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

Optimal. Leaf size=190 \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )+\frac{b e^7 n x^{2/3}}{6 d^7}+\frac{b e^5 n x^{4/3}}{12 d^5}-\frac{b e^4 n x^{5/3}}{15 d^4}+\frac{b e^3 n x^2}{18 d^3}-\frac{b e^2 n x^{7/3}}{21 d^2}-\frac{b e^8 n \sqrt [3]{x}}{3 d^8}-\frac{b e^6 n x}{9 d^6}+\frac{b e^9 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^9}+\frac{b e^9 n \log (x)}{9 d^9}+\frac{b e n x^{8/3}}{24 d} \]

[Out]

-(b*e^8*n*x^(1/3))/(3*d^8) + (b*e^7*n*x^(2/3))/(6*d^7) - (b*e^6*n*x)/(9*d^6) + (b*e^5*n*x^(4/3))/(12*d^5) - (b
*e^4*n*x^(5/3))/(15*d^4) + (b*e^3*n*x^2)/(18*d^3) - (b*e^2*n*x^(7/3))/(21*d^2) + (b*e*n*x^(8/3))/(24*d) + (b*e
^9*n*Log[d + e/x^(1/3)])/(3*d^9) + (x^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/3 + (b*e^9*n*Log[x])/(9*d^9)

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Rubi [A]  time = 0.130131, antiderivative size = 190, 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} \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )+\frac{b e^7 n x^{2/3}}{6 d^7}+\frac{b e^5 n x^{4/3}}{12 d^5}-\frac{b e^4 n x^{5/3}}{15 d^4}+\frac{b e^3 n x^2}{18 d^3}-\frac{b e^2 n x^{7/3}}{21 d^2}-\frac{b e^8 n \sqrt [3]{x}}{3 d^8}-\frac{b e^6 n x}{9 d^6}+\frac{b e^9 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^9}+\frac{b e^9 n \log (x)}{9 d^9}+\frac{b e n x^{8/3}}{24 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e^8*n*x^(1/3))/(3*d^8) + (b*e^7*n*x^(2/3))/(6*d^7) - (b*e^6*n*x)/(9*d^6) + (b*e^5*n*x^(4/3))/(12*d^5) - (b
*e^4*n*x^(5/3))/(15*d^4) + (b*e^3*n*x^2)/(18*d^3) - (b*e^2*n*x^(7/3))/(21*d^2) + (b*e*n*x^(8/3))/(24*d) + (b*e
^9*n*Log[d + e/x^(1/3)])/(3*d^9) + (x^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/3 + (b*e^9*n*Log[x])/(9*d^9)

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^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right ) \, dx &=-\left (3 \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^{10}} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \frac{1}{x^9 (d+e x)} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )-\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \left (\frac{1}{d x^9}-\frac{e}{d^2 x^8}+\frac{e^2}{d^3 x^7}-\frac{e^3}{d^4 x^6}+\frac{e^4}{d^5 x^5}-\frac{e^5}{d^6 x^4}+\frac{e^6}{d^7 x^3}-\frac{e^7}{d^8 x^2}+\frac{e^8}{d^9 x}-\frac{e^9}{d^9 (d+e x)}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\\ &=-\frac{b e^8 n \sqrt [3]{x}}{3 d^8}+\frac{b e^7 n x^{2/3}}{6 d^7}-\frac{b e^6 n x}{9 d^6}+\frac{b e^5 n x^{4/3}}{12 d^5}-\frac{b e^4 n x^{5/3}}{15 d^4}+\frac{b e^3 n x^2}{18 d^3}-\frac{b e^2 n x^{7/3}}{21 d^2}+\frac{b e n x^{8/3}}{24 d}+\frac{b e^9 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{3 d^9}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )+\frac{b e^9 n \log (x)}{9 d^9}\\ \end{align*}

Mathematica [A]  time = 0.132517, size = 175, normalized size = 0.92 \[ \frac{a x^3}{3}+\frac{1}{3} b x^3 \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-\frac{1}{3} b e n \left (-\frac{e^6 x^{2/3}}{2 d^7}-\frac{e^4 x^{4/3}}{4 d^5}+\frac{e^3 x^{5/3}}{5 d^4}-\frac{e^2 x^2}{6 d^3}+\frac{e^7 \sqrt [3]{x}}{d^8}+\frac{e^5 x}{3 d^6}-\frac{e^8 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{d^9}-\frac{e^8 \log (x)}{3 d^9}+\frac{e x^{7/3}}{7 d^2}-\frac{x^{8/3}}{8 d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^3)/3 + (b*x^3*Log[c*(d + e/x^(1/3))^n])/3 - (b*e*n*((e^7*x^(1/3))/d^8 - (e^6*x^(2/3))/(2*d^7) + (e^5*x)/(
3*d^6) - (e^4*x^(4/3))/(4*d^5) + (e^3*x^(5/3))/(5*d^4) - (e^2*x^2)/(6*d^3) + (e*x^(7/3))/(7*d^2) - x^(8/3)/(8*
d) - (e^8*Log[d + e/x^(1/3)])/d^9 - (e^8*Log[x])/(3*d^9)))/3

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Maple [F]  time = 0.335, size = 0, normalized size = 0. \begin{align*} \int{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(x^2*(a+b*ln(c*(d+e/x^(1/3))^n)),x)

