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

Optimal. Leaf size=234 \[ \frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac{b d^{10} n x^{2/3}}{8 e^{10}}-\frac{b d^8 n x^{4/3}}{16 e^8}+\frac{b d^7 n x^{5/3}}{20 e^7}-\frac{b d^6 n x^2}{24 e^6}+\frac{b d^5 n x^{7/3}}{28 e^5}-\frac{b d^4 n x^{8/3}}{32 e^4}+\frac{b d^3 n x^3}{36 e^3}-\frac{b d^2 n x^{10/3}}{40 e^2}+\frac{b d^{11} n \sqrt [3]{x}}{4 e^{11}}+\frac{b d^9 n x}{12 e^9}-\frac{b d^{12} n \log \left (d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac{b d n x^{11/3}}{44 e}-\frac{1}{48} b n x^4 \]

[Out]

(b*d^11*n*x^(1/3))/(4*e^11) - (b*d^10*n*x^(2/3))/(8*e^10) + (b*d^9*n*x)/(12*e^9) - (b*d^8*n*x^(4/3))/(16*e^8)
+ (b*d^7*n*x^(5/3))/(20*e^7) - (b*d^6*n*x^2)/(24*e^6) + (b*d^5*n*x^(7/3))/(28*e^5) - (b*d^4*n*x^(8/3))/(32*e^4
) + (b*d^3*n*x^3)/(36*e^3) - (b*d^2*n*x^(10/3))/(40*e^2) + (b*d*n*x^(11/3))/(44*e) - (b*n*x^4)/48 - (b*d^12*n*
Log[d + e*x^(1/3)])/(4*e^12) + (x^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/4

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Rubi [A]  time = 0.187993, antiderivative size = 234, 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} \[ \frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac{b d^{10} n x^{2/3}}{8 e^{10}}-\frac{b d^8 n x^{4/3}}{16 e^8}+\frac{b d^7 n x^{5/3}}{20 e^7}-\frac{b d^6 n x^2}{24 e^6}+\frac{b d^5 n x^{7/3}}{28 e^5}-\frac{b d^4 n x^{8/3}}{32 e^4}+\frac{b d^3 n x^3}{36 e^3}-\frac{b d^2 n x^{10/3}}{40 e^2}+\frac{b d^{11} n \sqrt [3]{x}}{4 e^{11}}+\frac{b d^9 n x}{12 e^9}-\frac{b d^{12} n \log \left (d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac{b d n x^{11/3}}{44 e}-\frac{1}{48} b n x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^11*n*x^(1/3))/(4*e^11) - (b*d^10*n*x^(2/3))/(8*e^10) + (b*d^9*n*x)/(12*e^9) - (b*d^8*n*x^(4/3))/(16*e^8)
+ (b*d^7*n*x^(5/3))/(20*e^7) - (b*d^6*n*x^2)/(24*e^6) + (b*d^5*n*x^(7/3))/(28*e^5) - (b*d^4*n*x^(8/3))/(32*e^4
) + (b*d^3*n*x^3)/(36*e^3) - (b*d^2*n*x^(10/3))/(40*e^2) + (b*d*n*x^(11/3))/(44*e) - (b*n*x^4)/48 - (b*d^12*n*
Log[d + e*x^(1/3)])/(4*e^12) + (x^4*(a + b*Log[c*(d + e*x^(1/3))^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 \sqrt [3]{x}\right )^n\right )\right ) \, dx &=3 \operatorname{Subst}\left (\int x^{11} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac{1}{4} (b e n) \operatorname{Subst}\left (\int \frac{x^{12}}{d+e x} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )-\frac{1}{4} (b e n) \operatorname{Subst}\left (\int \left (-\frac{d^{11}}{e^{12}}+\frac{d^{10} x}{e^{11}}-\frac{d^9 x^2}{e^{10}}+\frac{d^8 x^3}{e^9}-\frac{d^7 x^4}{e^8}+\frac{d^6 x^5}{e^7}-\frac{d^5 x^6}{e^6}+\frac{d^4 x^7}{e^5}-\frac{d^3 x^8}{e^4}+\frac{d^2 x^9}{e^3}-\frac{d x^{10}}{e^2}+\frac{x^{11}}{e}+\frac{d^{12}}{e^{12} (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{b d^{11} n \sqrt [3]{x}}{4 e^{11}}-\frac{b d^{10} n x^{2/3}}{8 e^{10}}+\frac{b d^9 n x}{12 e^9}-\frac{b d^8 n x^{4/3}}{16 e^8}+\frac{b d^7 n x^{5/3}}{20 e^7}-\frac{b d^6 n x^2}{24 e^6}+\frac{b d^5 n x^{7/3}}{28 e^5}-\frac{b d^4 n x^{8/3}}{32 e^4}+\frac{b d^3 n x^3}{36 e^3}-\frac{b d^2 n x^{10/3}}{40 e^2}+\frac{b d n x^{11/3}}{44 e}-\frac{1}{48} b n x^4-\frac{b d^{12} n \log \left (d+e \sqrt [3]{x}\right )}{4 e^{12}}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.229194, size = 219, normalized size = 0.94 \[ \frac{a x^4}{4}+\frac{1}{4} b x^4 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-\frac{1}{4} b e n \left (\frac{d^{10} x^{2/3}}{2 e^{11}}+\frac{d^8 x^{4/3}}{4 e^9}-\frac{d^7 x^{5/3}}{5 e^8}+\frac{d^6 x^2}{6 e^7}-\frac{d^5 x^{7/3}}{7 e^6}+\frac{d^4 x^{8/3}}{8 e^5}-\frac{d^3 x^3}{9 e^4}+\frac{d^2 x^{10/3}}{10 e^3}-\frac{d^{11} \sqrt [3]{x}}{e^{12}}-\frac{d^9 x}{3 e^{10}}+\frac{d^{12} \log \left (d+e \sqrt [3]{x}\right )}{e^{13}}-\frac{d x^{11/3}}{11 e^2}+\frac{x^4}{12 e}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^4)/4 - (b*e*n*(-((d^11*x^(1/3))/e^12) + (d^10*x^(2/3))/(2*e^11) - (d^9*x)/(3*e^10) + (d^8*x^(4/3))/(4*e^9
) - (d^7*x^(5/3))/(5*e^8) + (d^6*x^2)/(6*e^7) - (d^5*x^(7/3))/(7*e^6) + (d^4*x^(8/3))/(8*e^5) - (d^3*x^3)/(9*e
^4) + (d^2*x^(10/3))/(10*e^3) - (d*x^(11/3))/(11*e^2) + x^4/(12*e) + (d^12*Log[d + e*x^(1/3)])/e^13))/4 + (b*x
^4*Log[c*(d + e*x^(1/3))^n])/4

