3.510 \(\int x (a+b \log (c (d+\frac{e}{x^{2/3}})^n)) \, dx\)

Optimal. Leaf size=94 \[ \frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{b e^2 n x^{2/3}}{2 d^2}+\frac{b e^3 n \log \left (d+\frac{e}{x^{2/3}}\right )}{2 d^3}+\frac{b e^3 n \log (x)}{3 d^3}+\frac{b e n x^{4/3}}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.0624697, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2454, 2395, 44} \[ \frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{b e^2 n x^{2/3}}{2 d^2}+\frac{b e^3 n \log \left (d+\frac{e}{x^{2/3}}\right )}{2 d^3}+\frac{b e^3 n \log (x)}{3 d^3}+\frac{b e n x^{4/3}}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \, dx &=-\left (\frac{3}{2} \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^4} \, dx,x,\frac{1}{x^{2/3}}\right )\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \frac{1}{x^3 (d+e x)} \, dx,x,\frac{1}{x^{2/3}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \left (\frac{1}{d x^3}-\frac{e}{d^2 x^2}+\frac{e^2}{d^3 x}-\frac{e^3}{d^3 (d+e x)}\right ) \, dx,x,\frac{1}{x^{2/3}}\right )\\ &=-\frac{b e^2 n x^{2/3}}{2 d^2}+\frac{b e n x^{4/3}}{4 d}+\frac{b e^3 n \log \left (d+\frac{e}{x^{2/3}}\right )}{2 d^3}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+\frac{b e^3 n \log (x)}{3 d^3}\\ \end{align*}

Mathematica [A]  time = 0.0229221, size = 91, normalized size = 0.97 \[ \frac{a x^2}{2}+\frac{1}{2} b x^2 \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-\frac{1}{2} b e n \left (-\frac{e^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{d^3}-\frac{2 e^2 \log (x)}{3 d^3}+\frac{e x^{2/3}}{d^2}-\frac{x^{4/3}}{2 d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.358, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04099, size = 85, normalized size = 0.9 \begin{align*} \frac{1}{4} \, b e n{\left (\frac{2 \, e^{2} \log \left (d x^{\frac{2}{3}} + e\right )}{d^{3}} + \frac{d x^{\frac{4}{3}} - 2 \, e x^{\frac{2}{3}}}{d^{2}}\right )} + \frac{1}{2} \, b x^{2} \log \left (c{\left (d + \frac{e}{x^{\frac{2}{3}}}\right )}^{n}\right ) + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*(d+e/x^(2/3))^n)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.8801, size = 277, normalized size = 2.95 \begin{align*} \frac{2 \, b d^{3} x^{2} \log \left (c\right ) + b d^{2} e n x^{\frac{4}{3}} + 2 \, a d^{3} x^{2} - 4 \, b d^{3} n \log \left (x^{\frac{1}{3}}\right ) - 2 \, b d e^{2} n x^{\frac{2}{3}} + 2 \,{\left (b d^{3} + b e^{3}\right )} n \log \left (d x^{\frac{2}{3}} + e\right ) + 2 \,{\left (b d^{3} n x^{2} - b d^{3} n\right )} \log \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )}{4 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*(d+e/x^(2/3))^n)),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [A]  time = 1.26951, size = 97, normalized size = 1.03 \begin{align*} \frac{1}{2} \, b x^{2} \log \left (c\right ) + \frac{1}{4} \,{\left (2 \, x^{2} \log \left (d + \frac{e}{x^{\frac{2}{3}}}\right ) +{\left (\frac{d x^{\frac{4}{3}} - 2 \, x^{\frac{2}{3}} e}{d^{2}} + \frac{2 \, e^{2} \log \left ({\left | d x^{\frac{2}{3}} + e \right |}\right )}{d^{3}}\right )} e\right )} b n + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*(d+e/x^(2/3))^n)),x, algorithm="giac")

[Out]

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