3.495 \(\int \frac{a+b \log (c (d+\frac{e}{\sqrt [3]{x}})^n)}{x^3} \, dx\)

Optimal. Leaf size=138 \[ -\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{2 x^2}+\frac{b d^4 n}{4 e^4 x^{2/3}}+\frac{b d^2 n}{8 e^2 x^{4/3}}-\frac{b d^5 n}{2 e^5 \sqrt [3]{x}}-\frac{b d^3 n}{6 e^3 x}+\frac{b d^6 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{2 e^6}-\frac{b d n}{10 e x^{5/3}}+\frac{b n}{12 x^2} \]

[Out]

(b*n)/(12*x^2) - (b*d*n)/(10*e*x^(5/3)) + (b*d^2*n)/(8*e^2*x^(4/3)) - (b*d^3*n)/(6*e^3*x) + (b*d^4*n)/(4*e^4*x
^(2/3)) - (b*d^5*n)/(2*e^5*x^(1/3)) + (b*d^6*n*Log[d + e/x^(1/3)])/(2*e^6) - (a + b*Log[c*(d + e/x^(1/3))^n])/
(2*x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.097263, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2454, 2395, 43} \[ -\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{2 x^2}+\frac{b d^4 n}{4 e^4 x^{2/3}}+\frac{b d^2 n}{8 e^2 x^{4/3}}-\frac{b d^5 n}{2 e^5 \sqrt [3]{x}}-\frac{b d^3 n}{6 e^3 x}+\frac{b d^6 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{2 e^6}-\frac{b d n}{10 e x^{5/3}}+\frac{b n}{12 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e/x^(1/3))^n])/x^3,x]

[Out]

(b*n)/(12*x^2) - (b*d*n)/(10*e*x^(5/3)) + (b*d^2*n)/(8*e^2*x^(4/3)) - (b*d^3*n)/(6*e^3*x) + (b*d^4*n)/(4*e^4*x
^(2/3)) - (b*d^5*n)/(2*e^5*x^(1/3)) + (b*d^6*n*Log[d + e/x^(1/3)])/(2*e^6) - (a + b*Log[c*(d + e/x^(1/3))^n])/
(2*x^2)

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

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 \frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{x^3} \, dx &=-\left (3 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{2 x^2}+\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \frac{x^6}{d+e x} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\\ &=-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{2 x^2}+\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \left (-\frac{d^5}{e^6}+\frac{d^4 x}{e^5}-\frac{d^3 x^2}{e^4}+\frac{d^2 x^3}{e^3}-\frac{d x^4}{e^2}+\frac{x^5}{e}+\frac{d^6}{e^6 (d+e x)}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\\ &=\frac{b n}{12 x^2}-\frac{b d n}{10 e x^{5/3}}+\frac{b d^2 n}{8 e^2 x^{4/3}}-\frac{b d^3 n}{6 e^3 x}+\frac{b d^4 n}{4 e^4 x^{2/3}}-\frac{b d^5 n}{2 e^5 \sqrt [3]{x}}+\frac{b d^6 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{2 e^6}-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0894655, size = 135, normalized size = 0.98 \[ -\frac{a}{2 x^2}-\frac{b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{2 x^2}+\frac{1}{2} b e n \left (\frac{d^4}{2 e^5 x^{2/3}}+\frac{d^2}{4 e^3 x^{4/3}}-\frac{d^5}{e^6 \sqrt [3]{x}}-\frac{d^3}{3 e^4 x}+\frac{d^6 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{e^7}-\frac{d}{5 e^2 x^{5/3}}+\frac{1}{6 e x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/x^(1/3))^n])/x^3,x]

[Out]

-a/(2*x^2) + (b*e*n*(1/(6*e*x^2) - d/(5*e^2*x^(5/3)) + d^2/(4*e^3*x^(4/3)) - d^3/(3*e^4*x) + d^4/(2*e^5*x^(2/3
)) - d^5/(e^6*x^(1/3)) + (d^6*Log[d + e/x^(1/3)])/e^7))/2 - (b*Log[c*(d + e/x^(1/3))^n])/(2*x^2)

________________________________________________________________________________________

Maple [F]  time = 0.329, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt [3]{x}}}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e/x^(1/3))^n))/x^3,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/3))^n))/x^3,x)