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Maple [F]  time = 0.434, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \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^3*(a+b*ln(c*(d+e*x^(1/3))^n)),x)

[Out]

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

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Maxima [A]  time = 1.02377, size = 232, normalized size = 0.99 \begin{align*} \frac{1}{4} \, b x^{4} \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right ) + \frac{1}{4} \, a x^{4} - \frac{1}{110880} \, b e n{\left (\frac{27720 \, d^{12} \log \left (e x^{\frac{1}{3}} + d\right )}{e^{13}} + \frac{2310 \, e^{11} x^{4} - 2520 \, d e^{10} x^{\frac{11}{3}} + 2772 \, d^{2} e^{9} x^{\frac{10}{3}} - 3080 \, d^{3} e^{8} x^{3} + 3465 \, d^{4} e^{7} x^{\frac{8}{3}} - 3960 \, d^{5} e^{6} x^{\frac{7}{3}} + 4620 \, d^{6} e^{5} x^{2} - 5544 \, d^{7} e^{4} x^{\frac{5}{3}} + 6930 \, d^{8} e^{3} x^{\frac{4}{3}} - 9240 \, d^{9} e^{2} x + 13860 \, d^{10} e x^{\frac{2}{3}} - 27720 \, d^{11} x^{\frac{1}{3}}}{e^{12}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*x^4*log((e*x^(1/3) + d)^n*c) + 1/4*a*x^4 - 1/110880*b*e*n*(27720*d^12*log(e*x^(1/3) + d)/e^13 + (2310*e^
11*x^4 - 2520*d*e^10*x^(11/3) + 2772*d^2*e^9*x^(10/3) - 3080*d^3*e^8*x^3 + 3465*d^4*e^7*x^(8/3) - 3960*d^5*e^6
*x^(7/3) + 4620*d^6*e^5*x^2 - 5544*d^7*e^4*x^(5/3) + 6930*d^8*e^3*x^(4/3) - 9240*d^9*e^2*x + 13860*d^10*e*x^(2
/3) - 27720*d^11*x^(1/3))/e^12)

