3.449 \(\int \frac{a+b \log (c (d+e \sqrt [3]{x})^n)}{x^4} \, dx\)

Optimal. Leaf size=192 \[ -\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}-\frac{b e^7 n}{6 d^7 x^{2/3}}-\frac{b e^5 n}{12 d^5 x^{4/3}}+\frac{b e^4 n}{15 d^4 x^{5/3}}-\frac{b e^3 n}{18 d^3 x^2}+\frac{b e^2 n}{21 d^2 x^{7/3}}+\frac{b e^8 n}{3 d^8 \sqrt [3]{x}}+\frac{b e^6 n}{9 d^6 x}-\frac{b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}+\frac{b e^9 n \log (x)}{9 d^9}-\frac{b e n}{24 d x^{8/3}} \]

[Out]

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

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Rubi [A]  time = 0.125405, antiderivative size = 192, 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, 44} \[ -\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}-\frac{b e^7 n}{6 d^7 x^{2/3}}-\frac{b e^5 n}{12 d^5 x^{4/3}}+\frac{b e^4 n}{15 d^4 x^{5/3}}-\frac{b e^3 n}{18 d^3 x^2}+\frac{b e^2 n}{21 d^2 x^{7/3}}+\frac{b e^8 n}{3 d^8 \sqrt [3]{x}}+\frac{b e^6 n}{9 d^6 x}-\frac{b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}+\frac{b e^9 n \log (x)}{9 d^9}-\frac{b e n}{24 d x^{8/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x^(1/3))^n])/x^4,x]

[Out]

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

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^4} \, dx &=3 \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \frac{1}{x^9 (d+e x)} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \left (\frac{1}{d x^9}-\frac{e}{d^2 x^8}+\frac{e^2}{d^3 x^7}-\frac{e^3}{d^4 x^6}+\frac{e^4}{d^5 x^5}-\frac{e^5}{d^6 x^4}+\frac{e^6}{d^7 x^3}-\frac{e^7}{d^8 x^2}+\frac{e^8}{d^9 x}-\frac{e^9}{d^9 (d+e x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{b e n}{24 d x^{8/3}}+\frac{b e^2 n}{21 d^2 x^{7/3}}-\frac{b e^3 n}{18 d^3 x^2}+\frac{b e^4 n}{15 d^4 x^{5/3}}-\frac{b e^5 n}{12 d^5 x^{4/3}}+\frac{b e^6 n}{9 d^6 x}-\frac{b e^7 n}{6 d^7 x^{2/3}}+\frac{b e^8 n}{3 d^8 \sqrt [3]{x}}-\frac{b e^9 n \log \left (d+e \sqrt [3]{x}\right )}{3 d^9}-\frac{a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac{b e^9 n \log (x)}{9 d^9}\\ \end{align*}

Mathematica [A]  time = 0.191021, size = 177, normalized size = 0.92 \[ -\frac{a}{3 x^3}-\frac{b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{3 x^3}+\frac{1}{3} b e n \left (-\frac{e^6}{2 d^7 x^{2/3}}-\frac{e^4}{4 d^5 x^{4/3}}+\frac{e^3}{5 d^4 x^{5/3}}-\frac{e^2}{6 d^3 x^2}+\frac{e^7}{d^8 \sqrt [3]{x}}+\frac{e^5}{3 d^6 x}-\frac{e^8 \log \left (d+e \sqrt [3]{x}\right )}{d^9}+\frac{e^8 \log (x)}{3 d^9}+\frac{e}{7 d^2 x^{7/3}}-\frac{1}{8 d x^{8/3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x^(1/3))^n])/x^4,x]

[Out]

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

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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((a+b*ln(c*(d+e*x^(1/3))^n))/x^4,x)

[Out]

int((a+b*ln(c*(d+e*x^(1/3))^n))/x^4,x)

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Maxima [A]  time = 1.03738, size = 188, normalized size = 0.98 \begin{align*} -\frac{1}{2520} \, b e n{\left (\frac{840 \, e^{8} \log \left (e x^{\frac{1}{3}} + d\right )}{d^{9}} - \frac{280 \, e^{8} \log \left (x\right )}{d^{9}} - \frac{840 \, e^{7} x^{\frac{7}{3}} - 420 \, d e^{6} x^{2} + 280 \, d^{2} e^{5} x^{\frac{5}{3}} - 210 \, d^{3} e^{4} x^{\frac{4}{3}} + 168 \, d^{4} e^{3} x - 140 \, d^{5} e^{2} x^{\frac{2}{3}} + 120 \, d^{6} e x^{\frac{1}{3}} - 105 \, d^{7}}{d^{8} x^{\frac{8}{3}}}\right )} - \frac{b \log \left ({\left (e x^{\frac{1}{3}} + d\right )}^{n} c\right )}{3 \, x^{3}} - \frac{a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/3))^n))/x^4,x, algorithm="maxima")

[Out]

-1/2520*b*e*n*(840*e^8*log(e*x^(1/3) + d)/d^9 - 280*e^8*log(x)/d^9 - (840*e^7*x^(7/3) - 420*d*e^6*x^2 + 280*d^
2*e^5*x^(5/3) - 210*d^3*e^4*x^(4/3) + 168*d^4*e^3*x - 140*d^5*e^2*x^(2/3) + 120*d^6*e*x^(1/3) - 105*d^7)/(d^8*
x^(8/3))) - 1/3*b*log((e*x^(1/3) + d)^n*c)/x^3 - 1/3*a/x^3

