3.5.29 \(\int \frac {(d+e x^r) (a+b \log (c x^n))^2}{x} \, dx\) [429]

Optimal. Leaf size=80 \[ \frac {2 b^2 e n^2 x^r}{r^3}-\frac {2 b e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}+\frac {e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}+\frac {d \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2395, 2339, 30, 2342, 2341} \begin {gather*} \frac {d \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {2 b e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}+\frac {e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}+\frac {2 b^2 e n^2 x^r}{r^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

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

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Mathematica [A]
time = 0.09, size = 114, normalized size = 1.42 \begin {gather*} \frac {1}{3} b^2 d n^2 \log ^3(x)-b d n \log ^2(x) \left (a+b \log \left (c x^n\right )\right )+d \log (x) \left (a+b \log \left (c x^n\right )\right )^2+\frac {e x^r \left (2 b^2 n^2-2 a b n r+a^2 r^2+2 b r (-b n+a r) \log \left (c x^n\right )+b^2 r^2 \log ^2\left (c x^n\right )\right )}{r^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.18, size = 1712, normalized size = 21.40

method	result	size
risch	\(\text {Expression too large to display}\)	\(1712\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/r*ln(c)^2*b^2*e*x^r+ln(x)*a^2*d-b*(I*ln(x)*Pi*b*d*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*r^2-I*ln(x)*Pi*b*d*csg
n(I*c)*csgn(I*c*x^n)^2*r^2-I*ln(x)*Pi*b*d*csgn(I*x^n)*csgn(I*c*x^n)^2*r^2+I*ln(x)*Pi*b*d*csgn(I*c*x^n)^3*r^2+I
*Pi*b*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r*r-I*Pi*b*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r*r-I*Pi*b*e*csgn(I*x^n
)*csgn(I*c*x^n)^2*x^r*r+I*Pi*b*e*csgn(I*c*x^n)^3*x^r*r+b*d*n*ln(x)^2*r^2-2*ln(x)*ln(c)*b*d*r^2-2*ln(x)*a*d*r^2
-2*ln(c)*b*e*x^r*r-2*x^r*a*e*r+2*x^r*b*e*n)/r^2*ln(x^n)+b^2*(d*r*ln(x)+e*x^r)/r*ln(x^n)^2+a^2/r*x^r*e+ln(x)*ln
(c)^2*b^2*d+1/3*b^2*d*n^2*ln(x)^3-1/4*csgn(I*c*x^n)^6*d*b^2*Pi^2*ln(x)-1/4/r*Pi^2*b^2*e*csgn(I*c)^2*csgn(I*x^n
)^2*csgn(I*c*x^n)^2*x^r+1/2/r*Pi^2*b^2*e*csgn(I*c)^2*csgn(I*x^n)*csgn(I*c*x^n)^3*x^r+I/r*Pi*a*b*e*csgn(I*c)*cs
