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

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.09, 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 = {2454, 2395, 44} \[ -\frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}+\frac {b e^4 n}{4 d^4 x^{2/3}}+\frac {b e^2 n}{8 d^2 x^{4/3}}-\frac {b e^5 n}{2 d^5 \sqrt [3]{x}}-\frac {b e^3 n}{6 d^3 x}+\frac {b e^6 n \log \left (d+e \sqrt [3]{x}\right )}{2 d^6}-\frac {b e^6 n \log (x)}{6 d^6}-\frac {b e n}{10 d x^{5/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e*n)/(10*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 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])

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 \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^3} \, dx &=3 \operatorname {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) \operatorname {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) \operatorname {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*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 134, normalized size = 0.94 \[ -\frac {a}{2 x^2}-\frac {b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{2 x^2}+\frac {1}{2} b e n \left (\frac {e^5 \log \left (d+e \sqrt [3]{x}\right )}{d^6}-\frac {e^5 \log (x)}{3 d^6}-\frac {e^4}{d^5 \sqrt [3]{x}}+\frac {e^3}{2 d^4 x^{2/3}}-\frac {e^2}{3 d^3 x}+\frac {e}{4 d^2 x^{4/3}}-\frac {1}{5 d x^{5/3}}\right ) \]

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

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 125, normalized size = 0.87 \[ -\frac {60 \, b e^{6} n x^{2} \log \left (x^{\frac {1}{3}}\right ) + 20 \, b d^{3} e^{3} n x + 60 \, b d^{6} \log \relax (c) + 60 \, a d^{6} - 60 \, {\left (b e^{6} n x^{2} - b d^{6} n\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 15 \, {\left (4 \, b d e^{5} n x - b d^{4} e^{2} n\right )} x^{\frac {2}{3}} - 6 \, {\left (5 \, b d^{2} e^{4} n x - 2 \, b d^{5} e n\right )} x^{\frac {1}{3}}}{120 \, d^{6} x^{2}} \]

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

________________________________________________________________________________________

giac [B]  time = 0.21, size = 542, normalized size = 3.79 \[ \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 \relax (c) - 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 )}} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e \,x^{\frac {1}{3}}+d \right )^{n}\right )+a}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.72, size = 106, normalized size = 0.74 \[ \frac {1}{120} \, b e n {\left (\frac {60 \, e^{5} \log \left (e x^{\frac {1}{3}} + d\right )}{d^{6}} - \frac {20 \, e^{5} \log \relax (x)}{d^{6}} - \frac {60 \, e^{4} x^{\frac {4}{3}} - 30 \, d e^{3} x + 20 \, d^{2} e^{2} x^{\frac {2}{3}} - 15 \, d^{3} e x^{\frac {1}{3}} + 12 \, d^{4}}{d^{5} x^{\frac {5}{3}}}\right )} - \frac {b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )}{2 \, x^{2}} - \frac {a}{2 \, x^{2}} \]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________