*Pi^2*b^2*d
*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-54*Pi^2*b^2*d*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+108*Pi^2*b^2*
d*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-96*I*n*Pi*b^2*e*x*csgn(I*c*x^n)^3-288*I*ln(c)*Pi*b^2*e*x*csgn(I*c*x^
n)^3-216*Pi^2*b^2*d*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c))/x^4

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Maxima [A]  time = 1.15612, size = 204, normalized size = 1.87 \begin{align*} -\frac{2}{27} \, b^{2} e{\left (\frac{n^{2}}{x^{3}} + \frac{3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac{1}{32} \, b^{2} d{\left (\frac{n^{2}}{x^{4}} + \frac{4 \, n \log \left (c x^{n}\right )}{x^{4}}\right )} - \frac{b^{2} e \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac{2 \, a b e n}{9 \, x^{3}} - \frac{2 \, a b e \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{b^{2} d \log \left (c x^{n}\right )^{2}}{4 \, x^{4}} - \frac{a b d n}{8 \, x^{4}} - \frac{a^{2} e}{3 \, x^{3}} - \frac{a b d \log \left (c x^{n}\right )}{2 \, x^{4}} - \frac{a^{2} d}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*log(c*x^n))^2/x^5,x, algorithm="maxima")

[Out]

-2/27*b^2*e*(n^2/x^3 + 3*n*log(c*x^n)/x^3) - 1/32*b^2*d*(n^2/x^4 + 4*n*log(c*x^n)/x^4) - 1/3*b^2*e*log(c*x^n)^
2/x^3 - 2/9*a*b*e*n/x^3 - 2/3*a*b*e*log(c*x^n)/x^3 - 1/4*b^2*d*log(c*x^n)^2/x^4 - 1/8*a*b*d*n/x^4 - 1/3*a^2*e/
x^3 - 1/2*a*b*d*log(c*x^n)/x^4 - 1/4*a^2*d/x^4

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Fricas [A]  time = 1.02065, size = 467, normalized size = 4.28 \begin{align*} -\frac{27 \, b^{2} d n^{2} + 108 \, a b d n + 216 \, a^{2} d + 72 \,{\left (4 \, b^{2} e x + 3 \, b^{2} d\right )} \log \left (c\right )^{2} + 72 \,{\left (4 \, b^{2} e n^{2} x + 3 \, b^{2} d n^{2}\right )} \log \left (x\right )^{2} + 32 \,{\left (2 \, b^{2} e n^{2} + 6 \, a b e n + 9 \, a^{2} e\right )} x + 12 \,{\left (9 \, b^{2} d n + 36 \, a b d + 16 \,{\left (b^{2} e n + 3 \, a b e\right )} x\right )} \log \left (c\right ) + 12 \,{\left (9 \, b^{2} d n^{2} + 36 \, a b d n + 16 \,{\left (b^{2} e n^{2} + 3 \, a b e n\right )} x + 12 \,{\left (4 \, b^{2} e n x + 3 \, b^{2} d n\right )} \log \left (c\right )\right )} \log \left (x\right )}{864 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*log(c*x^n))^2/x^5,x, algorithm="fricas")

[Out]

-1/864*(27*b^2*d*n^2 + 108*a*b*d*n + 216*a^2*d + 72*(4*b^2*e*x + 3*b^2*d)*log(c)^2 + 72*(4*b^2*e*n^2*x + 3*b^2
*d*n^2)*log(x)^2 + 32*(2*b^2*e*n^2 + 6*a*b*e*n + 9*a^2*e)*x + 12*(9*b^2*d*n + 36*a*b*d + 16*(b^2*e*n + 3*a*b*e
)*x)*log(c) + 12*(9*b^2*d*n^2 + 36*a*b*d*n + 16*(b^2*e*n^2 + 3*a*b*e*n)*x + 12*(4*b^2*e*n*x + 3*b^2*d*n)*log(c
))*log(x))/x^4

