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

Optimal. Leaf size=136 \[ \frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac{b d^4 n x^{2/3}}{4 e^4}-\frac{b d^2 n x^{4/3}}{8 e^2}+\frac{b d^5 n \sqrt [3]{x}}{2 e^5}+\frac{b d^3 n x}{6 e^3}-\frac{b d^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 e^6}+\frac{b d n x^{5/3}}{10 e}-\frac{1}{12} b n x^2 \]

[Out]

(b*d^5*n*x^(1/3))/(2*e^5) - (b*d^4*n*x^(2/3))/(4*e^4) + (b*d^3*n*x)/(6*e^3) - (b*d^2*n*x^(4/3))/(8*e^2) + (b*d
*n*x^(5/3))/(10*e) - (b*n*x^2)/12 - (b*d^6*n*Log[d + e*x^(1/3)])/(2*e^6) + (x^2*(a + b*Log[c*(d + e*x^(1/3))^n
]))/2

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Rubi [A]  time = 0.0949436, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2454, 2395, 43} \[ \frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac{b d^4 n x^{2/3}}{4 e^4}-\frac{b d^2 n x^{4/3}}{8 e^2}+\frac{b d^5 n \sqrt [3]{x}}{2 e^5}+\frac{b d^3 n x}{6 e^3}-\frac{b d^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 e^6}+\frac{b d n x^{5/3}}{10 e}-\frac{1}{12} b n x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^5*n*x^(1/3))/(2*e^5) - (b*d^4*n*x^(2/3))/(4*e^4) + (b*d^3*n*x)/(6*e^3) - (b*d^2*n*x^(4/3))/(8*e^2) + (b*d
*n*x^(5/3))/(10*e) - (b*n*x^2)/12 - (b*d^6*n*Log[d + e*x^(1/3)])/(2*e^6) + (x^2*(a + b*Log[c*(d + e*x^(1/3))^n
]))/2

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

Mathematica [A]  time = 0.0927543, size = 133, normalized size = 0.98 \[ \frac{a x^2}{2}+\frac{1}{2} b x^2 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-\frac{1}{2} b e n \left (\frac{d^4 x^{2/3}}{2 e^5}+\frac{d^2 x^{4/3}}{4 e^3}-\frac{d^5 \sqrt [3]{x}}{e^6}-\frac{d^3 x}{3 e^4}+\frac{d^6 \log \left (d+e \sqrt [3]{x}\right )}{e^7}-\frac{d x^{5/3}}{5 e^2}+\frac{x^2}{6 e}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \begin{align*} \int x \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*(a+b*ln(c*(d+e*x^(1/3))^n)),x)

[Out]

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

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Maxima [A]  time = 1.00385, size = 143, normalized size = 1.05 \begin{align*} -\frac{1}{120} \, b e n{\left (\frac{60 \, d^{6} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{7}} + \frac{10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac{5}{3}} + 15 \, d^{2} e^{3} x^{\frac{4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac{2}{3}} - 60 \, d^{5} x^{\frac{1}{3}}}{e^{6}}\right )} + \frac{1}{2} \, b x^{2} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*b*e*n*(60*d^6*log(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 - 12*d*e^4*x^(5/3) + 15*d^2*e^3*x^(4/3) - 20*d^3*e^2
*x + 30*d^4*e*x^(2/3) - 60*d^5*x^(1/3))/e^6) + 1/2*b*x^2*log((e*x^(1/3) + d)^n*c) + 1/2*a*x^2

