3.1.27 \(\int x^3 \log (c (a+\frac {b}{x})^p) \, dx\) [27]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 = {2505, 269, 45} \begin {gather*} -\frac {b^4 p \log (a x+b)}{4 a^4}+\frac {b^3 p x}{4 a^3}-\frac {b^2 p x^2}{8 a^2}+\frac {1}{4} x^4 \log \left (c \left (a+\frac {b}{x}\right )^p\right )+\frac {b p x^3}{12 a} \end {gather*}

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 45

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])

Rule 269

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 2505

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]

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 = 0.02, size = 74, normalized size = 0.99 \begin {gather*} \frac {a b p x \left (6 b^2-3 a b x+2 a^2 x^2\right )-6 b^4 p \log \left (a+\frac {b}{x}\right )+6 a^4 x^4 \log \left (c \left (a+\frac {b}{x}\right )^p\right )-6 b^4 p \log (x)}{24 a^4} \end {gather*}

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 = 0.04, size = 0, normalized size = 0.00 \[\int x^{3} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )\, dx\]

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 = 0.28, size = 64, normalized size = 0.85 \begin {gather*} \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 {gather*}

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 = 0.37, size = 77, normalized size = 1.03 \begin {gather*} \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 {gather*}

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 = 1.35, size = 87, normalized size = 1.16 \begin {gather*} \begin {cases} \frac {x^{4} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \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}}{16} + \frac {x^{4} \log {\left (c \left (\frac {b}{x}\right )^{p} \right )}}{4} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**4*log(c*(a + b/x)**p)/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*lo
g(a*x + b)/(4*a**4), Ne(a, 0)), (p*x**4/16 + x**4*log(c*(b/x)**p)/4, True))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (65) = 130\).
time = 3.93, size = 257, normalized size = 3.43 \begin {gather*} \frac {\frac {6 \, b^{5} p \log \left (\frac {a x + b}{x}\right )}{a^{4} - \frac {4 \, {\left (a x + b\right )} a^{3}}{x} + \frac {6 \, {\left (a x + b\right )}^{2} a^{2}}{x^{2}} - \frac {4 \, {\left (a x + b\right )}^{3} a}{x^{3}} + \frac {{\left (a x + b\right )}^{4}}{x^{4}}} + \frac {6 \, b^{5} p \log \left (-a + \frac {a x + b}{x}\right )}{a^{4}} - \frac {6 \, b^{5} p \log \left (\frac {a x + b}{x}\right )}{a^{4}} - \frac {11 \, a^{3} b^{5} p - 6 \, a^{3} b^{5} \log \left (c\right ) - \frac {26 \, {\left (a x + b\right )} a^{2} b^{5} p}{x} + \frac {21 \, {\left (a x + b\right )}^{2} a b^{5} p}{x^{2}} - \frac {6 \, {\left (a x + b\right )}^{3} b^{5} p}{x^{3}}}{a^{7} - \frac {4 \, {\left (a x + b\right )} a^{6}}{x} + \frac {6 \, {\left (a x + b\right )}^{2} a^{5}}{x^{2}} - \frac {4 \, {\left (a x + b\right )}^{3} a^{4}}{x^{3}} + \frac {{\left (a x + b\right )}^{4} a^{3}}{x^{4}}}}{24 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(6*b^5*p*log((a*x + b)/x)/(a^4 - 4*(a*x + b)*a^3/x + 6*(a*x + b)^2*a^2/x^2 - 4*(a*x + b)^3*a/x^3 + (a*x +
b)^4/x^4) + 6*b^5*p*log(-a + (a*x + b)/x)/a^4 - 6*b^5*p*log((a*x + b)/x)/a^4 - (11*a^3*b^5*p - 6*a^3*b^5*log(
c) - 26*(a*x + b)*a^2*b^5*p/x + 21*(a*x + b)^2*a*b^5*p/x^2 - 6*(a*x + b)^3*b^5*p/x^3)/(a^7 - 4*(a*x + b)*a^6/x
+ 6*(a*x + b)^2*a^5/x^2 - 4*(a*x + b)^3*a^4/x^3 + (a*x + b)^4*a^3/x^4))/b

________________________________________________________________________________________

Mupad [B]
time = 0.21, size = 65, normalized size = 0.87 \begin {gather*} \frac {x^4\,\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{4}-\frac {b^2\,p\,x^2}{8\,a^2}-\frac {b^4\,p\,\ln \left (b+a\,x\right )}{4\,a^4}+\frac {b\,p\,x^3}{12\,a}+\frac {b^3\,p\,x}{4\,a^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*log(c*(a + b/x)^p),x)

[Out]

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

________________________________________________________________________________________