3.26 \(\int x^4 \log (c (a+\frac{b}{x})^p) \, dx\)

Optimal. Leaf size=89 \[ \frac{b^3 p x^2}{10 a^3}-\frac{b^2 p x^3}{15 a^2}-\frac{b^4 p x}{5 a^4}+\frac{b^5 p \log (a x+b)}{5 a^5}+\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x^4}{20 a} \]

[Out]

-(b^4*p*x)/(5*a^4) + (b^3*p*x^2)/(10*a^3) - (b^2*p*x^3)/(15*a^2) + (b*p*x^4)/(20*a) + (x^5*Log[c*(a + b/x)^p])
/5 + (b^5*p*Log[b + a*x])/(5*a^5)

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Rubi [A]  time = 0.0549788, antiderivative size = 89, 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} \[ \frac{b^3 p x^2}{10 a^3}-\frac{b^2 p x^3}{15 a^2}-\frac{b^4 p x}{5 a^4}+\frac{b^5 p \log (a x+b)}{5 a^5}+\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x^4}{20 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4*Log[c*(a + b/x)^p],x]

[Out]

-(b^4*p*x)/(5*a^4) + (b^3*p*x^2)/(10*a^3) - (b^2*p*x^3)/(15*a^2) + (b*p*x^4)/(20*a) + (x^5*Log[c*(a + b/x)^p])
/5 + (b^5*p*Log[b + a*x])/(5*a^5)

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^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx &=\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{5} (b p) \int \frac{x^3}{a+\frac{b}{x}} \, dx\\ &=\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{5} (b p) \int \frac{x^4}{b+a x} \, dx\\ &=\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{5} (b p) \int \left (-\frac{b^3}{a^4}+\frac{b^2 x}{a^3}-\frac{b x^2}{a^2}+\frac{x^3}{a}+\frac{b^4}{a^4 (b+a x)}\right ) \, dx\\ &=-\frac{b^4 p x}{5 a^4}+\frac{b^3 p x^2}{10 a^3}-\frac{b^2 p x^3}{15 a^2}+\frac{b p x^4}{20 a}+\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b^5 p \log (b+a x)}{5 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0457851, size = 85, normalized size = 0.96 \[ \frac{a b p x \left (-4 a^2 b x^2+3 a^3 x^3+6 a b^2 x-12 b^3\right )+12 a^5 x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+12 b^5 p \log \left (a+\frac{b}{x}\right )+12 b^5 p \log (x)}{60 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Log[c*(a + b/x)^p],x]

[Out]

(a*b*p*x*(-12*b^3 + 6*a*b^2*x - 4*a^2*b*x^2 + 3*a^3*x^3) + 12*b^5*p*Log[a + b/x] + 12*a^5*x^5*Log[c*(a + b/x)^
p] + 12*b^5*p*Log[x])/(60*a^5)

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Maple [F]  time = 0.358, size = 0, normalized size = 0. \begin{align*} \int{x}^{4}\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^4*ln(c*(a+b/x)^p),x)

[Out]

int(x^4*ln(c*(a+b/x)^p),x)

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Maxima [A]  time = 1.06121, size = 100, normalized size = 1.12 \begin{align*} \frac{1}{5} \, x^{5} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) + \frac{1}{60} \, b p{\left (\frac{12 \, b^{4} \log \left (a x + b\right )}{a^{5}} + \frac{3 \, a^{3} x^{4} - 4 \, a^{2} b x^{3} + 6 \, a b^{2} x^{2} - 12 \, b^{3} x}{a^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(c*(a+b/x)^p),x, algorithm="maxima")

[Out]

1/5*x^5*log((a + b/x)^p*c) + 1/60*b*p*(12*b^4*log(a*x + b)/a^5 + (3*a^3*x^4 - 4*a^2*b*x^3 + 6*a*b^2*x^2 - 12*b
^3*x)/a^4)

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Fricas [A]  time = 2.33376, size = 207, normalized size = 2.33 \begin{align*} \frac{12 \, a^{5} p x^{5} \log \left (\frac{a x + b}{x}\right ) + 12 \, a^{5} x^{5} \log \left (c\right ) + 3 \, a^{4} b p x^{4} - 4 \, a^{3} b^{2} p x^{3} + 6 \, a^{2} b^{3} p x^{2} - 12 \, a b^{4} p x + 12 \, b^{5} p \log \left (a x + b\right )}{60 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(c*(a+b/x)^p),x, algorithm="fricas")

[Out]

1/60*(12*a^5*p*x^5*log((a*x + b)/x) + 12*a^5*x^5*log(c) + 3*a^4*b*p*x^4 - 4*a^3*b^2*p*x^3 + 6*a^2*b^3*p*x^2 -
12*a*b^4*p*x + 12*b^5*p*log(a*x + b))/a^5

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Sympy [A]  time = 34.2231, size = 122, normalized size = 1.37 \begin{align*} \begin{cases} \frac{p x^{5} \log{\left (a + \frac{b}{x} \right )}}{5} + \frac{x^{5} \log{\left (c \right )}}{5} + \frac{b p x^{4}}{20 a} - \frac{b^{2} p x^{3}}{15 a^{2}} + \frac{b^{3} p x^{2}}{10 a^{3}} - \frac{b^{4} p x}{5 a^{4}} + \frac{b^{5} p \log{\left (a x + b \right )}}{5 a^{5}} & \text{for}\: a \neq 0 \\\frac{p x^{5} \log{\left (b \right )}}{5} - \frac{p x^{5} \log{\left (x \right )}}{5} + \frac{p x^{5}}{25} + \frac{x^{5} \log{\left (c \right )}}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*ln(c*(a+b/x)**p),x)

[Out]

Piecewise((p*x**5*log(a + b/x)/5 + x**5*log(c)/5 + b*p*x**4/(20*a) - b**2*p*x**3/(15*a**2) + b**3*p*x**2/(10*a
**3) - b**4*p*x/(5*a**4) + b**5*p*log(a*x + b)/(5*a**5), Ne(a, 0)), (p*x**5*log(b)/5 - p*x**5*log(x)/5 + p*x**
5/25 + x**5*log(c)/5, True))

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Giac [A]  time = 1.33013, size = 117, normalized size = 1.31 \begin{align*} \frac{1}{5} \, p x^{5} \log \left (a x + b\right ) - \frac{1}{5} \, p x^{5} \log \left (x\right ) + \frac{1}{5} \, x^{5} \log \left (c\right ) + \frac{b p x^{4}}{20 \, a} - \frac{b^{2} p x^{3}}{15 \, a^{2}} + \frac{b^{3} p x^{2}}{10 \, a^{3}} - \frac{b^{4} p x}{5 \, a^{4}} + \frac{b^{5} p \log \left (a x + b\right )}{5 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(c*(a+b/x)^p),x, algorithm="giac")

[Out]

1/5*p*x^5*log(a*x + b) - 1/5*p*x^5*log(x) + 1/5*x^5*log(c) + 1/20*b*p*x^4/a - 1/15*b^2*p*x^3/a^2 + 1/10*b^3*p*
x^2/a^3 - 1/5*b^4*p*x/a^4 + 1/5*b^5*p*log(a*x + b)/a^5