3.5.48 \(\int \frac {a+b \log (c (d+e \sqrt [3]{x})^n)}{x^3} \, dx\) [448]

Optimal. Leaf size=143 \[ -\frac {b e n}{10 d x^{5/3}}+\frac {b e^2 n}{8 d^2 x^{4/3}}-\frac {b e^3 n}{6 d^3 x}+\frac {b e^4 n}{4 d^4 x^{2/3}}-\frac {b e^5 n}{2 d^5 \sqrt [3]{x}}+\frac {b e^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 d^6}-\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}-\frac {b e^6 n \log (x)}{6 d^6} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 143, 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 = {2504, 2442, 46} \begin {gather*} -\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}+\frac {b e^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 d^6}-\frac {b e^6 n \log (x)}{6 d^6}-\frac {b e^5 n}{2 d^5 \sqrt [3]{x}}+\frac {b e^4 n}{4 d^4 x^{2/3}}-\frac {b e^3 n}{6 d^3 x}+\frac {b e^2 n}{8 d^2 x^{4/3}}-\frac {b e n}{10 d x^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 2442

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 2504

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

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Mathematica [A]
time = 0.10, size = 134, normalized size = 0.94 \begin {gather*} -\frac {a}{2 x^2}-\frac {b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}+\frac {1}{2} b e n \left (-\frac {1}{5 d x^{5/3}}+\frac {e}{4 d^2 x^{4/3}}-\frac {e^2}{3 d^3 x}+\frac {e^3}{2 d^4 x^{2/3}}-\frac {e^4}{d^5 \sqrt [3]{x}}+\frac {e^5 \log \left (d+e \sqrt [3]{x}\right )}{d^6}-\frac {e^5 \log (x)}{3 d^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 104, normalized size = 0.73 \begin {gather*} \frac {1}{120} \, b n {\left (\frac {60 \, e^{5} \log \left (x^{\frac {1}{3}} e + d\right )}{d^{6}} - \frac {20 \, e^{5} \log \left (x\right )}{d^{6}} + \frac {15 \, d^{3} x^{\frac {1}{3}} e - 12 \, d^{4} - 20 \, d^{2} x^{\frac {2}{3}} e^{2} + 30 \, d x e^{3} - 60 \, x^{\frac {4}{3}} e^{4}}{d^{5} x^{\frac {5}{3}}}\right )} e - \frac {b \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 120, normalized size = 0.84 \begin {gather*} -\frac {60 \, b d^{6} \log \left (c\right ) + 60 \, a d^{6} + 20 \, b d^{3} n x e^{3} + 60 \, b n x^{2} e^{6} \log \left (x^{\frac {1}{3}}\right ) + 60 \, {\left (b d^{6} n - b n x^{2} e^{6}\right )} \log \left (x^{\frac {1}{3}} e + d\right ) - 15 \, {\left (b d^{4} n e^{2} - 4 \, b d n x e^{5}\right )} x^{\frac {2}{3}} + 6 \, {\left (2 \, b d^{5} n e - 5 \, b d^{2} n x e^{4}\right )} x^{\frac {1}{3}}}{120 \, d^{6} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (112) = 224\).
time = 3.93, size = 542, normalized size = 3.79 \begin {gather*} \frac {{\left (60 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} b n e^{7} \log \left (x^{\frac {1}{3}} e + d\right ) - 360 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} b d n e^{7} \log \left (x^{\frac {1}{3}} e + d\right ) + 900 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} b d^{2} n e^{7} \log \left (x^{\frac {1}{3}} e + d\right ) - 1200 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} b d^{3} n e^{7} \log \left (x^{\frac {1}{3}} e + d\right ) + 900 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} b d^{4} n e^{7} \log \left (x^{\frac {1}{3}} e + d\right ) - 360 \, {\left (x^{\frac {1}{3}} e + d\right )} b d^{5} n e^{7} \log \left (x^{\frac {1}{3}} e + d\right ) - 60 \, {\left (x^{\frac {1}{3}} e + d\right )}^{6} b n e^{7} \log \left (x^{\frac {1}{3}} e\right ) + 360 