3.494 \(\int \frac{a+b \log (c (d+\frac{e}{\sqrt [3]{x}})^n)}{x^2} \, dx\)

Optimal. Leaf size=82 \[ -\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{x}+\frac{b d^2 n}{e^2 \sqrt [3]{x}}-\frac{b d^3 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}-\frac{b d n}{2 e x^{2/3}}+\frac{b n}{3 x} \]

[Out]

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

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Rubi [A]  time = 0.0682031, antiderivative size = 82, 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} \[ -\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{x}+\frac{b d^2 n}{e^2 \sqrt [3]{x}}-\frac{b d^3 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}-\frac{b d n}{2 e x^{2/3}}+\frac{b n}{3 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e/x^(1/3))^n])/x^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0325174, size = 85, normalized size = 1.04 \[ -\frac{a}{x}-\frac{b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{x}+\frac{b d^2 n}{e^2 \sqrt [3]{x}}-\frac{b d^3 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{e^3}-\frac{b d n}{2 e x^{2/3}}+\frac{b n}{3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/x^(1/3))^n])/x^2,x]

[Out]

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

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Maple [F]  time = 0.341, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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^2,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/3))^n))/x^2,x)

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Maxima [A]  time = 1.025, size = 116, normalized size = 1.41 \begin{align*} -\frac{1}{6} \, b e n{\left (\frac{6 \, d^{3} \log \left (d x^{\frac{1}{3}} + e\right )}{e^{4}} - \frac{2 \, d^{3} \log \left (x\right )}{e^{4}} - \frac{6 \, d^{2} x^{\frac{2}{3}} - 3 \, d e x^{\frac{1}{3}} + 2 \, e^{2}}{e^{3} x}\right )} - \frac{b \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right )}{x} - \frac{a}{x} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.76232, size = 250, normalized size = 3.05 \begin{align*} \frac{6 \, b d^{2} e n x^{\frac{2}{3}} - 3 \, b d e^{2} n x^{\frac{1}{3}} + 2 \, b e^{3} n - 6 \, a e^{3} - 2 \,{\left (b e^{3} n - 3 \, a e^{3}\right )} x + 6 \,{\left (b e^{3} x - b e^{3}\right )} \log \left (c\right ) - 6 \,{\left (b d^{3} n x + b e^{3} n\right )} \log \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right )}{6 \, e^{3} x} \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^2,x, algorithm="fricas")

[Out]

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

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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**2,x)

[Out]

Timed out

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Giac [A]  time = 1.34764, size = 128, normalized size = 1.56 \begin{align*} -\frac{1}{6} \,{\left ({\left (6 \, d^{3} e^{\left (-4\right )} \log \left ({\left | d x^{\frac{1}{3}} + e \right |}\right ) - 2 \, d^{3} e^{\left (-4\right )} \log \left ({\left | x \right |}\right ) - \frac{{\left (6 \, d^{2} x^{\frac{2}{3}} e - 3 \, d x^{\frac{1}{3}} e^{2} + 2 \, e^{3}\right )} e^{\left (-4\right )}}{x}\right )} e + \frac{6 \, \log \left (d + \frac{e}{x^{\frac{1}{3}}}\right )}{x}\right )} b n - \frac{b \log \left (c\right )}{x} - \frac{a}{x} \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^2,x, algorithm="giac")

[Out]

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