3.27 \(\int x^3 \log (c (a+\frac{b}{x})^p) \, dx\)

Optimal. Leaf size=75 \[ -\frac{b^2 p x^2}{8 a^2}+\frac{b^3 p x}{4 a^3}-\frac{b^4 p \log (a x+b)}{4 a^4}+\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x^3}{12 a} \]

[Out]

(b^3*p*x)/(4*a^3) - (b^2*p*x^2)/(8*a^2) + (b*p*x^3)/(12*a) + (x^4*Log[c*(a + b/x)^p])/4 - (b^4*p*Log[b + a*x])
/(4*a^4)

________________________________________________________________________________________

Rubi [A]  time = 0.040069, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2455, 263, 43} \[ -\frac{b^2 p x^2}{8 a^2}+\frac{b^3 p x}{4 a^3}-\frac{b^4 p \log (a x+b)}{4 a^4}+\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x^3}{12 a} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Log[c*(a + b/x)^p],x]

[Out]

(b^3*p*x)/(4*a^3) - (b^2*p*x^2)/(8*a^2) + (b*p*x^3)/(12*a) + (x^4*Log[c*(a + b/x)^p])/4 - (b^4*p*Log[b + a*x])
/(4*a^4)

Rule 2455

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx &=\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{4} (b p) \int \frac{x^2}{a+\frac{b}{x}} \, dx\\ &=\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{4} (b p) \int \frac{x^3}{b+a x} \, dx\\ &=\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{4} (b p) \int \left (\frac{b^2}{a^3}-\frac{b x}{a^2}+\frac{x^2}{a}-\frac{b^3}{a^3 (b+a x)}\right ) \, dx\\ &=\frac{b^3 p x}{4 a^3}-\frac{b^2 p x^2}{8 a^2}+\frac{b p x^3}{12 a}+\frac{1}{4} x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )-\frac{b^4 p \log (b+a x)}{4 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0319432, size = 74, normalized size = 0.99 \[ \frac{a b p x \left (2 a^2 x^2-3 a b x+6 b^2\right )+6 a^4 x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right )-6 b^4 p \log \left (a+\frac{b}{x}\right )-6 b^4 p \log (x)}{24 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Log[c*(a + b/x)^p],x]

[Out]

(a*b*p*x*(6*b^2 - 3*a*b*x + 2*a^2*x^2) - 6*b^4*p*Log[a + b/x] + 6*a^4*x^4*Log[c*(a + b/x)^p] - 6*b^4*p*Log[x])
/(24*a^4)

________________________________________________________________________________________

Maple [F]  time = 0.236, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*ln(c*(a+b/x)^p),x)

[Out]

int(x^3*ln(c*(a+b/x)^p),x)

________________________________________________________________________________________

Maxima [A]  time = 1.20977, size = 86, normalized size = 1.15 \begin{align*} \frac{1}{4} \, x^{4} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) - \frac{1}{24} \, b p{\left (\frac{6 \, b^{3} \log \left (a x + b\right )}{a^{4}} - \frac{2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(c*(a+b/x)^p),x, algorithm="maxima")

[Out]

1/4*x^4*log((a + b/x)^p*c) - 1/24*b*p*(6*b^3*log(a*x + b)/a^4 - (2*a^2*x^3 - 3*a*b*x^2 + 6*b^2*x)/a^3)

________________________________________________________________________________________

Fricas [A]  time = 2.28267, size = 177, normalized size = 2.36 \begin{align*} \frac{6 \, a^{4} p x^{4} \log \left (\frac{a x + b}{x}\right ) + 6 \, a^{4} x^{4} \log \left (c\right ) + 2 \, a^{3} b p x^{3} - 3 \, a^{2} b^{2} p x^{2} + 6 \, a b^{3} p x - 6 \, b^{4} p \log \left (a x + b\right )}{24 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(c*(a+b/x)^p),x, algorithm="fricas")

[Out]

1/24*(6*a^4*p*x^4*log((a*x + b)/x) + 6*a^4*x^4*log(c) + 2*a^3*b*p*x^3 - 3*a^2*b^2*p*x^2 + 6*a*b^3*p*x - 6*b^4*
p*log(a*x + b))/a^4

________________________________________________________________________________________

Sympy [A]  time = 16.4616, size = 109, normalized size = 1.45 \begin{align*} \begin{cases} \frac{p x^{4} \log{\left (a + \frac{b}{x} \right )}}{4} + \frac{x^{4} \log{\left (c \right )}}{4} + \frac{b p x^{3}}{12 a} - \frac{b^{2} p x^{2}}{8 a^{2}} + \frac{b^{3} p x}{4 a^{3}} - \frac{b^{4} p \log{\left (a x + b \right )}}{4 a^{4}} & \text{for}\: a \neq 0 \\\frac{p x^{4} \log{\left (b \right )}}{4} - \frac{p x^{4} \log{\left (x \right )}}{4} + \frac{p x^{4}}{16} + \frac{x^{4} \log{\left (c \right )}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*ln(c*(a+b/x)**p),x)

[Out]

Piecewise((p*x**4*log(a + b/x)/4 + x**4*log(c)/4 + b*p*x**3/(12*a) - b**2*p*x**2/(8*a**2) + b**3*p*x/(4*a**3)
- b**4*p*log(a*x + b)/(4*a**4), Ne(a, 0)), (p*x**4*log(b)/4 - p*x**4*log(x)/4 + p*x**4/16 + x**4*log(c)/4, Tru
e))

________________________________________________________________________________________

Giac [A]  time = 1.29949, size = 101, normalized size = 1.35 \begin{align*} \frac{1}{4} \, p x^{4} \log \left (a x + b\right ) - \frac{1}{4} \, p x^{4} \log \left (x\right ) + \frac{1}{4} \, x^{4} \log \left (c\right ) + \frac{b p x^{3}}{12 \, a} - \frac{b^{2} p x^{2}}{8 \, a^{2}} + \frac{b^{3} p x}{4 \, a^{3}} - \frac{b^{4} p \log \left (a x + b\right )}{4 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(c*(a+b/x)^p),x, algorithm="giac")

[Out]

1/4*p*x^4*log(a*x + b) - 1/4*p*x^4*log(x) + 1/4*x^4*log(c) + 1/12*b*p*x^3/a - 1/8*b^2*p*x^2/a^2 + 1/4*b^3*p*x/
a^3 - 1/4*b^4*p*log(a*x + b)/a^4