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} b d n e^{7} \log \left (x^{\frac {1}{3}} e\right ) - 900 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} b d^{2} n e^{7} \log \left (x^{\frac {1}{3}} e\right ) + 1200 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} b d^{3} n e^{7} \log \left (x^{\frac {1}{3}} e\right ) - 900 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} b d^{4} n e^{7} \log \left (x^{\frac {1}{3}} e\right ) + 360 \, {\left (x^{\frac {1}{3}} e + d\right )} b d^{5} n e^{7} \log \left (x^{\frac {1}{3}} e\right ) - 60 \, b d^{6} n e^{7} \log \left (x^{\frac {1}{3}} e\right ) - 60 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} b d n e^{7} + 330 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} b d^{2} n e^{7} - 740 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} b d^{3} n e^{7} + 855 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} b d^{4} n e^{7} - 522 \, {\left (x^{\frac {1}{3}} e + d\right )} b d^{5} n e^{7} + 137 \, b d^{6} n e^{7} - 60 \, b d^{6} e^{7} \log \left (c\right ) - 60 \, a d^{6} e^{7}\right )} e^{\left (-1\right )}}{120 \, {\left ({\left (x^{\frac {1}{3}} e + d\right )}^{6} d^{6} - 6 \, {\left (x^{\frac {1}{3}} e + d\right )}^{5} d^{7} + 15 \, {\left (x^{\frac {1}{3}} e + d\right )}^{4} d^{8} - 20 \, {\left (x^{\frac {1}{3}} e + d\right )}^{3} d^{9} + 15 \, {\left (x^{\frac {1}{3}} e + d\right )}^{2} d^{10} - 6 \, {\left (x^{\frac {1}{3}} e + d\right )} d^{11} + d^{12}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(60*(x^(1/3)*e + d)^6*b*n*e^7*log(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)^5*b*d*n*e^7*log(x^(1/3)*e + d) +
900*(x^(1/3)*e + d)^4*b*d^2*n*e^7*log(x^(1/3)*e + d) - 1200*(x^(1/3)*e + d)^3*b*d^3*n*e^7*log(x^(1/3)*e + d) +
900*(x^(1/3)*e + d)^2*b*d^4*n*e^7*log(x^(1/3)*e + d) - 360*(x^(1/3)*e + d)*b*d^5*n*e^7*log(x^(1/3)*e + d) - 6
0*(x^(1/3)*e + d)^6*b*n*e^7*log(x^(1/3)*e) + 360*(x^(1/3)*e + d)^5*b*d*n*e^7*log(x^(1/3)*e) - 900*(x^(1/3)*e +
d)^4*b*d^2*n*e^7*log(x^(1/3)*e) + 1200*(x^(1/3)*e + d)^3*b*d^3*n*e^7*log(x^(1/3)*e) - 900*(x^(1/3)*e + d)^2*b
*d^4*n*e^7*log(x^(1/3)*e) + 360*(x^(1/3)*e + d)*b*d^5*n*e^7*log(x^(1/3)*e) - 60*b*d^6*n*e^7*log(x^(1/3)*e) - 6
0*(x^(1/3)*e + d)^5*b*d*n*e^7 + 330*(x^(1/3)*e + d)^4*b*d^2*n*e^7 - 740*(x^(1/3)*e + d)^3*b*d^3*n*e^7 + 855*(x
^(1/3)*e + d)^2*b*d^4*n*e^7 - 522*(x^(1/3)*e + d)*b*d^5*n*e^7 + 137*b*d^6*n*e^7 - 60*b*d^6*e^7*log(c) - 60*a*d
^6*e^7)*e^(-1)/((x^(1/3)*e + d)^6*d^6 - 6*(x^(1/3)*e + d)^5*d^7 + 15*(x^(1/3)*e + d)^4*d^8 - 20*(x^(1/3)*e + d
)^3*d^9 + 15*(x^(1/3)*e + d)^2*d^10 - 6*(x^(1/3)*e + d)*d^11 + d^12)

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Mupad [B]
time = 0.66, size = 109, normalized size = 0.76 \begin {gather*} \frac {b\,e^6\,n\,\mathrm {atanh}\left (\frac {2\,e\,x^{1/3}}{d}+1\right )}{d^6}-\frac {\frac {b\,e\,n}{5\,d}-\frac {b\,e^4\,n\,x}{2\,d^4}-\frac {b\,e^2\,n\,x^{1/3}}{4\,d^2}+\frac {b\,e^3\,n\,x^{2/3}}{3\,d^3}+\frac {b\,e^5\,n\,x^{4/3}}{d^5}}{2\,x^{5/3}}-\frac {b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2\,x^2}-\frac {a}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*e^6*n*atanh((2*e*x^(1/3))/d + 1))/d^6 - ((b*e*n)/(5*d) - (b*e^4*n*x)/(2*d^4) - (b*e^2*n*x^(1/3))/(4*d^2) +
(b*e^3*n*x^(2/3))/(3*d^3) + (b*e^5*n*x^(4/3))/d^5)/(2*x^(5/3)) - (b*log(c*(d + e*x^(1/3))^n))/(2*x^2) - a/(2*x
^2)

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