3.5.43 \(\int x^2 (a+b \log (c (d+e \sqrt [3]{x})^n)) \, dx\) [443]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 185, normalized size of antiderivative = 1.00, 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 = {2504, 2442, 45} \begin {gather*} \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+\frac {b d^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 e^9}-\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 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(b*d^8*n*x^(1/3))/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 45

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])

Rule 2442

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 2504

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])

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 \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.10, size = 176, normalized size = 0.95 \begin {gather*} \frac {a x^3}{3}-\frac {1}{3} b e n \left (\frac {d^8 \sqrt [3]{x}}{e^9}-\frac {d^7 x^{2/3}}{2 e^8}+\frac {d^6 x}{3 e^7}-\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 x^{8/3}}{8 e^2}+\frac {x^3}{9 e}-\frac {d^9 \log \left (d+e \sqrt [3]{x}\right )}{e^{10}}\right )+\frac {1}{3} b x^3 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )\, dx\]

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 = 0.27, size = 134, normalized size = 0.72 \begin {gather*} \frac {1}{3} \, b x^{3} \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right ) + \frac {1}{3} \, a x^{3} + \frac {1}{7560} \, {\left (2520 \, d^{9} e^{\left (-10\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (1260 \, d^{7} x^{\frac {2}{3}} e - 2520 \, d^{8} x^{\frac {1}{3}} - 840 \, d^{6} x e^{2} + 630 \, d^{5} x^{\frac {4}{3}} e^{3} - 504 \, d^{4} x^{\frac {5}{3}} e^{4} + 420 \, d^{3} x^{2} e^{5} - 360 \, d^{2} x^{\frac {7}{3}} e^{6} + 315 \, d x^{\frac {8}{3}} e^{7} - 280 \, x^{3} e^{8}\right )} e^{\left (-9\right )}\right )} b n e \end {gather*}

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((x^(1/3)*e + d)^n*c) + 1/3*a*x^3 + 1/7560*(2520*d^9*e^(-10)*log(x^(1/3)*e + d) + (1260*d^7*x^(2/
3)*e - 2520*d^8*x^(1/3) - 840*d^6*x*e^2 + 630*d^5*x^(4/3)*e^3 - 504*d^4*x^(5/3)*e^4 + 420*d^3*x^2*e^5 - 360*d^
2*x^(7/3)*e^6 + 315*d*x^(8/3)*e^7 - 280*x^3*e^8)*e^(-9))*b*n*e

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Fricas [A]
time = 0.38, size = 149, normalized size = 0.81 \begin {gather*} -\frac {1}{7560} \, {\left (840 \, b d^{6} n x e^{3} - 420 \, b d^{3} n x^{2} e^{6} - 2520 \, b x^{3} e^{9} \log \left (c\right ) + 280 \, {\left (b n - 9 \, a\right )} x^{3} e^{9} - 2520 \, {\left (b d^{9} n + b n x^{3} e^{9}\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 63 \, {\left (20 \, b d^{7} n e^{2} - 8 \, b d^{4} n x e^{5} + 5 \, b d n x^{2} e^{8}\right )} x^{\frac {2}{3}} + 90 \, {\left (28 \, b d^{8} n e - 7 \, b d^{5} n x e^{4} + 4 \, b d^{2} n x^{2} e^{7}\right )} x^{\frac {1}{3}}\right )} e^{\left (-9\right )} \end {gather*}

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*(840*b*d^6*n*x*e^3 - 420*b*d^3*n*x^2*e^6 - 2520*b*x^3*e^9*log(c) + 280*(b*n - 9*a)*x^3*e^9 - 2520*(b*d
^9*n + b*n*x^3*e^9)*log(x^(1/3)*e + d) - 63*(20*b*d^7*n*e^2 - 8*b*d^4*n*x*e^5 + 5*b*d*n*x^2*e^8)*x^(2/3) + 90*
(28*b*d^8*n*e - 7*b*d^5*n*x*e^4 + 4*b*d^2*n*x^2*e^7)*x^(1/3))*e^(-9)

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Sympy [A]
time = 9.84, size = 173, normalized size = 0.94 \begin {gather*} \frac {a x^{3}}{3} + b \left (- \frac {e n \left (- \frac {3 d^{9} \left (\begin {cases} \frac {\sqrt [3]{x}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e \sqrt [3]{x} \right )}}{e} & \text {otherwise} \end {cases}\right )}{e^{9}} + \frac {3 d^{8} \sqrt [3]{x}}{e^{9}} - \frac {3 d^{7} x^{\frac {2}{3}}}{2 e^{8}} + \frac {d^{6} x}{e^{7}} - \frac {3 d^{5} x^{\frac {4}{3}}}{4 e^{6}} + \frac {3 d^{4} x^{\frac {5}{3}}}{5 e^{5}} - \frac {d^{3} x^{2}}{2 e^{4}} + \frac {3 d^{2} x^{\frac {7}{3}}}{7 e^{3}} - \frac {3 d x^{\frac {8}{3}}}{8 e^{2}} + \frac {x^{3}}{3 e}\right )}{9} + \frac {x^{3} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{3}\right ) \end {gather*}

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]

a*x**3/3 + b*(-e*n*(-3*d**9*Piecewise((x**(1/3)/d, Eq(e, 0)), (log(d + e*x**(1/3))/e, True))/e**9 + 3*d**8*x**
(1/3)/e**9 - 3*d**7*x**(2/3)/(2*e**8) + d**6*x/e**7 - 3*d**5*x**(4/3)/(4*e**6) + 3*d**4*x**(5/3)/(5*e**5) - d*
*3*x**2/(2*e**4) + 3*d**2*x**(7/3)/(7*e**3) - 3*d*x**(8/3)/(8*e**2) + x**3/(3*e))/9 + x**3*log(c*(d + e*x**(1/
3))**n)/3)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (140) = 280\).
time = 5.90, size = 400, normalized size = 2.16 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.51, size = 150, normalized size = 0.81 \begin {gather*} \frac {a\,x^3}{3}-\frac {b\,n\,x^3}{27}+\frac {b\,x^3\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3}+\frac {b\,d\,n\,x^{8/3}}{24\,e}-\frac {b\,d^6\,n\,x}{9\,e^6}+\frac {b\,d^9\,n\,\ln \left (d+e\,x^{1/3}\right )}{3\,e^9}+\frac {b\,d^3\,n\,x^2}{18\,e^3}-\frac {b\,d^2\,n\,x^{7/3}}{21\,e^2}-\frac {b\,d^4\,n\,x^{5/3}}{15\,e^4}+\frac {b\,d^5\,n\,x^{4/3}}{12\,e^5}+\frac {b\,d^7\,n\,x^{2/3}}{6\,e^7}-\frac {b\,d^8\,n\,x^{1/3}}{3\,e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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