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Fricas [A]  time = 1.8302, size = 292, normalized size = 2.15 \begin{align*} \frac{60 \, b e^{6} x^{2} \log \left (c\right ) + 20 \, b d^{3} e^{3} n x - 10 \,{\left (b e^{6} n - 6 \, a e^{6}\right )} x^{2} + 60 \,{\left (b e^{6} n x^{2} - b d^{6} n\right )} \log \left (e x^{\frac{1}{3}} + d\right ) + 6 \,{\left (2 \, b d e^{5} n x - 5 \, b d^{4} e^{2} n\right )} x^{\frac{2}{3}} - 15 \,{\left (b d^{2} e^{4} n x - 4 \, b d^{5} e n\right )} x^{\frac{1}{3}}}{120 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(60*b*e^6*x^2*log(c) + 20*b*d^3*e^3*n*x - 10*(b*e^6*n - 6*a*e^6)*x^2 + 60*(b*e^6*n*x^2 - b*d^6*n)*log(e*
x^(1/3) + d) + 6*(2*b*d*e^5*n*x - 5*b*d^4*e^2*n)*x^(2/3) - 15*(b*d^2*e^4*n*x - 4*b*d^5*e*n)*x^(1/3))/e^6

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Sympy [A]  time = 11.8602, size = 131, normalized size = 0.96 \begin{align*} \frac{a x^{2}}{2} + b \left (- \frac{e n \left (\frac{3 d^{6} \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^{6}} - \frac{3 d^{5} \sqrt [3]{x}}{e^{6}} + \frac{3 d^{4} x^{\frac{2}{3}}}{2 e^{5}} - \frac{d^{3} x}{e^{4}} + \frac{3 d^{2} x^{\frac{4}{3}}}{4 e^{3}} - \frac{3 d x^{\frac{5}{3}}}{5 e^{2}} + \frac{x^{2}}{2 e}\right )}{6} + \frac{x^{2} \log{\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.30847, size = 366, normalized size = 2.69 \begin{align*} \frac{1}{120} \,{\left (60 \, b x^{2} e \log \left (c\right ) + 60 \, a x^{2} e +{\left (60 \,{\left (x^{\frac{1}{3}} e + d\right )}^{6} e^{\left (-5\right )} \log \left (x^{\frac{1}{3}} e + d\right ) - 360 \,{\left (x^{\frac{1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (x^{\frac{1}{3}} e + d\right ) + 900 \,{\left (x^{\frac{1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (x^{\frac{1}{3}} e + d\right ) - 1200 \,{\left (x^{\frac{1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (x^{\frac{1}{3}} e + d\right ) + 900 \,{\left (x^{\frac{1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (x^{\frac{1}{3}} e + d\right ) - 360 \,{\left (x^{\frac{1}{3}} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (x^{\frac{1}{3}} e + d\right ) - 10 \,{\left (x^{\frac{1}{3}} e + d\right )}^{6} e^{\left (-5\right )} + 72 \,{\left (x^{\frac{1}{3}} e + d\right )}^{5} d e^{\left (-5\right )} - 225 \,{\left (x^{\frac{1}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} + 400 \,{\left (x^{\frac{1}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} - 450 \,{\left (x^{\frac{1}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} + 360 \,{\left (x^{\frac{1}{3}} e + d\right )} d^{5} e^{\left (-5\right )}\right )} b n\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(60*b*x^2*e*log(c) + 60*a*x^2*e + (60*(x^(1/3)*e + d)^6*e^(-5)*log(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)^
5*d*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^4*d^2*e^(-5)*log(x^(1/3)*e + d) - 1200*(x^(1/3)*e + d)^3*d
^3*e^(-5)*log(x^(1/3)*e + d) + 900*(x^(1/3)*e + d)^2*d^4*e^(-5)*log(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)*d^5*e
^(-5)*log(x^(1/3)*e + d) - 10*(x^(1/3)*e + d)^6*e^(-5) + 72*(x^(1/3)*e + d)^5*d*e^(-5) - 225*(x^(1/3)*e + d)^4
*d^2*e^(-5) + 400*(x^(1/3)*e + d)^3*d^3*e^(-5) - 450*(x^(1/3)*e + d)^2*d^4*e^(-5) + 360*(x^(1/3)*e + d)*d^5*e^
(-5))*b*n)*e^(-1)