________________________________________________________________________________________

Maxima [A]  time = 1.03237, size = 158, normalized size = 1.14 \begin{align*} \frac{1}{120} \, b e n{\left (\frac{60 \, d^{6} \log \left (d x^{\frac{1}{3}} + e\right )}{e^{7}} - \frac{20 \, d^{6} \log \left (x\right )}{e^{7}} - \frac{60 \, d^{5} x^{\frac{5}{3}} - 30 \, d^{4} e x^{\frac{4}{3}} + 20 \, d^{3} e^{2} x - 15 \, d^{2} e^{3} x^{\frac{2}{3}} + 12 \, d e^{4} x^{\frac{1}{3}} - 10 \, e^{5}}{e^{6} x^{2}}\right )} - \frac{b \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right )}{2 \, x^{2}} - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^n))/x^3,x, algorithm="maxima")

[Out]

1/120*b*e*n*(60*d^6*log(d*x^(1/3) + e)/e^7 - 20*d^6*log(x)/e^7 - (60*d^5*x^(5/3) - 30*d^4*e*x^(4/3) + 20*d^3*e
^2*x - 15*d^2*e^3*x^(2/3) + 12*d*e^4*x^(1/3) - 10*e^5)/(e^6*x^2)) - 1/2*b*log(c*(d + e/x^(1/3))^n)/x^2 - 1/2*a
/x^2

________________________________________________________________________________________

Fricas [A]  time = 1.81379, size = 377, normalized size = 2.73 \begin{align*} -\frac{20 \, b d^{3} e^{3} n x - 10 \, b e^{6} n + 60 \, a e^{6} - 10 \,{\left (6 \, a e^{6} +{\left (2 \, b d^{3} e^{3} - b e^{6}\right )} n\right )} x^{2} - 60 \,{\left (b e^{6} x^{2} - b e^{6}\right )} \log \left (c\right ) - 60 \,{\left (b d^{6} n x^{2} - b e^{6} n\right )} \log \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right ) + 15 \,{\left (4 \, b d^{5} e n x - b d^{2} e^{4} n\right )} x^{\frac{2}{3}} - 6 \,{\left (5 \, b d^{4} e^{2} n x - 2 \, b d e^{5} n\right )} x^{\frac{1}{3}}}{120 \, e^{6} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^n))/x^3,x, algorithm="fricas")

[Out]

-1/120*(20*b*d^3*e^3*n*x - 10*b*e^6*n + 60*a*e^6 - 10*(6*a*e^6 + (2*b*d^3*e^3 - b*e^6)*n)*x^2 - 60*(b*e^6*x^2
- b*e^6)*log(c) - 60*(b*d^6*n*x^2 - b*e^6*n)*log((d*x + e*x^(2/3))/x) + 15*(4*b*d^5*e*n*x - b*d^2*e^4*n)*x^(2/
3) - 6*(5*b*d^4*e^2*n*x - 2*b*d*e^5*n)*x^(1/3))/(e^6*x^2)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d+e/x**(1/3))**n))/x**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.33372, size = 166, normalized size = 1.2 \begin{align*} \frac{1}{120} \,{\left ({\left (60 \, d^{6} e^{\left (-7\right )} \log \left ({\left | d x^{\frac{1}{3}} + e \right |}\right ) - 20 \, d^{6} e^{\left (-7\right )} \log \left ({\left | x \right |}\right ) - \frac{{\left (60 \, d^{5} x^{\frac{5}{3}} e - 30 \, d^{4} x^{\frac{4}{3}} e^{2} + 20 \, d^{3} x e^{3} - 15 \, d^{2} x^{\frac{2}{3}} e^{4} + 12 \, d x^{\frac{1}{3}} e^{5} - 10 \, e^{6}\right )} e^{\left (-7\right )}}{x^{2}}\right )} e - \frac{60 \, \log \left (d + \frac{e}{x^{\frac{1}{3}}}\right )}{x^{2}}\right )} b n - \frac{b \log \left (c\right )}{2 \, x^{2}} - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^n))/x^3,x, algorithm="giac")

[Out]

1/120*((60*d^6*e^(-7)*log(abs(d*x^(1/3) + e)) - 20*d^6*e^(-7)*log(abs(x)) - (60*d^5*x^(5/3)*e - 30*d^4*x^(4/3)
*e^2 + 20*d^3*x*e^3 - 15*d^2*x^(2/3)*e^4 + 12*d*x^(1/3)*e^5 - 10*e^6)*e^(-7)/x^2)*e - 60*log(d + e/x^(1/3))/x^
2)*b*n - 1/2*b*log(c)/x^2 - 1/2*a/x^2