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

Optimal. Leaf size=185 \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac{b d^7 n x^{2/3}}{6 e^7}+\frac{b d^5 n x^{4/3}}{12 e^5}-\frac{b d^4 n x^{5/3}}{15 e^4}+\frac{b d^3 n x^2}{18 e^3}-\frac{b d^2 n x^{7/3}}{21 e^2}-\frac{b d^8 n \sqrt [3]{x}}{3 e^8}-\frac{b d^6 n x}{9 e^6}+\frac{b d^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 e^9}+\frac{b d n x^{8/3}}{24 e}-\frac{1}{27} b n x^3 \]

[Out]

-(b*d^8*n*x^(1/3))/(3*e^8) + (b*d^7*n*x^(2/3))/(6*e^7) - (b*d^6*n*x)/(9*e^6) + (b*d^5*n*x^(4/3))/(12*e^5) - (b
*d^4*n*x^(5/3))/(15*e^4) + (b*d^3*n*x^2)/(18*e^3) - (b*d^2*n*x^(7/3))/(21*e^2) + (b*d*n*x^(8/3))/(24*e) - (b*n
*x^3)/27 + (b*d^9*n*Log[d + e*x^(1/3)])/(3*e^9) + (x^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/3

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Rubi [A]  time = 0.134155, antiderivative size = 185, 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} \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac{b d^7 n x^{2/3}}{6 e^7}+\frac{b d^5 n x^{4/3}}{12 e^5}-\frac{b d^4 n x^{5/3}}{15 e^4}+\frac{b d^3 n x^2}{18 e^3}-\frac{b d^2 n x^{7/3}}{21 e^2}-\frac{b d^8 n \sqrt [3]{x}}{3 e^8}-\frac{b d^6 n x}{9 e^6}+\frac{b d^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 e^9}+\frac{b d n x^{8/3}}{24 e}-\frac{1}{27} b n x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d^8*n*x^(1/3))/(3*e^8) + (b*d^7*n*x^(2/3))/(6*e^7) - (b*d^6*n*x)/(9*e^6) + (b*d^5*n*x^(4/3))/(12*e^5) - (b
*d^4*n*x^(5/3))/(15*e^4) + (b*d^3*n*x^2)/(18*e^3) - (b*d^2*n*x^(7/3))/(21*e^2) + (b*d*n*x^(8/3))/(24*e) - (b*n
*x^3)/27 + (b*d^9*n*Log[d + e*x^(1/3)])/(3*e^9) + (x^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/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 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 x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \, dx &=3 \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \frac{x^9}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \left (\frac{d^8}{e^9}-\frac{d^7 x}{e^8}+\frac{d^6 x^2}{e^7}-\frac{d^5 x^3}{e^6}+\frac{d^4 x^4}{e^5}-\frac{d^3 x^5}{e^4}+\frac{d^2 x^6}{e^3}-\frac{d x^7}{e^2}+\frac{x^8}{e}-\frac{d^9}{e^9 (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{b d^8 n \sqrt [3]{x}}{3 e^8}+\frac{b d^7 n x^{2/3}}{6 e^7}-\frac{b d^6 n x}{9 e^6}+\frac{b d^5 n x^{4/3}}{12 e^5}-\frac{b d^4 n x^{5/3}}{15 e^4}+\frac{b d^3 n x^2}{18 e^3}-\frac{b d^2 n x^{7/3}}{21 e^2}+\frac{b d n x^{8/3}}{24 e}-\frac{1}{27} b n x^3+\frac{b d^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 e^9}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.13615, size = 176, normalized size = 0.95 \[ \frac{a x^3}{3}+\frac{1}{3} b x^3 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-\frac{1}{3} b e n \left (-\frac{d^7 x^{2/3}}{2 e^8}-\frac{d^5 x^{4/3}}{4 e^6}+\frac{d^4 x^{5/3}}{5 e^5}-\frac{d^3 x^2}{6 e^4}+\frac{d^2 x^{7/3}}{7 e^3}+\frac{d^8 \sqrt [3]{x}}{e^9}+\frac{d^6 x}{3 e^7}-\frac{d^9 \log \left (d+e \sqrt [3]{x}\right )}{e^{10}}-\frac{d x^{8/3}}{8 e^2}+\frac{x^3}{9 e}\right ) \]