gn(I*c*x^n)^2*x^r+I/r*Pi*a*b*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-1/4/r*Pi^2*b^2*e*csgn(I*c*x^n)^6*x^r+2/r*ln(c)*
a*b*e*x^r-2/r^2*ln(c)*b^2*e*n*x^r-2/r^2*a*b*e*n*x^r-1/4*csgn(I*c*x^n)^2*csgn(I*x^n)^2*csgn(I*c)^2*d*b^2*Pi^2*l
n(x)+2*b^2*e*n^2*x^r/r^3+2*ln(x)*ln(c)*a*b*d-ln(x)^2*ln(c)*b^2*d*n-ln(x)^2*a*d*b*n-csgn(I*c*x^n)^4*csgn(I*x^n)
*csgn(I*c)*d*b^2*Pi^2*ln(x)+I/r*Pi*ln(c)*b^2*e*csgn(I*c)*csgn(I*c*x^n)^2*x^r+I/r*Pi*ln(c)*b^2*e*csgn(I*x^n)*cs
gn(I*c*x^n)^2*x^r-I/r^2*Pi*b^2*e*n*csgn(I*c)*csgn(I*c*x^n)^2*x^r-I/r^2*Pi*b^2*e*n*csgn(I*x^n)*csgn(I*c*x^n)^2*
x^r+1/2*I*Pi*b^2*d*n*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*ln(x)^2-I*ln(x)*Pi*ln(c)*b^2*d*csgn(I*c)*csgn(I*x^n)*
csgn(I*c*x^n)-I*ln(x)*Pi*a*b*d*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I/r^2*Pi*b^2*e*n*csgn(I*c)*csgn(I*x^n)*csgn
(I*c*x^n)*x^r-I/r*Pi*ln(c)*b^2*e*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x^r-I/r*Pi*a*b*e*csgn(I*c)*csgn(I*x^n)*cs
gn(I*c*x^n)*x^r+1/2/r*Pi^2*b^2*e*csgn(I*c)*csgn(I*x^n)^2*csgn(I*c*x^n)^3*x^r-1/r*Pi^2*b^2*e*csgn(I*c)*csgn(I*x
^n)*csgn(I*c*x^n)^4*x^r+I/r^2*Pi*b^2*e*n*csgn(I*c*x^n)^3*x^r+I*ln(x)*Pi*ln(c)*b^2*d*csgn(I*c)*csgn(I*c*x^n)^2+
I*ln(x)*Pi*ln(c)*b^2*d*csgn(I*x^n)*csgn(I*c*x^n)^2+I*ln(x)*Pi*a*b*d*csgn(I*c)*csgn(I*c*x^n)^2+I*ln(x)*Pi*a*b*d
*csgn(I*x^n)*csgn(I*c*x^n)^2-I/r*Pi*ln(c)*b^2*e*csgn(I*c*x^n)^3*x^r-I/r*Pi*a*b*e*csgn(I*c*x^n)^3*x^r-1/2*I*Pi*
b^2*d*n*csgn(I*c)*csgn(I*c*x^n)^2*ln(x)^2-1/2*I*Pi*b^2*d*n*csgn(I*x^n)*csgn(I*c*x^n)^2*ln(x)^2-1/4*csgn(I*c*x^
n)^4*csgn(I*x^n)^2*d*b^2*Pi^2*ln(x)+1/2*csgn(I*c*x^n)^5*csgn(I*x^n)*d*b^2*Pi^2*ln(x)-1/4*csgn(I*c*x^n)^4*csgn(
I*c)^2*d*b^2*Pi^2*ln(x)+1/2*csgn(I*c*x^n)^5*csgn(I*c)*d*b^2*Pi^2*ln(x)+1/2*csgn(I*c*x^n)^3*csgn(I*x^n)*csgn(I*
c)^2*d*b^2*Pi^2*ln(x)-1/4/r*Pi^2*b^2*e*csgn(I*c)^2*csgn(I*c*x^n)^4*x^r+1/2/r*Pi^2*b^2*e*csgn(I*c)*csgn(I*c*x^n
)^5*x^r-1/4/r*Pi^2*b^2*e*csgn(I*x^n)^2*csgn(I*c*x^n)^4*x^r+1/2/r*Pi^2*b^2*e*csgn(I*x^n)*csgn(I*c*x^n)^5*x^r+1/
2*I*Pi*b^2*d*n*csgn(I*c*x^n)^3*ln(x)^2-I*ln(x)*Pi*ln(c)*b^2*d*csgn(I*c*x^n)^3-I*ln(x)*Pi*a*b*d*csgn(I*c*x^n)^3
+1/2*csgn(I*c*x^n)^3*csgn(I*x^n)^2*csgn(I*c)*d*b^2*Pi^2*ln(x)

