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

Optimal. Leaf size=187 \[ -\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3}-\frac{b d^7 n}{6 e^7 x^{2/3}}-\frac{b d^5 n}{12 e^5 x^{4/3}}+\frac{b d^4 n}{15 e^4 x^{5/3}}-\frac{b d^3 n}{18 e^3 x^2}+\frac{b d^2 n}{21 e^2 x^{7/3}}+\frac{b d^8 n}{3 e^8 \sqrt [3]{x}}+\frac{b d^6 n}{9 e^6 x}-\frac{b d^9 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{3 e^9}-\frac{b d n}{24 e x^{8/3}}+\frac{b n}{27 x^3} \]

[Out]

(b*n)/(27*x^3) - (b*d*n)/(24*e*x^(8/3)) + (b*d^2*n)/(21*e^2*x^(7/3)) - (b*d^3*n)/(18*e^3*x^2) + (b*d^4*n)/(15*
e^4*x^(5/3)) - (b*d^5*n)/(12*e^5*x^(4/3)) + (b*d^6*n)/(9*e^6*x) - (b*d^7*n)/(6*e^7*x^(2/3)) + (b*d^8*n)/(3*e^8
*x^(1/3)) - (b*d^9*n*Log[d + e/x^(1/3)])/(3*e^9) - (a + b*Log[c*(d + e/x^(1/3))^n])/(3*x^3)

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Rubi [A]  time = 0.133426, antiderivative size = 187, 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} \[ -\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3}-\frac{b d^7 n}{6 e^7 x^{2/3}}-\frac{b d^5 n}{12 e^5 x^{4/3}}+\frac{b d^4 n}{15 e^4 x^{5/3}}-\frac{b d^3 n}{18 e^3 x^2}+\frac{b d^2 n}{21 e^2 x^{7/3}}+\frac{b d^8 n}{3 e^8 \sqrt [3]{x}}+\frac{b d^6 n}{9 e^6 x}-\frac{b d^9 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{3 e^9}-\frac{b d n}{24 e x^{8/3}}+\frac{b n}{27 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*n)/(27*x^3) - (b*d*n)/(24*e*x^(8/3)) + (b*d^2*n)/(21*e^2*x^(7/3)) - (b*d^3*n)/(18*e^3*x^2) + (b*d^4*n)/(15*
e^4*x^(5/3)) - (b*d^5*n)/(12*e^5*x^(4/3)) + (b*d^6*n)/(9*e^6*x) - (b*d^7*n)/(6*e^7*x^(2/3)) + (b*d^8*n)/(3*e^8
*x^(1/3)) - (b*d^9*n*Log[d + e/x^(1/3)])/(3*e^9) - (a + b*Log[c*(d + e/x^(1/3))^n])/(3*x^3)

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 \frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{x^4} \, dx &=-\left (3 \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\right )\\ &=-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3}+\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \frac{x^9}{d+e x} \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\\ &=-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3}+\frac{1}{3} (b e n) \operatorname{Subst}\left (\int \left (\frac{d^8}{e^9}-\frac{d^7 x}{e^8}+\frac{d^6 x^2}{e^7}-\frac{d^5 x^3}{e^6}+\frac{d^4 x^4}{e^5}-\frac{d^3 x^5}{e^4}+\frac{d^2 x^6}{e^3}-\frac{d x^7}{e^2}+\frac{x^8}{e}-\frac{d^9}{e^9 (d+e x)}\right ) \, dx,x,\frac{1}{\sqrt [3]{x}}\right )\\ &=\frac{b n}{27 x^3}-\frac{b d n}{24 e x^{8/3}}+\frac{b d^2 n}{21 e^2 x^{7/3}}-\frac{b d^3 n}{18 e^3 x^2}+\frac{b d^4 n}{15 e^4 x^{5/3}}-\frac{b d^5 n}{12 e^5 x^{4/3}}+\frac{b d^6 n}{9 e^6 x}-\frac{b d^7 n}{6 e^7 x^{2/3}}+\frac{b d^8 n}{3 e^8 \sqrt [3]{x}}-\frac{b d^9 n \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{3 e^9}-\frac{a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.150398, size = 178, normalized size = 0.95 \[ -\frac{a}{3 x^3}-\frac{b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3}+\frac{1}{3} b e n \left (-\frac{d^7}{2 e^8 x^{2/3}}-\frac{d^5}{4 e^6 x^{4/3}}+\frac{d^4}{5 e^5 x^{5/3}}-\frac{d^3}{6 e^4 x^2}+\frac{d^2}{7 e^3 x^{7/3}}+\frac{d^8}{e^9 \sqrt [3]{x}}+\frac{d^6}{3 e^7 x}-\frac{d^9 \log \left (d+\frac{e}{\sqrt [3]{x}}\right )}{e^{10}}-\frac{d}{8 e^2 x^{8/3}}+\frac{1}{9 e x^3}\right ) \]

