\(\int \frac {a+b \log (c (d+\frac {e}{x^{2/3}})^n)}{x^4} \, dx\) [515]

Optimal result

Integrand size = 22, antiderivative size = 132 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=\frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}+\frac {2 b d^{9/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{9/2}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2505, 269, 348, 331, 211} \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}+\frac {2 b d^{9/2} n \arctan \left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{9/2}}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b n}{27 x^3} \]

[In]

Int[(a + b*Log[c*(d + e/x^(2/3))^n])/x^4,x]

[Out]

(2*b*n)/(27*x^3) - (2*b*d*n)/(21*e*x^(7/3)) + (2*b*d^2*n)/(15*e^2*x^(5/3)) - (2*b*d^3*n)/(9*e^3*x) + (2*b*d^4*
n)/(3*e^4*x^(1/3)) + (2*b*d^(9/2)*n*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(3*e^(9/2)) - (a + b*Log[c*(d + e/x^(2/
3))^n])/(3*x^3)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 348

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {1}{9} (2 b e n) \int \frac {1}{\left (d+\frac {e}{x^{2/3}}\right ) x^{14/3}} \, dx \\ & = -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {1}{9} (2 b e n) \int \frac {1}{\left (e+d x^{2/3}\right ) x^4} \, dx \\ & = -\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {1}{3} (2 b e n) \text {Subst}\left (\int \frac {1}{x^{10} \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {2 b n}{27 x^3}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}+\frac {1}{3} (2 b d n) \text {Subst}\left (\int \frac {1}{x^8 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {\left (2 b d^2 n\right ) \text {Subst}\left (\int \frac {1}{x^6 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 e} \\ & = \frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}+\frac {\left (2 b d^3 n\right ) \text {Subst}\left (\int \frac {1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 e^2} \\ & = \frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d^3 n}{9 e^3 x}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}-\frac {\left (2 b d^4 n\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 e^3} \\ & = \frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3}+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4} \\ & = \frac {2 b n}{27 x^3}-\frac {2 b d n}{21 e x^{7/3}}+\frac {2 b d^2 n}{15 e^2 x^{5/3}}-\frac {2 b d^3 n}{9 e^3 x}+\frac {2 b d^4 n}{3 e^4 \sqrt [3]{x}}+\frac {2 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {d} \sqrt [3]{x}}{\sqrt {e}}\right )}{3 e^{9/2}}-\frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {2}{9} b e n \left (-\frac {1}{3 e x^3}+\frac {3 d}{7 e^2 x^{7/3}}-\frac {3 d^2}{5 e^3 x^{5/3}}+\frac {d^3}{e^4 x}-\frac {3 d^4}{e^5 \sqrt [3]{x}}+\frac {3 d^{9/2} \arctan \left (\frac {\sqrt {e}}{\sqrt {d} \sqrt [3]{x}}\right )}{e^{11/2}}\right )-\frac {b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{3 x^3} \]

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])/x^4,x]

[Out]

-1/3*a/x^3 - (2*b*e*n*(-1/3*1/(e*x^3) + (3*d)/(7*e^2*x^(7/3)) - (3*d^2)/(5*e^3*x^(5/3)) + d^3/(e^4*x) - (3*d^4
)/(e^5*x^(1/3)) + (3*d^(9/2)*ArcTan[Sqrt[e]/(Sqrt[d]*x^(1/3))])/e^(11/2)))/9 - (b*Log[c*(d + e/x^(2/3))^n])/(3
*x^3)

Maple [F]

\[\int \frac {a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )}{x^{4}}d x\]

[In]

int((a+b*ln(c*(d+e/x^(2/3))^n))/x^4,x)