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Maxima [A]
time = 0.28, size = 131, normalized size = 1.64 \begin {gather*} \frac {b^{2} e x^{r} \log \left (c x^{n}\right )^{2}}{r} + \frac {b^{2} d \log \left (c x^{n}\right )^{3}}{3 \, n} - 2 \, b^{2} e {\left (\frac {n x^{r} \log \left (c x^{n}\right )}{r^{2}} - \frac {n^{2} x^{r}}{r^{3}}\right )} + \frac {2 \, a b e x^{r} \log \left (c x^{n}\right )}{r} + \frac {a b d \log \left (c x^{n}\right )^{2}}{n} + a^{2} d \log \left (x\right ) - \frac {2 \, a b e n x^{r}}{r^{2}} + \frac {a^{2} e x^{r}}{r} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (81) = 162\).
time = 0.36, size = 200, normalized size = 2.50 \begin {gather*} \frac {b^{2} d n^{2} r^{3} \log \left (x\right )^{3} + 3 \, {\left (b^{2} d n r^{3} \log \left (c\right ) + a b d n r^{3}\right )} \log \left (x\right )^{2} + 3 \, {\left (b^{2} n^{2} r^{2} e \log \left (x\right )^{2} + b^{2} r^{2} e \log \left (c\right )^{2} - 2 \, {\left (b^{2} n r - a b r^{2}\right )} e \log \left (c\right ) + {\left (2 \, b^{2} n^{2} - 2 \, a b n r + a^{2} r^{2}\right )} e + 2 \, {\left (b^{2} n r^{2} e \log \left (c\right ) - {\left (b^{2} n^{2} r - a b n r^{2}\right )} e\right )} \log \left (x\right )\right )} x^{r} + 3 \, {\left (b^{2} d r^{3} \log \left (c\right )^{2} + 2 \, a b d r^{3} \log \left (c\right ) + a^{2} d r^{3}\right )} \log \left (x\right )}{3 \, r^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (76) = 152\).
time = 12.87, size = 245, normalized size = 3.06 \begin {gather*} \begin {cases} \left (a + b \log {\left (c \right )}\right )^{2} \left (d + e\right ) \log {\left (x \right )} & \text {for}\: n = 0 \wedge r = 0 \\\left (d + e\right ) \left (\begin {cases} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\left (c \right )} + b^{2} \log {\left (c \right )}^{2}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) & \text {for}\: r = 0 \\\left (a + b \log {\left (c \right )}\right )^{2} \left (d \log {\left (x \right )} + \frac {e x^{r}}{r}\right ) & \text {for}\: n = 0 \\\frac {a^{2} d \log {\left (c x^{n} \right )}}{n} + \frac {a^{2} e x^{r}}{r} + \frac {a b d \log {\left (c x^{n} \right )}^{2}}{n} - \frac {2 a b e n x^{r}}{r^{2}} + \frac {2 a b e x^{r} \log {\left (c x^{n} \right )}}{r} + \frac {b^{2} d \log {\left (c x^{n} \right )}^{3}}{3 n} + \frac {2 b^{2} e n^{2} x^{r}}{r^{3}} - \frac {2 b^{2} e n x^{r} \log {\left (c x^{n} \right )}}{r^{2}} + \frac {b^{2} e x^{r} \log {\left (c x^{n} \right )}^{2}}{r} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (81) = 162\).
time = 6.29, size = 219, normalized size = 2.74 \begin {gather*} \frac {1}{3} \, b^{2} d n^{2} \log \left (x\right )^{3} + \frac {b^{2} n^{2} x^{r} e \log \left (x\right )^{2}}{r} + b^{2} d n \log \left (c\right ) \log \left (x\right )^{2} + \frac {2 \, b^{2} n x^{r} e \log \left (c\right ) \log \left (x\right )}{r} + b^{2} d \log \left (c\right )^{2} \log \left (x\right ) + a b d n \log \left (x\right )^{2} + \frac {b^{2} x^{r} e \log \left (c\right )^{2}}{r} - \frac {2 \, b^{2} n^{2} x^{r} e \log \left (x\right )}{r^{2}} + \frac {2 \, a b n x^{r} e \log \left (x\right )}{r} + 2 \, a b d \log \left (c\right ) \log \left (x\right ) - \frac {2 \, b^{2} n x^{r} e \log \left (c\right )}{r^{2}} + \frac {2 \, a b x^{r} e \log \left (c\right )}{r} + a^{2} d \log \left (x\right ) + \frac {2 \, b^{2} n^{2} x^{r} e}{r^{3}} - \frac {2 \, a b n x^{r} e}{r^{2}} + \frac {a^{2} x^{r} e}{r} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (d+e\,x^r\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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