3.1.82 \(\int \frac {(d+e x) (a+b \log (c x^n))^2}{x^4} \, dx\) [82]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 109, normalized size of antiderivative = 1.00, 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 = {2395, 2342, 2341} \begin {gather*} -\frac {2 b d 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}{3 x^3}-\frac {b e n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {2 b^2 d n^2}{27 x^3}-\frac {b^2 e n^2}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(a + b*Log[c*x^n])^2)/x^4,x]

[Out]

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

Rule 2341

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
]

Rule 2342

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 2395

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
]))

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 82, normalized size = 0.75 \begin {gather*} -\frac {36 d \left (a+b \log \left (c x^n\right )\right )^2+54 e x \left (a+b \log \left (c x^n\right )\right )^2+27 b e n x \left (2 a+b n+2 b \log \left (c x^n\right )\right )+8 b d n \left (3 a+b n+3 b \log \left (c x^n\right )\right )}{108 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(a + b*Log[c*x^n])^2)/x^4,x]

[Out]

-1/108*(36*d*(a + b*Log[c*x^n])^2 + 54*e*x*(a + b*Log[c*x^n])^2 + 27*b*e*n*x*(2*a + b*n + 2*b*Log[c*x^n]) + 8*
b*d*n*(3*a + b*n + 3*b*Log[c*x^n]))/x^3

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.11, size = 1486, normalized size = 13.63

method	result	size
risch	\(\text {Expression too large to display}\)	\(1486\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(a+b*ln(c*x^n))^2/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 156, normalized size = 1.43 \begin {gather*} -\frac {2}{27} \, b^{2} d {\left (\frac {n^{2}}{x^{3}} + \frac {3 \, n \log \left (c x^{n}\right )}{x^{3}}\right )} - \frac {1}{4} \, b^{2} {\left (\frac {n^{2}}{x^{2}} + \frac {2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} e - \frac {b^{2} e \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac {a b n e}{2 \, x^{2}} - \frac {a b e \log \left (c x^{n}\right )}{x^{2}} - \frac {b^{2} d \log \left (c x^{n}\right )^{2}}{3 \, x^{3}} - \frac {2 \, a b d n}{9 \, x^{3}} - \frac {a^{2} e}{2 \, x^{2}} - \frac {2 \, a b d \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac {a^{2} d}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*log(c*x^n))^2/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 189, normalized size = 1.73 \begin {gather*} -\frac {8 \, b^{2} d n^{2} + 24 \, a b d n + 36 \, a^{2} d + 27 \, {\left (b^{2} n^{2} + 2 \, a b n + 2 \, a^{2}\right )} x e + 18 \, {\left (3 \, b^{2} x e + 2 \, b^{2} d\right )} \log \left (c\right )^{2} + 18 \, {\left (3 \, b^{2} n^{2} x e + 2 \, b^{2} d n^{2}\right )} \log \left (x\right )^{2} + 6 \, {\left (4 \, b^{2} d n + 12 \, a b d + 9 \, {\left (b^{2} n + 2 \, a b\right )} x e\right )} \log \left (c\right ) + 6 \, {\left (4 \, b^{2} d n^{2} + 12 \, a b d n + 9 \, {\left (b^{2} n^{2} + 2 \, a b n\right )} x e + 6 \, {\left (3 \, b^{2} n x e + 2 \, b^{2} d n\right )} \log \left (c\right )\right )} \log \left (x\right )}{108 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*log(c*x^n))^2/x^4,x, algorithm="fricas")

[Out]

-1/108*(8*b^2*d*n^2 + 24*a*b*d*n + 36*a^2*d + 27*(b^2*n^2 + 2*a*b*n + 2*a^2)*x*e + 18*(3*b^2*x*e + 2*b^2*d)*lo
g(c)^2 + 18*(3*b^2*n^2*x*e + 2*b^2*d*n^2)*log(x)^2 + 6*(4*b^2*d*n + 12*a*b*d + 9*(b^2*n + 2*a*b)*x*e)*log(c) +
6*(4*b^2*d*n^2 + 12*a*b*d*n + 9*(b^2*n^2 + 2*a*b*n)*x*e + 6*(3*b^2*n*x*e + 2*b^2*d*n)*log(c))*log(x))/x^3