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

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Maple [F]  time = 0.38, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\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 = 1.04253, size = 203, normalized size = 1.09 \begin{align*} -\frac{1}{7560} \, b e n{\left (\frac{2520 \, d^{9} \log \left (d x^{\frac{1}{3}} + e\right )}{e^{10}} - \frac{840 \, d^{9} \log \left (x\right )}{e^{10}} - \frac{2520 \, d^{8} x^{\frac{8}{3}} - 1260 \, d^{7} e x^{\frac{7}{3}} + 840 \, d^{6} e^{2} x^{2} - 630 \, d^{5} e^{3} x^{\frac{5}{3}} + 504 \, d^{4} e^{4} x^{\frac{4}{3}} - 420 \, d^{3} e^{5} x + 360 \, d^{2} e^{6} x^{\frac{2}{3}} - 315 \, d e^{7} x^{\frac{1}{3}} + 280 \, e^{8}}{e^{9} x^{3}}\right )} - \frac{b \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\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]

-1/7560*b*e*n*(2520*d^9*log(d*x^(1/3) + e)/e^10 - 840*d^9*log(x)/e^10 - (2520*d^8*x^(8/3) - 1260*d^7*e*x^(7/3)
+ 840*d^6*e^2*x^2 - 630*d^5*e^3*x^(5/3) + 504*d^4*e^4*x^(4/3) - 420*d^3*e^5*x + 360*d^2*e^6*x^(2/3) - 315*d*e
^7*x^(1/3) + 280*e^8)/(e^9*x^3)) - 1/3*b*log(c*(d + e/x^(1/3))^n)/x^3 - 1/3*a/x^3

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Fricas [A]  time = 1.81207, size = 502, normalized size = 2.68 \begin{align*} \frac{840 \, b d^{6} e^{3} n x^{2} - 420 \, b d^{3} e^{6} n x + 280 \, b e^{9} n - 2520 \, a e^{9} + 140 \,{\left (18 \, a e^{9} -{\left (6 \, b d^{6} e^{3} - 3 \, b d^{3} e^{6} + 2 \, b e^{9}\right )} n\right )} x^{3} + 2520 \,{\left (b e^{9} x^{3} - b e^{9}\right )} \log \left (c\right ) - 2520 \,{\left (b d^{9} n x^{3} + b e^{9} n\right )} \log \left (\frac{d x + e x^{\frac{2}{3}}}{x}\right ) + 90 \,{\left (28 \, b d^{8} e n x^{2} - 7 \, b d^{5} e^{4} n x + 4 \, b d^{2} e^{7} n\right )} x^{\frac{2}{3}} - 63 \,{\left (20 \, b d^{7} e^{2} n x^{2} - 8 \, b d^{4} e^{5} n x + 5 \, b d e^{8} n\right )} x^{\frac{1}{3}}}{7560 \, e^{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]

1/7560*(840*b*d^6*e^3*n*x^2 - 420*b*d^3*e^6*n*x + 280*b*e^9*n - 2520*a*e^9 + 140*(18*a*e^9 - (6*b*d^6*e^3 - 3*
b*d^3*e^6 + 2*b*e^9)*n)*x^3 + 2520*(b*e^9*x^3 - b*e^9)*log(c) - 2520*(b*d^9*n*x^3 + b*e^9*n)*log((d*x + e*x^(2
/3))/x) + 90*(28*b*d^8*e*n*x^2 - 7*b*d^5*e^4*n*x + 4*b*d^2*e^7*n)*x^(2/3) - 63*(20*b*d^7*e^2*n*x^2 - 8*b*d^4*e
^5*n*x + 5*b*d*e^8*n)*x^(1/3))/(e^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 [A]  time = 1.38991, size = 207, normalized size = 1.11 \begin{align*} -\frac{1}{7560} \,{\left ({\left (2520 \, d^{9} e^{\left (-10\right )} \log \left ({\left | d x^{\frac{1}{3}} + e \right |}\right ) - 840 \, d^{9} e^{\left (-10\right )} \log \left ({\left | x \right |}\right ) - \frac{{\left (2520 \, d^{8} x^{\frac{8}{3}} e - 1260 \, d^{7} x^{\frac{7}{3}} e^{2} + 840 \, d^{6} x^{2} e^{3} - 630 \, d^{5} x^{\frac{5}{3}} e^{4} + 504 \, d^{4} x^{\frac{4}{3}} e^{5} - 420 \, d^{3} x e^{6} + 360 \, d^{2} x^{\frac{2}{3}} e^{7} - 315 \, d x^{\frac{1}{3}} e^{8} + 280 \, e^{9}\right )} e^{\left (-10\right )}}{x^{3}}\right )} e + \frac{2520 \, \log \left (d + \frac{e}{x^{\frac{1}{3}}}\right )}{x^{3}}\right )} b n - \frac{b \log \left (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="giac")

[Out]

-1/7560*((2520*d^9*e^(-10)*log(abs(d*x^(1/3) + e)) - 840*d^9*e^(-10)*log(abs(x)) - (2520*d^8*x^(8/3)*e - 1260*
d^7*x^(7/3)*e^2 + 840*d^6*x^2*e^3 - 630*d^5*x^(5/3)*e^4 + 504*d^4*x^(4/3)*e^5 - 420*d^3*x*e^6 + 360*d^2*x^(2/3
)*e^7 - 315*d*x^(1/3)*e^8 + 280*e^9)*e^(-10)/x^3)*e + 2520*log(d + e/x^(1/3))/x^3)*b*n - 1/3*b*log(c)/x^3 - 1/
3*a/x^3