[Out]

int((a+b*ln(c*(d+e/x^(2/3))^n))/x^4,x)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.57 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=\left [\frac {315 \, b d^{4} n x^{3} \sqrt {-\frac {d}{e}} \log \left (\frac {d^{3} x^{2} - 2 \, d e^{2} x \sqrt {-\frac {d}{e}} - e^{3} + 2 \, {\left (d^{2} e x \sqrt {-\frac {d}{e}} + d e^{2}\right )} x^{\frac {2}{3}} - 2 \, {\left (d^{2} e x - e^{3} \sqrt {-\frac {d}{e}}\right )} x^{\frac {1}{3}}}{d^{3} x^{2} + e^{3}}\right ) - 210 \, b d^{3} e n x^{2} + 126 \, b d^{2} e^{2} n x^{\frac {4}{3}} - 315 \, b e^{4} n \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) + 70 \, b e^{4} n - 315 \, b e^{4} \log \left (c\right ) - 315 \, a e^{4} + 90 \, {\left (7 \, b d^{4} n x^{2} - b d e^{3} n\right )} x^{\frac {2}{3}}}{945 \, e^{4} x^{3}}, \frac {630 \, b d^{4} n x^{3} \sqrt {\frac {d}{e}} \arctan \left (x^{\frac {1}{3}} \sqrt {\frac {d}{e}}\right ) - 210 \, b d^{3} e n x^{2} + 126 \, b d^{2} e^{2} n x^{\frac {4}{3}} - 315 \, b e^{4} n \log \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right ) + 70 \, b e^{4} n - 315 \, b e^{4} \log \left (c\right ) - 315 \, a e^{4} + 90 \, {\left (7 \, b d^{4} n x^{2} - b d e^{3} n\right )} x^{\frac {2}{3}}}{945 \, e^{4} x^{3}}\right ] \]

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))/x^4,x, algorithm="fricas")

[Out]

[1/945*(315*b*d^4*n*x^3*sqrt(-d/e)*log((d^3*x^2 - 2*d*e^2*x*sqrt(-d/e) - e^3 + 2*(d^2*e*x*sqrt(-d/e) + d*e^2)*
x^(2/3) - 2*(d^2*e*x - e^3*sqrt(-d/e))*x^(1/3))/(d^3*x^2 + e^3)) - 210*b*d^3*e*n*x^2 + 126*b*d^2*e^2*n*x^(4/3)
- 315*b*e^4*n*log((d*x + e*x^(1/3))/x) + 70*b*e^4*n - 315*b*e^4*log(c) - 315*a*e^4 + 90*(7*b*d^4*n*x^2 - b*d*
e^3*n)*x^(2/3))/(e^4*x^3), 1/945*(630*b*d^4*n*x^3*sqrt(d/e)*arctan(x^(1/3)*sqrt(d/e)) - 210*b*d^3*e*n*x^2 + 12
6*b*d^2*e^2*n*x^(4/3) - 315*b*e^4*n*log((d*x + e*x^(1/3))/x) + 70*b*e^4*n - 315*b*e^4*log(c) - 315*a*e^4 + 90*
(7*b*d^4*n*x^2 - b*d*e^3*n)*x^(2/3))/(e^4*x^3)]

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=\text {Timed out} \]

[In]

integrate((a+b*ln(c*(d+e/x**(2/3))**n))/x**4,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=\frac {1}{945} \, {\left (2 \, e {\left (\frac {315 \, d^{5} \arctan \left (\frac {d x^{\frac {1}{3}}}{\sqrt {d e}}\right )}{\sqrt {d e} e^{5}} + \frac {315 \, d^{4} x^{\frac {8}{3}} - 105 \, d^{3} e x^{2} + 63 \, d^{2} e^{2} x^{\frac {4}{3}} - 45 \, d e^{3} x^{\frac {2}{3}} + 35 \, e^{4}}{e^{5} x^{3}}\right )} - \frac {315 \, \log \left (d + \frac {e}{x^{\frac {2}{3}}}\right )}{x^{3}}\right )} b n - \frac {b \log \left (c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \]

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))/x^4,x, algorithm="giac")

[Out]

1/945*(2*e*(315*d^5*arctan(d*x^(1/3)/sqrt(d*e))/(sqrt(d*e)*e^5) + (315*d^4*x^(8/3) - 105*d^3*e*x^2 + 63*d^2*e^
2*x^(4/3) - 45*d*e^3*x^(2/3) + 35*e^4)/(e^5*x^3)) - 315*log(d + e/x^(2/3))/x^3)*b*n - 1/3*b*log(c)/x^3 - 1/3*a
/x^3

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )}{x^4} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )}{x^4} \,d x \]

[In]

int((a + b*log(c*(d + e/x^(2/3))^n))/x^4,x)

[Out]

int((a + b*log(c*(d + e/x^(2/3))^n))/x^4, x)