3.150 \(\int (d x)^m (a+b \log (c x^n))^3 \, dx\)

Optimal. Leaf size=116 \[ \frac{6 b^2 n^2 (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^3}+\frac{(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^3}{d (m+1)}-\frac{3 b n (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)^2}-\frac{6 b^3 n^3 (d x)^{m+1}}{d (m+1)^4} \]

[Out]

(-6*b^3*n^3*(d*x)^(1 + m))/(d*(1 + m)^4) + (6*b^2*n^2*(d*x)^(1 + m)*(a + b*Log[c*x^n]))/(d*(1 + m)^3) - (3*b*n
*(d*x)^(1 + m)*(a + b*Log[c*x^n])^2)/(d*(1 + m)^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])^3)/(d*(1 + m))

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Rubi [A]  time = 0.0905633, antiderivative size = 116, 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 = {2305, 2304} \[ \frac{6 b^2 n^2 (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{d (m+1)^3}+\frac{(d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^3}{d (m+1)}-\frac{3 b n (d x)^{m+1} \left (a+b \log \left (c x^n\right )\right )^2}{d (m+1)^2}-\frac{6 b^3 n^3 (d x)^{m+1}}{d (m+1)^4} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*Log[c*x^n])^3,x]

[Out]

(-6*b^3*n^3*(d*x)^(1 + m))/(d*(1 + m)^4) + (6*b^2*n^2*(d*x)^(1 + m)*(a + b*Log[c*x^n]))/(d*(1 + m)^3) - (3*b*n
*(d*x)^(1 + m)*(a + b*Log[c*x^n])^2)/(d*(1 + m)^2) + ((d*x)^(1 + m)*(a + b*Log[c*x^n])^3)/(d*(1 + m))

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

Mathematica [A]  time = 0.0453227, size = 76, normalized size = 0.66 \[ \frac{x (d x)^m \left (\left (a+b \log \left (c x^n\right )\right )^3-\frac{3 b n \left ((m+1)^2 \left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (b n-(m+1) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{(m+1)^3}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*Log[c*x^n])^3,x]

[Out]

(x*(d*x)^m*((a + b*Log[c*x^n])^3 - (3*b*n*((1 + m)^2*(a + b*Log[c*x^n])^2 + 2*b*n*(b*n - (1 + m)*(a + b*Log[c*
x^n]))))/(1 + m)^3))/(1 + m)

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Maple [C]  time = 0.698, size = 9684, normalized size = 83.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+b*ln(c*x^n))^3,x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.05821, size = 1242, normalized size = 10.71 \begin{align*} \frac{{\left ({\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} n^{3} x \log \left (x\right )^{3} +{\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} x \log \left (c\right )^{3} + 3 \,{\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2} -{\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n\right )} x \log \left (c\right )^{2} + 3 \,{\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b + 2 \,{\left (b^{3} m + b^{3}\right )} n^{2} - 2 \,{\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n\right )} x \log \left (c\right ) + 3 \,{\left ({\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} n^{2} x \log \left (c\right ) -{\left ({\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n^{3} -{\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2}\right )} n^{2}\right )} x\right )} \log \left (x\right )^{2} +{\left (a^{3} m^{3} - 6 \, b^{3} n^{3} + 3 \, a^{3} m^{2} + 3 \, a^{3} m + a^{3} + 6 \,{\left (a b^{2} m + a b^{2}\right )} n^{2} - 3 \,{\left (a^{2} b m^{2} + 2 \, a^{2} b m + a^{2} b\right )} n\right )} x + 3 \,{\left ({\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3}\right )} n x \log \left (c\right )^{2} - 2 \,{\left ({\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n^{2} -{\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2}\right )} n\right )} x \log \left (c\right ) +{\left (2 \,{\left (b^{3} m + b^{3}\right )} n^{3} - 2 \,{\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n^{2} +{\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b\right )} n\right )} x\right )} \log \left (x\right )\right )} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{4} + 4 \, m^{3} + 6 \, m^{2} + 4 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*log(c*x^n))^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+b*ln(c*x**n))**3,x)

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.93021, size = 1530, normalized size = 13.19 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*log(c*x^n))^3,x, algorithm="giac")

[Out]

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