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

Optimal. Leaf size=70 \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-\frac{b e^2 n \sqrt [3]{x}}{d^2}+\frac{b e^3 n \log \left (d \sqrt [3]{x}+e\right )}{d^3}+\frac{b e n x^{2/3}}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0501666, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2448, 263, 190, 43} \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-\frac{b e^2 n \sqrt [3]{x}}{d^2}+\frac{b e^3 n \log \left (d \sqrt [3]{x}+e\right )}{d^3}+\frac{b e n x^{2/3}}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Log[c*(d + e/x^(1/3))^n],x]

[Out]

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

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

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 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/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 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right ) \, dx\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+\frac{1}{3} (b e n) \int \frac{1}{\left (d+\frac{e}{\sqrt [3]{x}}\right ) \sqrt [3]{x}} \, dx\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+\frac{1}{3} (b e n) \int \frac{1}{e+d \sqrt [3]{x}} \, dx\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+(b e n) \operatorname{Subst}\left (\int \frac{x^2}{e+d x} \, dx,x,\sqrt [3]{x}\right )\\ &=a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+(b e n) \operatorname{Subst}\left (\int \left (-\frac{e}{d^2}+\frac{x}{d}+\frac{e^2}{d^2 (e+d x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{b e^2 n \sqrt [3]{x}}{d^2}+\frac{b e n x^{2/3}}{2 d}+a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+\frac{b e^3 n \log \left (e+d \sqrt [3]{x}\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.0470079, size = 79, normalized size = 1.13 \[ a x+b x \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-b e n \left (-\frac{e^2 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{d^3}-\frac{e^2 \log (x)}{3 d^3}+\frac{e \sqrt [3]{x}}{d^2}-\frac{x^{2/3}}{2 d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.102, size = 115, normalized size = 1.6 \begin{align*} ax+xb\ln \left ( c \left ({ \left ( e+d\sqrt [3]{x} \right ){\frac{1}{\sqrt [3]{x}}}} \right ) ^{n} \right ) +{\frac{b{e}^{3}n\ln \left ({d}^{3}x+{e}^{3} \right ) }{3\,{d}^{3}}}+{\frac{enb}{2\,d}{x}^{{\frac{2}{3}}}}-{\frac{b{e}^{3}n}{3\,{d}^{3}}\ln \left ({d}^{2}{x}^{{\frac{2}{3}}}-ed\sqrt [3]{x}+{e}^{2} \right ) }+{\frac{2\,b{e}^{3}n}{3\,{d}^{3}}\ln \left ( e+d\sqrt [3]{x} \right ) }-{\frac{b{e}^{2}n}{{d}^{2}}\sqrt [3]{x}} \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)

[Out]

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

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Maxima [A]  time = 1.03024, size = 80, normalized size = 1.14 \begin{align*} \frac{1}{2} \,{\left (e n{\left (\frac{2 \, e^{2} \log \left (d x^{\frac{1}{3}} + e\right )}{d^{3}} + \frac{d x^{\frac{2}{3}} - 2 \, e x^{\frac{1}{3}}}{d^{2}}\right )} + 2 \, x \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right )\right )} b + 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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.15552, size = 269, normalized size = 3.84 \begin{align*} \frac{2 \, b d^{3} x \log \left (c\right ) - 2 \, b d^{3} n \log \left (x^{\frac{1}{3}}\right ) + b d^{2} e n x^{\frac{2}{3}} - 2 \, b d e^{2} n x^{\frac{1}{3}} + 2 \, a d^{3} x + 2 \,{\left (b d^{3} + b e^{3}\right )} n \log \left (d x^{\frac{1}{3}} + e\right ) + 2 \,{\left (b d^{3} n x - b d^{3} n\right )} \log \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right )}{2 \, d^{3}} \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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 15.0529, size = 92, normalized size = 1.31 \begin{align*} a x + b \left (\frac{e n \left (\frac{3 x^{\frac{2}{3}}}{2 d} - \frac{3 e \sqrt [3]{x}}{d^{2}} + \frac{3 e^{3} \left (\begin{cases} \frac{1}{d \sqrt [3]{x}} & \text{for}\: e = 0 \\\frac{\log{\left (d + \frac{e}{\sqrt [3]{x}} \right )}}{e} & \text{otherwise} \end{cases}\right )}{d^{3}} - \frac{3 e^{2} \log{\left (\frac{1}{\sqrt [3]{x}} \right )}}{d^{3}}\right )}{3} + x \log{\left (c \left (d + \frac{e}{\sqrt [3]{x}}\right )^{n} \right )}\right ) \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)

[Out]

a*x + b*(e*n*(3*x**(2/3)/(2*d) - 3*e*x**(1/3)/d**2 + 3*e**3*Piecewise((1/(d*x**(1/3)), Eq(e, 0)), (log(d + e/x
**(1/3))/e, True))/d**3 - 3*e**2*log(x**(-1/3))/d**3)/3 + x*log(c*(d + e/x**(1/3))**n))

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Giac [A]  time = 1.38545, size = 89, normalized size = 1.27 \begin{align*} \frac{1}{2} \,{\left ({\left ({\left (\frac{d x^{\frac{2}{3}} - 2 \, x^{\frac{1}{3}} e}{d^{2}} + \frac{2 \, e^{2} \log \left ({\left | d x^{\frac{1}{3}} + e \right |}\right )}{d^{3}}\right )} e + 2 \, x \log \left (d + \frac{e}{x^{\frac{1}{3}}}\right )\right )} n + 2 \, x \log \left (c\right )\right )} b + 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, algorithm="giac")

[Out]

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