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Fricas [A]  time = 1.81819, size = 513, normalized size = 2.19 \begin{align*} \frac{27720 \, b e^{12} x^{4} \log \left (c\right ) + 3080 \, b d^{3} e^{9} n x^{3} - 4620 \, b d^{6} e^{6} n x^{2} + 9240 \, b d^{9} e^{3} n x - 2310 \,{\left (b e^{12} n - 12 \, a e^{12}\right )} x^{4} + 27720 \,{\left (b e^{12} n x^{4} - b d^{12} n\right )} \log \left (e x^{\frac{1}{3}} + d\right ) + 63 \,{\left (40 \, b d e^{11} n x^{3} - 55 \, b d^{4} e^{8} n x^{2} + 88 \, b d^{7} e^{5} n x - 220 \, b d^{10} e^{2} n\right )} x^{\frac{2}{3}} - 198 \,{\left (14 \, b d^{2} e^{10} n x^{3} - 20 \, b d^{5} e^{7} n x^{2} + 35 \, b d^{8} e^{4} n x - 140 \, b d^{11} e n\right )} x^{\frac{1}{3}}}{110880 \, e^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/110880*(27720*b*e^12*x^4*log(c) + 3080*b*d^3*e^9*n*x^3 - 4620*b*d^6*e^6*n*x^2 + 9240*b*d^9*e^3*n*x - 2310*(b
*e^12*n - 12*a*e^12)*x^4 + 27720*(b*e^12*n*x^4 - b*d^12*n)*log(e*x^(1/3) + d) + 63*(40*b*d*e^11*n*x^3 - 55*b*d
^4*e^8*n*x^2 + 88*b*d^7*e^5*n*x - 220*b*d^10*e^2*n)*x^(2/3) - 198*(14*b*d^2*e^10*n*x^3 - 20*b*d^5*e^7*n*x^2 +
35*b*d^8*e^4*n*x - 140*b*d^11*e*n)*x^(1/3))/e^12

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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**(1/3))**n)),x)

[Out]

Timed out

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Giac [B]  time = 1.2498, size = 714, normalized size = 3.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/110880*(27720*b*x^4*e*log(c) + 27720*a*x^4*e + (27720*(x^(1/3)*e + d)^12*e^(-11)*log(x^(1/3)*e + d) - 332640
*(x^(1/3)*e + d)^11*d*e^(-11)*log(x^(1/3)*e + d) + 1829520*(x^(1/3)*e + d)^10*d^2*e^(-11)*log(x^(1/3)*e + d) -
6098400*(x^(1/3)*e + d)^9*d^3*e^(-11)*log(x^(1/3)*e + d) + 13721400*(x^(1/3)*e + d)^8*d^4*e^(-11)*log(x^(1/3)
*e + d) - 21954240*(x^(1/3)*e + d)^7*d^5*e^(-11)*log(x^(1/3)*e + d) + 25613280*(x^(1/3)*e + d)^6*d^6*e^(-11)*l
og(x^(1/3)*e + d) - 21954240*(x^(1/3)*e + d)^5*d^7*e^(-11)*log(x^(1/3)*e + d) + 13721400*(x^(1/3)*e + d)^4*d^8
*e^(-11)*log(x^(1/3)*e + d) - 6098400*(x^(1/3)*e + d)^3*d^9*e^(-11)*log(x^(1/3)*e + d) + 1829520*(x^(1/3)*e +
d)^2*d^10*e^(-11)*log(x^(1/3)*e + d) - 332640*(x^(1/3)*e + d)*d^11*e^(-11)*log(x^(1/3)*e + d) - 2310*(x^(1/3)*
e + d)^12*e^(-11) + 30240*(x^(1/3)*e + d)^11*d*e^(-11) - 182952*(x^(1/3)*e + d)^10*d^2*e^(-11) + 677600*(x^(1/
3)*e + d)^9*d^3*e^(-11) - 1715175*(x^(1/3)*e + d)^8*d^4*e^(-11) + 3136320*(x^(1/3)*e + d)^7*d^5*e^(-11) - 4268
880*(x^(1/3)*e + d)^6*d^6*e^(-11) + 4390848*(x^(1/3)*e + d)^5*d^7*e^(-11) - 3430350*(x^(1/3)*e + d)^4*d^8*e^(-
11) + 2032800*(x^(1/3)*e + d)^3*d^9*e^(-11) - 914760*(x^(1/3)*e + d)^2*d^10*e^(-11) + 332640*(x^(1/3)*e + d)*d
^11*e^(-11))*b*n)*e^(-1)