[Out]

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

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Maxima [A]  time = 1.0329, size = 173, normalized size = 0.91 \begin{align*} \frac{1}{3} \, b x^{3} \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right ) + \frac{1}{3} \, a x^{3} + \frac{1}{2520} \, b e n{\left (\frac{840 \, e^{8} \log \left (d x^{\frac{1}{3}} + e\right )}{d^{9}} + \frac{105 \, d^{7} x^{\frac{8}{3}} - 120 \, d^{6} e x^{\frac{7}{3}} + 140 \, d^{5} e^{2} x^{2} - 168 \, d^{4} e^{3} x^{\frac{5}{3}} + 210 \, d^{3} e^{4} x^{\frac{4}{3}} - 280 \, d^{2} e^{5} x + 420 \, d e^{6} x^{\frac{2}{3}} - 840 \, e^{7} x^{\frac{1}{3}}}{d^{8}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e/x^(1/3))^n)),x, algorithm="maxima")

[Out]

1/3*b*x^3*log(c*(d + e/x^(1/3))^n) + 1/3*a*x^3 + 1/2520*b*e*n*(840*e^8*log(d*x^(1/3) + e)/d^9 + (105*d^7*x^(8/
3) - 120*d^6*e*x^(7/3) + 140*d^5*e^2*x^2 - 168*d^4*e^3*x^(5/3) + 210*d^3*e^4*x^(4/3) - 280*d^2*e^5*x + 420*d*e
^6*x^(2/3) - 840*e^7*x^(1/3))/d^8)

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Fricas [A]  time = 2.19577, size = 473, normalized size = 2.49 \begin{align*} \frac{840 \, b d^{9} x^{3} \log \left (c\right ) + 140 \, b d^{6} e^{3} n x^{2} + 840 \, a d^{9} x^{3} - 280 \, b d^{3} e^{6} n x - 840 \, b d^{9} n \log \left (x^{\frac{1}{3}}\right ) + 840 \,{\left (b d^{9} + b e^{9}\right )} n \log \left (d x^{\frac{1}{3}} + e\right ) + 840 \,{\left (b d^{9} n x^{3} - b d^{9} n\right )} \log \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right ) + 21 \,{\left (5 \, b d^{8} e n x^{2} - 8 \, b d^{5} e^{4} n x + 20 \, b d^{2} e^{7} n\right )} x^{\frac{2}{3}} - 30 \,{\left (4 \, b d^{7} e^{2} n x^{2} - 7 \, b d^{4} e^{5} n x + 28 \, b d e^{8} n\right )} x^{\frac{1}{3}}}{2520 \, d^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e/x^(1/3))^n)),x, algorithm="fricas")

[Out]

1/2520*(840*b*d^9*x^3*log(c) + 140*b*d^6*e^3*n*x^2 + 840*a*d^9*x^3 - 280*b*d^3*e^6*n*x - 840*b*d^9*n*log(x^(1/
3)) + 840*(b*d^9 + b*e^9)*n*log(d*x^(1/3) + e) + 840*(b*d^9*n*x^3 - b*d^9*n)*log((d*x + e*x^(2/3))/x) + 21*(5*
b*d^8*e*n*x^2 - 8*b*d^5*e^4*n*x + 20*b*d^2*e^7*n)*x^(2/3) - 30*(4*b*d^7*e^2*n*x^2 - 7*b*d^4*e^5*n*x + 28*b*d*e
^8*n)*x^(1/3))/d^9

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

[Out]

Timed out

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Giac [A]  time = 1.39846, size = 177, normalized size = 0.93 \begin{align*} \frac{1}{3} \, b x^{3} \log \left (c\right ) + \frac{1}{3} \, a x^{3} + \frac{1}{2520} \,{\left (840 \, x^{3} \log \left (d + \frac{e}{x^{\frac{1}{3}}}\right ) +{\left (\frac{105 \, d^{7} x^{\frac{8}{3}} - 120 \, d^{6} x^{\frac{7}{3}} e + 140 \, d^{5} x^{2} e^{2} - 168 \, d^{4} x^{\frac{5}{3}} e^{3} + 210 \, d^{3} x^{\frac{4}{3}} e^{4} - 280 \, d^{2} x e^{5} + 420 \, d x^{\frac{2}{3}} e^{6} - 840 \, x^{\frac{1}{3}} e^{7}}{d^{8}} + \frac{840 \, e^{8} \log \left ({\left | d x^{\frac{1}{3}} + e \right |}\right )}{d^{9}}\right )} e\right )} b n \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e/x^(1/3))^n)),x, algorithm="giac")

[Out]

1/3*b*x^3*log(c) + 1/3*a*x^3 + 1/2520*(840*x^3*log(d + e/x^(1/3)) + ((105*d^7*x^(8/3) - 120*d^6*x^(7/3)*e + 14
0*d^5*x^2*e^2 - 168*d^4*x^(5/3)*e^3 + 210*d^3*x^(4/3)*e^4 - 280*d^2*x*e^5 + 420*d*x^(2/3)*e^6 - 840*x^(1/3)*e^
7)/d^8 + 840*e^8*log(abs(d*x^(1/3) + e))/d^9)*e)*b*n