3.463 \(\int x^3 (a+b \log (c (d+e x^{2/3})^n)) \, dx\)

Optimal. Leaf size=138 \[ \frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{b d^5 n x^{2/3}}{4 e^5}-\frac{b d^4 n x^{4/3}}{8 e^4}+\frac{b d^3 n x^2}{12 e^3}-\frac{b d^2 n x^{8/3}}{16 e^2}-\frac{b d^6 n \log \left (d+e x^{2/3}\right )}{4 e^6}+\frac{b d n x^{10/3}}{20 e}-\frac{1}{24} b n x^4 \]

[Out]

(b*d^5*n*x^(2/3))/(4*e^5) - (b*d^4*n*x^(4/3))/(8*e^4) + (b*d^3*n*x^2)/(12*e^3) - (b*d^2*n*x^(8/3))/(16*e^2) +
(b*d*n*x^(10/3))/(20*e) - (b*n*x^4)/24 - (b*d^6*n*Log[d + e*x^(2/3)])/(4*e^6) + (x^4*(a + b*Log[c*(d + e*x^(2/
3))^n]))/4

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Rubi [A]  time = 0.106875, antiderivative size = 138, 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} \[ \frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{b d^5 n x^{2/3}}{4 e^5}-\frac{b d^4 n x^{4/3}}{8 e^4}+\frac{b d^3 n x^2}{12 e^3}-\frac{b d^2 n x^{8/3}}{16 e^2}-\frac{b d^6 n \log \left (d+e x^{2/3}\right )}{4 e^6}+\frac{b d n x^{10/3}}{20 e}-\frac{1}{24} b n x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^5*n*x^(2/3))/(4*e^5) - (b*d^4*n*x^(4/3))/(8*e^4) + (b*d^3*n*x^2)/(12*e^3) - (b*d^2*n*x^(8/3))/(16*e^2) +
(b*d*n*x^(10/3))/(20*e) - (b*n*x^4)/24 - (b*d^6*n*Log[d + e*x^(2/3)])/(4*e^6) + (x^4*(a + b*Log[c*(d + e*x^(2/
3))^n]))/4

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^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx &=\frac{3}{2} \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,x^{2/3}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{1}{4} (b e n) \operatorname{Subst}\left (\int \frac{x^6}{d+e x} \, dx,x,x^{2/3}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{1}{4} (b e n) \operatorname{Subst}\left (\int \left (-\frac{d^5}{e^6}+\frac{d^4 x}{e^5}-\frac{d^3 x^2}{e^4}+\frac{d^2 x^3}{e^3}-\frac{d x^4}{e^2}+\frac{x^5}{e}+\frac{d^6}{e^6 (d+e x)}\right ) \, dx,x,x^{2/3}\right )\\ &=\frac{b d^5 n x^{2/3}}{4 e^5}-\frac{b d^4 n x^{4/3}}{8 e^4}+\frac{b d^3 n x^2}{12 e^3}-\frac{b d^2 n x^{8/3}}{16 e^2}+\frac{b d n x^{10/3}}{20 e}-\frac{1}{24} b n x^4-\frac{b d^6 n \log \left (d+e x^{2/3}\right )}{4 e^6}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.109732, size = 135, normalized size = 0.98 \[ \frac{a x^4}{4}+\frac{1}{4} b x^4 \log \left (c \left (d+e x^{2/3}\right )^n\right )-\frac{1}{4} b e n \left (-\frac{d^5 x^{2/3}}{e^6}+\frac{d^4 x^{4/3}}{2 e^5}-\frac{d^3 x^2}{3 e^4}+\frac{d^2 x^{8/3}}{4 e^3}+\frac{d^6 \log \left (d+e x^{2/3}\right )}{e^7}-\frac{d x^{10/3}}{5 e^2}+\frac{x^4}{6 e}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.416, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03905, size = 146, normalized size = 1.06 \begin{align*} \frac{1}{4} \, b x^{4} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + \frac{1}{4} \, a x^{4} - \frac{1}{240} \, b e n{\left (\frac{60 \, d^{6} \log \left (e x^{\frac{2}{3}} + d\right )}{e^{7}} + \frac{10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac{10}{3}} + 15 \, d^{2} e^{3} x^{\frac{8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac{4}{3}} - 60 \, d^{5} x^{\frac{2}{3}}}{e^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4*log((e*x^(2/3) + d)^n*c) + 1/4*a*x^4 - 1/240*b*e*n*(60*d^6*log(e*x^(2/3) + d)/e^7 + (10*e^5*x^4 - 12
*d*e^4*x^(10/3) + 15*d^2*e^3*x^(8/3) - 20*d^3*e^2*x^2 + 30*d^4*e*x^(4/3) - 60*d^5*x^(2/3))/e^6)