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*e*n*((d^8*x^(1/3))/e^9 - (d^7*x^(2/3))/(2*e^8) + (d^6*x)/(3*e^7) - (d^5*x^(4/3))/(4*e^6) + (d^4
*x^(5/3))/(5*e^5) - (d^3*x^2)/(6*e^4) + (d^2*x^(7/3))/(7*e^3) - (d*x^(8/3))/(8*e^2) + x^3/(9*e) - (d^9*Log[d +
e*x^(1/3)])/e^10))/3 + (b*x^3*Log[c*(d + e*x^(1/3))^n])/3

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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(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 = 1.04077, size = 189, normalized size = 1.02 \begin{align*} \frac{1}{3} \, b x^{3} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + \frac{1}{3} \, a x^{3} + \frac{1}{7560} \, b e n{\left (\frac{2520 \, d^{9} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{10}} - \frac{280 \, e^{8} x^{3} - 315 \, d e^{7} x^{\frac{8}{3}} + 360 \, d^{2} e^{6} x^{\frac{7}{3}} - 420 \, d^{3} e^{5} x^{2} + 504 \, d^{4} e^{4} x^{\frac{5}{3}} - 630 \, d^{5} e^{3} x^{\frac{4}{3}} + 840 \, d^{6} e^{2} x - 1260 \, d^{7} e x^{\frac{2}{3}} + 2520 \, d^{8} x^{\frac{1}{3}}}{e^{9}}\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((e*x^(1/3) + d)^n*c) + 1/3*a*x^3 + 1/7560*b*e*n*(2520*d^9*log(e*x^(1/3) + d)/e^10 - (280*e^8*x^3
- 315*d*e^7*x^(8/3) + 360*d^2*e^6*x^(7/3) - 420*d^3*e^5*x^2 + 504*d^4*e^4*x^(5/3) - 630*d^5*e^3*x^(4/3) + 840
*d^6*e^2*x - 1260*d^7*e*x^(2/3) + 2520*d^8*x^(1/3))/e^9)

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Fricas [A]  time = 1.84729, size = 392, normalized size = 2.12 \begin{align*} \frac{2520 \, b e^{9} x^{3} \log \left (c\right ) + 420 \, b d^{3} e^{6} n x^{2} - 840 \, b d^{6} e^{3} n x - 280 \,{\left (b e^{9} n - 9 \, a e^{9}\right )} x^{3} + 2520 \,{\left (b e^{9} n x^{3} + b d^{9} n\right )} \log \left (e x^{\frac{1}{3}} + d\right ) + 63 \,{\left (5 \, b d e^{8} n x^{2} - 8 \, b d^{4} e^{5} n x + 20 \, b d^{7} e^{2} n\right )} x^{\frac{2}{3}} - 90 \,{\left (4 \, b d^{2} e^{7} n x^{2} - 7 \, b d^{5} e^{4} n x + 28 \, b d^{8} e n\right )} x^{\frac{1}{3}}}{7560 \, e^{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]