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Sympy [B]  time = 4.39674, size = 311, normalized size = 2.85 \begin{align*} - \frac{a^{2} d}{4 x^{4}} - \frac{a^{2} e}{3 x^{3}} - \frac{a b d n \log{\left (x \right )}}{2 x^{4}} - \frac{a b d n}{8 x^{4}} - \frac{a b d \log{\left (c \right )}}{2 x^{4}} - \frac{2 a b e n \log{\left (x \right )}}{3 x^{3}} - \frac{2 a b e n}{9 x^{3}} - \frac{2 a b e \log{\left (c \right )}}{3 x^{3}} - \frac{b^{2} d n^{2} \log{\left (x \right )}^{2}}{4 x^{4}} - \frac{b^{2} d n^{2} \log{\left (x \right )}}{8 x^{4}} - \frac{b^{2} d n^{2}}{32 x^{4}} - \frac{b^{2} d n \log{\left (c \right )} \log{\left (x \right )}}{2 x^{4}} - \frac{b^{2} d n \log{\left (c \right )}}{8 x^{4}} - \frac{b^{2} d \log{\left (c \right )}^{2}}{4 x^{4}} - \frac{b^{2} e n^{2} \log{\left (x \right )}^{2}}{3 x^{3}} - \frac{2 b^{2} e n^{2} \log{\left (x \right )}}{9 x^{3}} - \frac{2 b^{2} e n^{2}}{27 x^{3}} - \frac{2 b^{2} e n \log{\left (c \right )} \log{\left (x \right )}}{3 x^{3}} - \frac{2 b^{2} e n \log{\left (c \right )}}{9 x^{3}} - \frac{b^{2} e \log{\left (c \right )}^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*ln(c*x**n))**2/x**5,x)

[Out]

-a**2*d/(4*x**4) - a**2*e/(3*x**3) - a*b*d*n*log(x)/(2*x**4) - a*b*d*n/(8*x**4) - a*b*d*log(c)/(2*x**4) - 2*a*
b*e*n*log(x)/(3*x**3) - 2*a*b*e*n/(9*x**3) - 2*a*b*e*log(c)/(3*x**3) - b**2*d*n**2*log(x)**2/(4*x**4) - b**2*d
*n**2*log(x)/(8*x**4) - b**2*d*n**2/(32*x**4) - b**2*d*n*log(c)*log(x)/(2*x**4) - b**2*d*n*log(c)/(8*x**4) - b
**2*d*log(c)**2/(4*x**4) - b**2*e*n**2*log(x)**2/(3*x**3) - 2*b**2*e*n**2*log(x)/(9*x**3) - 2*b**2*e*n**2/(27*
x**3) - 2*b**2*e*n*log(c)*log(x)/(3*x**3) - 2*b**2*e*n*log(c)/(9*x**3) - b**2*e*log(c)**2/(3*x**3)

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Giac [B]  time = 1.38664, size = 278, normalized size = 2.55 \begin{align*} -\frac{288 \, b^{2} n^{2} x e \log \left (x\right )^{2} + 192 \, b^{2} n^{2} x e \log \left (x\right ) + 576 \, b^{2} n x e \log \left (c\right ) \log \left (x\right ) + 216 \, b^{2} d n^{2} \log \left (x\right )^{2} + 64 \, b^{2} n^{2} x e + 192 \, b^{2} n x e \log \left (c\right ) + 288 \, b^{2} x e \log \left (c\right )^{2} + 108 \, b^{2} d n^{2} \log \left (x\right ) + 576 \, a b n x e \log \left (x\right ) + 432 \, b^{2} d n \log \left (c\right ) \log \left (x\right ) + 27 \, b^{2} d n^{2} + 192 \, a b n x e + 108 \, b^{2} d n \log \left (c\right ) + 576 \, a b x e \log \left (c\right ) + 216 \, b^{2} d \log \left (c\right )^{2} + 432 \, a b d n \log \left (x\right ) + 108 \, a b d n + 288 \, a^{2} x e + 432 \, a b d \log \left (c\right ) + 216 \, a^{2} d}{864 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*log(c*x^n))^2/x^5,x, algorithm="giac")

[Out]

-1/864*(288*b^2*n^2*x*e*log(x)^2 + 192*b^2*n^2*x*e*log(x) + 576*b^2*n*x*e*log(c)*log(x) + 216*b^2*d*n^2*log(x)
^2 + 64*b^2*n^2*x*e + 192*b^2*n*x*e*log(c) + 288*b^2*x*e*log(c)^2 + 108*b^2*d*n^2*log(x) + 576*a*b*n*x*e*log(x
) + 432*b^2*d*n*log(c)*log(x) + 27*b^2*d*n^2 + 192*a*b*n*x*e + 108*b^2*d*n*log(c) + 576*a*b*x*e*log(c) + 216*b
^2*d*log(c)^2 + 432*a*b*d*n*log(x) + 108*a*b*d*n + 288*a^2*x*e + 432*a*b*d*log(c) + 216*a^2*d)/x^4