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Fricas [A]  time = 1.83991, size = 409, normalized size = 2.13 \begin{align*} \frac{840 \, b e^{9} n x^{3} \log \left (x^{\frac{1}{3}}\right ) + 280 \, b d^{3} e^{6} n x^{2} - 140 \, b d^{6} e^{3} n x - 840 \, b d^{9} \log \left (c\right ) - 840 \, a d^{9} - 840 \,{\left (b e^{9} n x^{3} + b d^{9} n\right )} \log \left (e x^{\frac{1}{3}} + d\right ) + 30 \,{\left (28 \, b d e^{8} n x^{2} - 7 \, b d^{4} e^{5} n x + 4 \, b d^{7} e^{2} n\right )} x^{\frac{2}{3}} - 21 \,{\left (20 \, b d^{2} e^{7} n x^{2} - 8 \, b d^{5} e^{4} n x + 5 \, b d^{8} e n\right )} x^{\frac{1}{3}}}{2520 \, d^{9} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/3))^n))/x^4,x, algorithm="fricas")

[Out]

1/2520*(840*b*e^9*n*x^3*log(x^(1/3)) + 280*b*d^3*e^6*n*x^2 - 140*b*d^6*e^3*n*x - 840*b*d^9*log(c) - 840*a*d^9
- 840*(b*e^9*n*x^3 + b*d^9*n)*log(e*x^(1/3) + d) + 30*(28*b*d*e^8*n*x^2 - 7*b*d^4*e^5*n*x + 4*b*d^7*e^2*n)*x^(
2/3) - 21*(20*b*d^2*e^7*n*x^2 - 8*b*d^5*e^4*n*x + 5*b*d^8*e*n)*x^(1/3))/(d^9*x^3)

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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((a+b*ln(c*(d+e*x**(1/3))**n))/x**4,x)

[Out]

Timed out

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Giac [B]  time = 1.31043, size = 1091, normalized size = 5.68 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^(1/3))^n))/x^4,x, algorithm="giac")

[Out]

-1/2520*(840*(x^(1/3)*e + d)^9*b*n*e^10*log(x^(1/3)*e + d) - 7560*(x^(1/3)*e + d)^8*b*d*n*e^10*log(x^(1/3)*e +
d) + 30240*(x^(1/3)*e + d)^7*b*d^2*n*e^10*log(x^(1/3)*e + d) - 70560*(x^(1/3)*e + d)^6*b*d^3*n*e^10*log(x^(1/
3)*e + d) + 105840*(x^(1/3)*e + d)^5*b*d^4*n*e^10*log(x^(1/3)*e + d) - 105840*(x^(1/3)*e + d)^4*b*d^5*n*e^10*l
og(x^(1/3)*e + d) + 70560*(x^(1/3)*e + d)^3*b*d^6*n*e^10*log(x^(1/3)*e + d) - 30240*(x^(1/3)*e + d)^2*b*d^7*n*
e^10*log(x^(1/3)*e + d) + 7560*(x^(1/3)*e + d)*b*d^8*n*e^10*log(x^(1/3)*e + d) - 840*(x^(1/3)*e + d)^9*b*n*e^1
0*log(x^(1/3)*e) + 7560*(x^(1/3)*e + d)^8*b*d*n*e^10*log(x^(1/3)*e) - 30240*(x^(1/3)*e + d)^7*b*d^2*n*e^10*log
(x^(1/3)*e) + 70560*(x^(1/3)*e + d)^6*b*d^3*n*e^10*log(x^(1/3)*e) - 105840*(x^(1/3)*e + d)^5*b*d^4*n*e^10*log(
x^(1/3)*e) + 105840*(x^(1/3)*e + d)^4*b*d^5*n*e^10*log(x^(1/3)*e) - 70560*(x^(1/3)*e + d)^3*b*d^6*n*e^10*log(x
^(1/3)*e) + 30240*(x^(1/3)*e + d)^2*b*d^7*n*e^10*log(x^(1/3)*e) - 7560*(x^(1/3)*e + d)*b*d^8*n*e^10*log(x^(1/3
)*e) + 840*b*d^9*n*e^10*log(x^(1/3)*e) - 840*(x^(1/3)*e + d)^8*b*d*n*e^10 + 7140*(x^(1/3)*e + d)^7*b*d^2*n*e^1
0 - 26740*(x^(1/3)*e + d)^6*b*d^3*n*e^10 + 57750*(x^(1/3)*e + d)^5*b*d^4*n*e^10 - 78918*(x^(1/3)*e + d)^4*b*d^
5*n*e^10 + 70252*(x^(1/3)*e + d)^3*b*d^6*n*e^10 - 40188*(x^(1/3)*e + d)^2*b*d^7*n*e^10 + 13827*(x^(1/3)*e + d)
*b*d^8*n*e^10 - 2283*b*d^9*n*e^10 + 840*b*d^9*e^10*log(c) + 840*a*d^9*e^10)*e^(-1)/((x^(1/3)*e + d)^9*d^9 - 9*
(x^(1/3)*e + d)^8*d^10 + 36*(x^(1/3)*e + d)^7*d^11 - 84*(x^(1/3)*e + d)^6*d^12 + 126*(x^(1/3)*e + d)^5*d^13 -
126*(x^(1/3)*e + d)^4*d^14 + 84*(x^(1/3)*e + d)^3*d^15 - 36*(x^(1/3)*e + d)^2*d^16 + 9*(x^(1/3)*e + d)*d^17 -
d^18)