3.5.96 \(\int \frac {a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n)}{x^4} \, dx\) [496]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 187, 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, 45} \begin {gather*} -\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3}-\frac {b d^9 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{3 e^9}+\frac {b d^8 n}{3 e^8 \sqrt [3]{x}}-\frac {b d^7 n}{6 e^7 x^{2/3}}+\frac {b d^6 n}{9 e^6 x}-\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 n}{24 e x^{8/3}}+\frac {b n}{27 x^3} \end {gather*}

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 45

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])

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+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^4} \, dx &=-\left (3 \text {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) \text {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) \text {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*}

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Mathematica [A]
time = 0.11, size = 178, normalized size = 0.95 \begin {gather*} -\frac {a}{3 x^3}+\frac {1}{3} b e n \left (\frac {1}{9 e x^3}-\frac {d}{8 e^2 x^{8/3}}+\frac {d^2}{7 e^3 x^{7/3}}-\frac {d^3}{6 e^4 x^2}+\frac {d^4}{5 e^5 x^{5/3}}-\frac {d^5}{4 e^6 x^{4/3}}+\frac {d^6}{3 e^7 x}-\frac {d^7}{2 e^8 x^{2/3}}+\frac {d^8}{e^9 \sqrt [3]{x}}-\frac {d^9 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^{10}}\right )-\frac {b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*a/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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )}{x^{4}}\, dx\]

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 = 0.29, size = 144, normalized size = 0.77 \begin {gather*} -\frac {1}{7560} \, {\left (2520 \, d^{9} e^{\left (-10\right )} \log \left (d x^{\frac {1}{3}} + e\right ) - 840 \, d^{9} e^{\left (-10\right )} \log \left (x\right ) - \frac {{\left (2520 \, d^{8} x^{\frac {8}{3}} - 1260 \, d^{7} x^{\frac {7}{3}} e + 840 \, d^{6} x^{2} e^{2} - 630 \, d^{5} x^{\frac {5}{3}} e^{3} + 504 \, d^{4} x^{\frac {4}{3}} e^{4} - 420 \, d^{3} x e^{5} + 360 \, d^{2} x^{\frac {2}{3}} e^{6} - 315 \, d x^{\frac {1}{3}} e^{7} + 280 \, e^{8}\right )} e^{\left (-9\right )}}{x^{3}}\right )} b n e - \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 {gather*}

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*(2520*d^9*e^(-10)*log(d*x^(1/3) + e) - 840*d^9*e^(-10)*log(x) - (2520*d^8*x^(8/3) - 1260*d^7*x^(7/3)*e
+ 840*d^6*x^2*e^2 - 630*d^5*x^(5/3)*e^3 + 504*d^4*x^(4/3)*e^4 - 420*d^3*x*e^5 + 360*d^2*x^(2/3)*e^6 - 315*d*x
^(1/3)*e^7 + 280*e^8)*e^(-9)/x^3)*b*n*e - 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 = 0.39, size = 196, normalized size = 1.05 \begin {gather*} \frac {{\left (2520 \, {\left (b x^{3} - b\right )} e^{9} \log \left (c\right ) - 280 \, {\left ({\left (b n - 9 \, a\right )} x^{3} - b n + 9 \, a\right )} e^{9} + 420 \, {\left (b d^{3} n x^{3} - b d^{3} n x\right )} e^{6} - 840 \, {\left (b d^{6} n x^{3} - b d^{6} n x^{2}\right )} e^{3} - 2520 \, {\left (b d^{9} n x^{3} + b n e^{9}\right )} \log \left (\frac {d x + x^{\frac {2}{3}} e}{x}\right ) + 90 \, {\left (28 \, b d^{8} n x^{2} e - 7 \, b d^{5} n x e^{4} + 4 \, b d^{2} n e^{7}\right )} x^{\frac {2}{3}} - 63 \, {\left (20 \, b d^{7} n x^{2} e^{2} - 8 \, b d^{4} n x e^{5} + 5 \, b d n e^{8}\right )} x^{\frac {1}{3}}\right )} e^{\left (-9\right )}}{7560 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

1/7560*(2520*(b*x^3 - b)*e^9*log(c) - 280*((b*n - 9*a)*x^3 - b*n + 9*a)*e^9 + 420*(b*d^3*n*x^3 - b*d^3*n*x)*e^
6 - 840*(b*d^6*n*x^3 - b*d^6*n*x^2)*e^3 - 2520*(b*d^9*n*x^3 + b*n*e^9)*log((d*x + x^(2/3)*e)/x) + 90*(28*b*d^8
*n*x^2*e - 7*b*d^5*n*x*e^4 + 4*b*d^2*n*e^7)*x^(2/3) - 63*(20*b*d^7*n*x^2*e^2 - 8*b*d^4*n*x*e^5 + 5*b*d*n*e^8)*
x^(1/3))*e^(-9)/x^3

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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**4,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5987 deep

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Giac [A]
time = 3.43, size = 153, normalized size = 0.82 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.53, size = 152, normalized size = 0.81 \begin {gather*} \frac {b\,n}{27\,x^3}-\frac {a}{3\,x^3}-\frac {b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}{3\,x^3}-\frac {b\,d\,n}{24\,e\,x^{8/3}}-\frac {b\,d^9\,n\,\ln \left (d+\frac {e}{x^{1/3}}\right )}{3\,e^9}-\frac {b\,d^3\,n}{18\,e^3\,x^2}+\frac {b\,d^6\,n}{9\,e^6\,x}+\frac {b\,d^2\,n}{21\,e^2\,x^{7/3}}+\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^7\,n}{6\,e^7\,x^{2/3}}+\frac {b\,d^8\,n}{3\,e^8\,x^{1/3}} \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^4,x)

[Out]

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

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