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Fricas [A]  time = 1.86777, size = 302, normalized size = 2.19 \begin{align*} \frac{60 \, b e^{6} x^{4} \log \left (c\right ) + 20 \, b d^{3} e^{3} n x^{2} - 10 \,{\left (b e^{6} n - 6 \, a e^{6}\right )} x^{4} + 60 \,{\left (b e^{6} n x^{4} - b d^{6} n\right )} \log \left (e x^{\frac{2}{3}} + d\right ) - 15 \,{\left (b d^{2} e^{4} n x^{2} - 4 \, b d^{5} e n\right )} x^{\frac{2}{3}} + 6 \,{\left (2 \, b d e^{5} n x^{3} - 5 \, b d^{4} e^{2} n x\right )} x^{\frac{1}{3}}}{240 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(60*b*e^6*x^4*log(c) + 20*b*d^3*e^3*n*x^2 - 10*(b*e^6*n - 6*a*e^6)*x^4 + 60*(b*e^6*n*x^4 - b*d^6*n)*log(
e*x^(2/3) + d) - 15*(b*d^2*e^4*n*x^2 - 4*b*d^5*e*n)*x^(2/3) + 6*(2*b*d*e^5*n*x^3 - 5*b*d^4*e^2*n*x)*x^(1/3))/e
^6

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

[Out]

Timed out

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Giac [B]  time = 1.303, size = 359, normalized size = 2.6 \begin{align*} \frac{1}{4} \, b x^{4} \log \left (c\right ) + \frac{1}{4} \, a x^{4} + \frac{1}{240} \,{\left (60 \,{\left (x^{\frac{2}{3}} e + d\right )}^{6} e^{\left (-5\right )} \log \left (x^{\frac{2}{3}} e + d\right ) - 360 \,{\left (x^{\frac{2}{3}} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (x^{\frac{2}{3}} e + d\right ) + 900 \,{\left (x^{\frac{2}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (x^{\frac{2}{3}} e + d\right ) - 1200 \,{\left (x^{\frac{2}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (x^{\frac{2}{3}} e + d\right ) + 900 \,{\left (x^{\frac{2}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (x^{\frac{2}{3}} e + d\right ) - 360 \,{\left (x^{\frac{2}{3}} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (x^{\frac{2}{3}} e + d\right ) - 10 \,{\left (x^{\frac{2}{3}} e + d\right )}^{6} e^{\left (-5\right )} + 72 \,{\left (x^{\frac{2}{3}} e + d\right )}^{5} d e^{\left (-5\right )} - 225 \,{\left (x^{\frac{2}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} + 400 \,{\left (x^{\frac{2}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} - 450 \,{\left (x^{\frac{2}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} + 360 \,{\left (x^{\frac{2}{3}} e + d\right )} d^{5} e^{\left (-5\right )}\right )} b n e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4*log(c) + 1/4*a*x^4 + 1/240*(60*(x^(2/3)*e + d)^6*e^(-5)*log(x^(2/3)*e + d) - 360*(x^(2/3)*e + d)^5*d
*e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^4*d^2*e^(-5)*log(x^(2/3)*e + d) - 1200*(x^(2/3)*e + d)^3*d^3*
e^(-5)*log(x^(2/3)*e + d) + 900*(x^(2/3)*e + d)^2*d^4*e^(-5)*log(x^(2/3)*e + d) - 360*(x^(2/3)*e + d)*d^5*e^(-
5)*log(x^(2/3)*e + d) - 10*(x^(2/3)*e + d)^6*e^(-5) + 72*(x^(2/3)*e + d)^5*d*e^(-5) - 225*(x^(2/3)*e + d)^4*d^
2*e^(-5) + 400*(x^(2/3)*e + d)^3*d^3*e^(-5) - 450*(x^(2/3)*e + d)^2*d^4*e^(-5) + 360*(x^(2/3)*e + d)*d^5*e^(-5
))*b*n*e^(-1)