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

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

[Out]

1/4*b*d^5*n*x^(2/3)/e^5-1/8*b*d^4*n*x^(4/3)/e^4+1/12*b*d^3*n*x^2/e^3-1/16*b*d^2*n*x^(8/3)/e^2+1/20*b*d*n*x^(10
/3)/e-1/24*b*n*x^4-1/4*b*d^6*n*ln(d+e*x^(2/3))/e^6+1/4*x^4*(a+b*ln(c*(d+e*x^(2/3))^n))

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Rubi [A]  time = 0.11, antiderivative size = 138, 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 = {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 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])

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 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])

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*}

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Mathematica [A]  time = 0.11, size = 135, normalized size = 0.98 \[ \frac {a x^4}{4}+\frac {1}{4} b x^4 \log \left (c \left (d+e x^{2/3}\right )^n\right )-\frac {1}{4} b e n \left (\frac {d^6 \log \left (d+e x^{2/3}\right )}{e^7}-\frac {d^5 x^{2/3}}{e^6}+\frac {d^4 x^{4/3}}{2 e^5}-\frac {d^3 x^2}{3 e^4}+\frac {d^2 x^{8/3}}{4 e^3}-\frac {d x^{10/3}}{5 e^2}+\frac {x^4}{6 e}\right ) \]

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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fricas [A]  time = 0.46, size = 129, normalized size = 0.93 \[ \frac {60 \, b e^{6} x^{4} \log \relax (c) + 20 \, b d^{3} e^{3} n x^{2} - 10 \, {\left (b e^{6} n - 6 \, a e^{6}\right )} x^{4} + 60 \, {\left (b e^{6} n x^{4} - b d^{6} n\right )} \log \left (e x^{\frac {2}{3}} + d\right ) - 15 \, {\left (b d^{2} e^{4} n x^{2} - 4 \, b d^{5} e n\right )} x^{\frac {2}{3}} + 6 \, {\left (2 \, b d e^{5} n x^{3} - 5 \, b d^{4} e^{2} n x\right )} x^{\frac {1}{3}}}{240 \, e^{6}} \]

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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giac [B]  time = 0.34, size = 266, normalized size = 1.93 \[ \frac {1}{4} \, b x^{4} \log \relax (c) + \frac {1}{4} \, a x^{4} + \frac {1}{240} \, {\left (60 \, {\left (x^{\frac {2}{3}} e + d\right )}^{6} e^{\left (-5\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 360 \, {\left (x^{\frac {2}{3}} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 900 \, {\left (x^{\frac {2}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 1200 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + 900 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 360 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 10 \, {\left (x^{\frac {2}{3}} e + d\right )}^{6} e^{\left (-5\right )} + 72 \, {\left (x^{\frac {2}{3}} e + d\right )}^{5} d e^{\left (-5\right )} - 225 \, {\left (x^{\frac {2}{3}} e + d\right )}^{4} d^{2} e^{\left (-5\right )} + 400 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} d^{3} e^{\left (-5\right )} - 450 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d^{4} e^{\left (-5\right )} + 360 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{5} e^{\left (-5\right )}\right )} b n e^{\left (-1\right )} \]

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)

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maple [F]  time = 0.17, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right ) x^{3}\, dx \]

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 = 0.49, size = 108, normalized size = 0.78 \[ \frac {1}{4} \, b x^{4} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + \frac {1}{4} \, a x^{4} - \frac {1}{240} \, b e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac {10}{3}} + 15 \, d^{2} e^{3} x^{\frac {8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac {4}{3}} - 60 \, d^{5} x^{\frac {2}{3}}}{e^{6}}\right )} \]

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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mupad [B]  time = 0.45, size = 113, normalized size = 0.82 \[ \frac {a\,x^4}{4}-\frac {b\,n\,x^4}{24}+\frac {b\,x^4\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}{4}+\frac {b\,d\,n\,x^{10/3}}{20\,e}-\frac {b\,d^6\,n\,\ln \left (d+e\,x^{2/3}\right )}{4\,e^6}+\frac {b\,d^3\,n\,x^2}{12\,e^3}-\frac {b\,d^2\,n\,x^{8/3}}{16\,e^2}-\frac {b\,d^4\,n\,x^{4/3}}{8\,e^4}+\frac {b\,d^5\,n\,x^{2/3}}{4\,e^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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