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Sympy [A]
time = 0.40, size = 185, normalized size = 1.70 \begin {gather*} - \frac {a^{2} d}{3 x^{3}} - \frac {a^{2} e}{2 x^{2}} - \frac {2 a b d n}{9 x^{3}} - \frac {2 a b d \log {\left (c x^{n} \right )}}{3 x^{3}} - \frac {a b e n}{2 x^{2}} - \frac {a b e \log {\left (c x^{n} \right )}}{x^{2}} - \frac {2 b^{2} d n^{2}}{27 x^{3}} - \frac {2 b^{2} d n \log {\left (c x^{n} \right )}}{9 x^{3}} - \frac {b^{2} d \log {\left (c x^{n} \right )}^{2}}{3 x^{3}} - \frac {b^{2} e n^{2}}{4 x^{2}} - \frac {b^{2} e n \log {\left (c x^{n} \right )}}{2 x^{2}} - \frac {b^{2} e \log {\left (c x^{n} \right )}^{2}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*ln(c*x**n))**2/x**4,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (100) = 200\).
time = 2.93, size = 206, normalized size = 1.89 \begin {gather*} -\frac {54 \, b^{2} n^{2} x e \log \left (x\right )^{2} + 54 \, b^{2} n^{2} x e \log \left (x\right ) + 108 \, b^{2} n x e \log \left (c\right ) \log \left (x\right ) + 36 \, b^{2} d n^{2} \log \left (x\right )^{2} + 27 \, b^{2} n^{2} x e + 54 \, b^{2} n x e \log \left (c\right ) + 54 \, b^{2} x e \log \left (c\right )^{2} + 24 \, b^{2} d n^{2} \log \left (x\right ) + 108 \, a b n x e \log \left (x\right ) + 72 \, b^{2} d n \log \left (c\right ) \log \left (x\right ) + 8 \, b^{2} d n^{2} + 54 \, a b n x e + 24 \, b^{2} d n \log \left (c\right ) + 108 \, a b x e \log \left (c\right ) + 36 \, b^{2} d \log \left (c\right )^{2} + 72 \, a b d n \log \left (x\right ) + 24 \, a b d n + 54 \, a^{2} x e + 72 \, a b d \log \left (c\right ) + 36 \, a^{2} d}{108 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*log(c*x^n))^2/x^4,x, algorithm="giac")

[Out]

-1/108*(54*b^2*n^2*x*e*log(x)^2 + 54*b^2*n^2*x*e*log(x) + 108*b^2*n*x*e*log(c)*log(x) + 36*b^2*d*n^2*log(x)^2
+ 27*b^2*n^2*x*e + 54*b^2*n*x*e*log(c) + 54*b^2*x*e*log(c)^2 + 24*b^2*d*n^2*log(x) + 108*a*b*n*x*e*log(x) + 72
*b^2*d*n*log(c)*log(x) + 8*b^2*d*n^2 + 54*a*b*n*x*e + 24*b^2*d*n*log(c) + 108*a*b*x*e*log(c) + 36*b^2*d*log(c)
^2 + 72*a*b*d*n*log(x) + 24*a*b*d*n + 54*a^2*x*e + 72*a*b*d*log(c) + 36*a^2*d)/x^3

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Mupad [B]
time = 3.79, size = 114, normalized size = 1.05 \begin {gather*} -\frac {x\,\left (9\,e\,a^2+9\,e\,a\,b\,n+\frac {9\,e\,b^2\,n^2}{2}\right )+6\,a^2\,d+\frac {4\,b^2\,d\,n^2}{3}+4\,a\,b\,d\,n}{18\,x^3}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {2\,b\,d\,\left (3\,a+b\,n\right )}{3}+\frac {3\,b\,e\,x\,\left (2\,a+b\,n\right )}{2}\right )}{3\,x^3}-\frac {{\ln \left (c\,x^n\right )}^2\,\left (\frac {b^2\,d}{3}+\frac {b^2\,e\,x}{2}\right )}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (x*(9*a^2*e + (9*b^2*e*n^2)/2 + 9*a*b*e*n) + 6*a^2*d + (4*b^2*d*n^2)/3 + 4*a*b*d*n)/(18*x^3) - (log(c*x^n)*(
(2*b*d*(3*a + b*n))/3 + (3*b*e*x*(2*a + b*n))/2))/(3*x^3) - (log(c*x^n)^2*((b^2*d)/3 + (b^2*e*x)/2))/x^3

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