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

Optimal. Leaf size=143 \[ -\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}+\frac{b e^4 n}{4 d^4 x^{2/3}}+\frac{b e^2 n}{8 d^2 x^{4/3}}-\frac{b e^5 n}{2 d^5 \sqrt [3]{x}}-\frac{b e^3 n}{6 d^3 x}+\frac{b e^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 d^6}-\frac{b e^6 n \log (x)}{6 d^6}-\frac{b e n}{10 d x^{5/3}} \]

[Out]

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

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Rubi [A]  time = 0.0927983, antiderivative size = 143, 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, 44} \[ -\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}+\frac{b e^4 n}{4 d^4 x^{2/3}}+\frac{b e^2 n}{8 d^2 x^{4/3}}-\frac{b e^5 n}{2 d^5 \sqrt [3]{x}}-\frac{b e^3 n}{6 d^3 x}+\frac{b e^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 d^6}-\frac{b e^6 n \log (x)}{6 d^6}-\frac{b e n}{10 d x^{5/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}+\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \frac{1}{x^6 (d+e x)} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}+\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \left (\frac{1}{d x^6}-\frac{e}{d^2 x^5}+\frac{e^2}{d^3 x^4}-\frac{e^3}{d^4 x^3}+\frac{e^4}{d^5 x^2}-\frac{e^5}{d^6 x}+\frac{e^6}{d^6 (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{b e n}{10 d x^{5/3}}+\frac{b e^2 n}{8 d^2 x^{4/3}}-\frac{b e^3 n}{6 d^3 x}+\frac{b e^4 n}{4 d^4 x^{2/3}}-\frac{b e^5 n}{2 d^5 \sqrt [3]{x}}+\frac{b e^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 d^6}-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}-\frac{b e^6 n \log (x)}{6 d^6}\\ \end{align*}

Mathematica [A]  time = 0.138207, size = 134, normalized size = 0.94 \[ -\frac{a}{2 x^2}-\frac{b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}+\frac{1}{2} b e n \left (\frac{e^3}{2 d^4 x^{2/3}}-\frac{e^4}{d^5 \sqrt [3]{x}}-\frac{e^2}{3 d^3 x}+\frac{e^5 \log \left (d+e \sqrt [3]{x}\right )}{d^6}-\frac{e^5 \log (x)}{3 d^6}+\frac{e}{4 d^2 x^{4/3}}-\frac{1}{5 d x^{5/3}}\right ) \]

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

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Maple [F]  time = 0.101, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( a+b\ln \left ( c \left ( d+e\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)

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Maxima [A]  time = 1.03044, size = 143, normalized size = 1. \begin{align*} \frac{1}{120} \, b e n{\left (\frac{60 \, e^{5} \log \left (e x^{\frac{1}{3}} + d\right )}{d^{6}} - \frac{20 \, e^{5} \log \left (x\right )}{d^{6}} - \frac{60 \, e^{4} x^{\frac{4}{3}} - 30 \, d e^{3} x + 20 \, d^{2} e^{2} x^{\frac{2}{3}} - 15 \, d^{3} e x^{\frac{1}{3}} + 12 \, d^{4}}{d^{5} x^{\frac{5}{3}}}\right )} - \frac{b \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} 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="maxima")

[Out]

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

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Fricas [A]  time = 1.93037, size = 312, normalized size = 2.18 \begin{align*} -\frac{60 \, b e^{6} n x^{2} \log \left (x^{\frac{1}{3}}\right ) + 20 \, b d^{3} e^{3} n x + 60 \, b d^{6} \log \left (c\right ) + 60 \, a d^{6} - 60 \,{\left (b e^{6} n x^{2} - b d^{6} n\right )} \log \left (e x^{\frac{1}{3}} + d\right ) + 15 \,{\left (4 \, b d e^{5} n x - b d^{4} e^{2} n\right )} x^{\frac{2}{3}} - 6 \,{\left (5 \, b d^{2} e^{4} n x - 2 \, b d^{5} e n\right )} x^{\frac{1}{3}}}{120 \, d^{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*(60*b*e^6*n*x^2*log(x^(1/3)) + 20*b*d^3*e^3*n*x + 60*b*d^6*log(c) + 60*a*d^6 - 60*(b*e^6*n*x^2 - b*d^6*
n)*log(e*x^(1/3) + d) + 15*(4*b*d*e^5*n*x - b*d^4*e^2*n)*x^(2/3) - 6*(5*b*d^2*e^4*n*x - 2*b*d^5*e*n)*x^(1/3))/
(d^6*x^2)

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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**3,x)

[Out]

Timed out

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Giac [B]  time = 1.30122, size = 732, normalized size = 5.12 \begin{align*} \text{result too large to display} \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*(x^(1/3)*e + d)^6*b*n*e^7*log(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)^5*b*d*n*e^7*log(x^(1/3)*e + d) +
900*(x^(1/3)*e + d)^4*b*d^2*n*e^7*log(x^(1/3)*e + d) - 1200*(x^(1/3)*e + d)^3*b*d^3*n*e^7*log(x^(1/3)*e + d) +
900*(x^(1/3)*e + d)^2*b*d^4*n*e^7*log(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)*b*d^5*n*e^7*log(x^(1/3)*e + d) - 6
0*(x^(1/3)*e + d)^6*b*n*e^7*log(x^(1/3)*e) + 360*(x^(1/3)*e + d)^5*b*d*n*e^7*log(x^(1/3)*e) - 900*(x^(1/3)*e +
d)^4*b*d^2*n*e^7*log(x^(1/3)*e) + 1200*(x^(1/3)*e + d)^3*b*d^3*n*e^7*log(x^(1/3)*e) - 900*(x^(1/3)*e + d)^2*b
*d^4*n*e^7*log(x^(1/3)*e) + 360*(x^(1/3)*e + d)*b*d^5*n*e^7*log(x^(1/3)*e) - 60*b*d^6*n*e^7*log(x^(1/3)*e) - 6
0*(x^(1/3)*e + d)^5*b*d*n*e^7 + 330*(x^(1/3)*e + d)^4*b*d^2*n*e^7 - 740*(x^(1/3)*e + d)^3*b*d^3*n*e^7 + 855*(x
^(1/3)*e + d)^2*b*d^4*n*e^7 - 522*(x^(1/3)*e + d)*b*d^5*n*e^7 + 137*b*d^6*n*e^7 - 60*b*d^6*e^7*log(c) - 60*a*d
^6*e^7)*e^(-1)/((x^(1/3)*e + d)^6*d^6 - 6*(x^(1/3)*e + d)^5*d^7 + 15*(x^(1/3)*e + d)^4*d^8 - 20*(x^(1/3)*e + d
)^3*d^9 + 15*(x^(1/3)*e + d)^2*d^10 - 6*(x^(1/3)*e + d)*d^11 + d^12)