1/7560*(2520*b*e^9*x^3*log(c) + 420*b*d^3*e^6*n*x^2 - 840*b*d^6*e^3*n*x - 280*(b*e^9*n - 9*a*e^9)*x^3 + 2520*(
b*e^9*n*x^3 + b*d^9*n)*log(e*x^(1/3) + d) + 63*(5*b*d*e^8*n*x^2 - 8*b*d^4*e^5*n*x + 20*b*d^7*e^2*n)*x^(2/3) -
90*(4*b*d^2*e^7*n*x^2 - 7*b*d^5*e^4*n*x + 28*b*d^8*e*n)*x^(1/3))/e^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 [B]  time = 1.34357, size = 540, normalized size = 2.92 \begin{align*} \frac{1}{7560} \,{\left (2520 \, b x^{3} e \log \left (c\right ) + 2520 \, a x^{3} e +{\left (2520 \,{\left (x^{\frac{1}{3}} e + d\right )}^{9} e^{\left (-8\right )} \log \left (x^{\frac{1}{3}} e + d\right ) - 22680 \,{\left (x^{\frac{1}{3}} e + d\right )}^{8} d e^{\left (-8\right )} \log \left (x^{\frac{1}{3}} e + d\right ) + 90720 \,{\left (x^{\frac{1}{3}} e + d\right )}^{7} d^{2} e^{\left (-8\right )} \log \left (x^{\frac{1}{3}} e + d\right ) - 211680 \,{\left (x^{\frac{1}{3}} e + d\right )}^{6} d^{3} e^{\left (-8\right )} \log \left (x^{\frac{1}{3}} e + d\right ) + 317520 \,{\left (x^{\frac{1}{3}} e + d\right )}^{5} d^{4} e^{\left (-8\right )} \log \left (x^{\frac{1}{3}} e + d\right ) - 317520 \,{\left (x^{\frac{1}{3}} e + d\right )}^{4} d^{5} e^{\left (-8\right )} \log \left (x^{\frac{1}{3}} e + d\right ) + 211680 \,{\left (x^{\frac{1}{3}} e + d\right )}^{3} d^{6} e^{\left (-8\right )} \log \left (x^{\frac{1}{3}} e + d\right ) - 90720 \,{\left (x^{\frac{1}{3}} e + d\right )}^{2} d^{7} e^{\left (-8\right )} \log \left (x^{\frac{1}{3}} e + d\right ) + 22680 \,{\left (x^{\frac{1}{3}} e + d\right )} d^{8} e^{\left (-8\right )} \log \left (x^{\frac{1}{3}} e + d\right ) - 280 \,{\left (x^{\frac{1}{3}} e + d\right )}^{9} e^{\left (-8\right )} + 2835 \,{\left (x^{\frac{1}{3}} e + d\right )}^{8} d e^{\left (-8\right )} - 12960 \,{\left (x^{\frac{1}{3}} e + d\right )}^{7} d^{2} e^{\left (-8\right )} + 35280 \,{\left (x^{\frac{1}{3}} e + d\right )}^{6} d^{3} e^{\left (-8\right )} - 63504 \,{\left (x^{\frac{1}{3}} e + d\right )}^{5} d^{4} e^{\left (-8\right )} + 79380 \,{\left (x^{\frac{1}{3}} e + d\right )}^{4} d^{5} e^{\left (-8\right )} - 70560 \,{\left (x^{\frac{1}{3}} e + d\right )}^{3} d^{6} e^{\left (-8\right )} + 45360 \,{\left (x^{\frac{1}{3}} e + d\right )}^{2} d^{7} e^{\left (-8\right )} - 22680 \,{\left (x^{\frac{1}{3}} e + d\right )} d^{8} e^{\left (-8\right )}\right )} b n\right )} e^{\left (-1\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="giac")

[Out]

1/7560*(2520*b*x^3*e*log(c) + 2520*a*x^3*e + (2520*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d) - 22680*(x^(1/3
)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d) + 90720*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d) - 211680*(x^(1/
3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d) + 317520*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d) - 317520*(x
^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d) + 211680*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d) - 90720
*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d) + 22680*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) - 280*(
x^(1/3)*e + d)^9*e^(-8) + 2835*(x^(1/3)*e + d)^8*d*e^(-8) - 12960*(x^(1/3)*e + d)^7*d^2*e^(-8) + 35280*(x^(1/3
)*e + d)^6*d^3*e^(-8) - 63504*(x^(1/3)*e + d)^5*d^4*e^(-8) + 79380*(x^(1/3)*e + d)^4*d^5*e^(-8) - 70560*(x^(1/
3)*e + d)^3*d^6*e^(-8) + 45360*(x^(1/3)*e + d)^2*d^7*e^(-8) - 22680*(x^(1/3)*e + d)*d^8*e^(-8))*b*n)*e^(-1)