Integrand size = 22, antiderivative size = 136 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx=\frac {b n}{18 x^3}-\frac {b d n}{15 e x^{5/2}}+\frac {b d^2 n}{12 e^2 x^2}-\frac {b d^3 n}{9 e^3 x^{3/2}}+\frac {b d^4 n}{6 e^4 x}-\frac {b d^5 n}{3 e^5 \sqrt {x}}+\frac {b d^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 e^6}-\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3} \] Output:
1/18*b*n/x^3-1/15*b*d*n/e/x^(5/2)+1/12*b*d^2*n/e^2/x^2-1/9*b*d^3*n/e^3/x^( 3/2)+1/6*b*d^4*n/e^4/x-1/3*b*d^5*n/e^5/x^(1/2)+1/3*b*d^6*n*ln(d+e/x^(1/2)) /e^6-1/3*(a+b*ln(c*(d+e/x^(1/2))^n))/x^3
Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {1}{6} b e n \left (-\frac {1}{3 e x^3}+\frac {2 d}{5 e^2 x^{5/2}}-\frac {d^2}{2 e^3 x^2}+\frac {2 d^3}{3 e^4 x^{3/2}}-\frac {d^4}{e^5 x}+\frac {2 d^5}{e^6 \sqrt {x}}-\frac {2 d^6 \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^7}\right )-\frac {b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{3 x^3} \] Input:
Integrate[(a + b*Log[c*(d + e/Sqrt[x])^n])/x^4,x]
Output:
-1/3*a/x^3 - (b*e*n*(-1/3*1/(e*x^3) + (2*d)/(5*e^2*x^(5/2)) - d^2/(2*e^3*x ^2) + (2*d^3)/(3*e^4*x^(3/2)) - d^4/(e^5*x) + (2*d^5)/(e^6*Sqrt[x]) - (2*d ^6*Log[d + e/Sqrt[x]])/e^7))/6 - (b*Log[c*(d + e/Sqrt[x])^n])/(3*x^3)
Time = 0.58 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2904, 2842, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle -2 \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^{5/2}}d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle -2 \left (\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{6 x^3}-\frac {1}{6} b e n \int \frac {1}{\left (d+\frac {e}{\sqrt {x}}\right ) x^3}d\frac {1}{\sqrt {x}}\right )\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -2 \left (\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{6 x^3}-\frac {1}{6} b e n \int \left (\frac {d^6}{e^6 \left (d+\frac {e}{\sqrt {x}}\right )}-\frac {d^5}{e^6}+\frac {d^4}{e^5 \sqrt {x}}-\frac {d^3}{e^4 x}+\frac {d^2}{e^3 x^{3/2}}-\frac {d}{e^2 x^2}+\frac {1}{e x^{5/2}}\right )d\frac {1}{\sqrt {x}}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (\frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{6 x^3}-\frac {1}{6} b e n \left (\frac {d^6 \log \left (d+\frac {e}{\sqrt {x}}\right )}{e^7}-\frac {d^5}{e^6 \sqrt {x}}+\frac {d^4}{2 e^5 x}-\frac {d^3}{3 e^4 x^{3/2}}+\frac {d^2}{4 e^3 x^2}-\frac {d}{5 e^2 x^{5/2}}+\frac {1}{6 e x^3}\right )\right )\) |
Input:
Int[(a + b*Log[c*(d + e/Sqrt[x])^n])/x^4,x]
Output:
-2*(-1/6*(b*e*n*(1/(6*e*x^3) - d/(5*e^2*x^(5/2)) + d^2/(4*e^3*x^2) - d^3/( 3*e^4*x^(3/2)) + d^4/(2*e^5*x) - d^5/(e^6*Sqrt[x]) + (d^6*Log[d + e/Sqrt[x ]])/e^7)) + (a + b*Log[c*(d + e/Sqrt[x])^n])/(6*x^3))
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] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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] - Simp[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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[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])
\[\int \frac {a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )}{x^{4}}d x\]
Input:
int((a+b*ln(c*(d+e/x^(1/2))^n))/x^4,x)
Output:
int((a+b*ln(c*(d+e/x^(1/2))^n))/x^4,x)
Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx=\frac {30 \, b d^{4} e^{2} n x^{2} + 15 \, b d^{2} e^{4} n x + 10 \, b e^{6} n - 60 \, b e^{6} \log \left (c\right ) - 60 \, a e^{6} + 60 \, {\left (b d^{6} n x^{3} - b e^{6} n\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) - 4 \, {\left (15 \, b d^{5} e n x^{2} + 5 \, b d^{3} e^{3} n x + 3 \, b d e^{5} n\right )} \sqrt {x}}{180 \, e^{6} x^{3}} \] Input:
integrate((a+b*log(c*(d+e/x^(1/2))^n))/x^4,x, algorithm="fricas")
Output:
1/180*(30*b*d^4*e^2*n*x^2 + 15*b*d^2*e^4*n*x + 10*b*e^6*n - 60*b*e^6*log(c ) - 60*a*e^6 + 60*(b*d^6*n*x^3 - b*e^6*n)*log((d*x + e*sqrt(x))/x) - 4*(15 *b*d^5*e*n*x^2 + 5*b*d^3*e^3*n*x + 3*b*d*e^5*n)*sqrt(x))/(e^6*x^3)
Timed out. \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(d+e/x**(1/2))**n))/x**4,x)
Output:
Timed out
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx=\frac {1}{180} \, b e n {\left (\frac {60 \, d^{6} \log \left (d \sqrt {x} + e\right )}{e^{7}} - \frac {30 \, d^{6} \log \left (x\right )}{e^{7}} - \frac {60 \, d^{5} x^{\frac {5}{2}} - 30 \, d^{4} e x^{2} + 20 \, d^{3} e^{2} x^{\frac {3}{2}} - 15 \, d^{2} e^{3} x + 12 \, d e^{4} \sqrt {x} - 10 \, e^{5}}{e^{6} x^{3}}\right )} - \frac {b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \] Input:
integrate((a+b*log(c*(d+e/x^(1/2))^n))/x^4,x, algorithm="maxima")
Output:
1/180*b*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^5*x ^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt( x) - 10*e^5)/(e^6*x^3)) - 1/3*b*log(c*(d + e/sqrt(x))^n)/x^3 - 1/3*a/x^3
Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (110) = 220\).
Time = 0.14 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.62 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx=\frac {60 \, {\left (\frac {6 \, {\left (d \sqrt {x} + e\right )} b d^{5} n}{e^{5} \sqrt {x}} - \frac {15 \, {\left (d \sqrt {x} + e\right )}^{2} b d^{4} n}{e^{5} x} + \frac {20 \, {\left (d \sqrt {x} + e\right )}^{3} b d^{3} n}{e^{5} x^{\frac {3}{2}}} - \frac {15 \, {\left (d \sqrt {x} + e\right )}^{4} b d^{2} n}{e^{5} x^{2}} + \frac {6 \, {\left (d \sqrt {x} + e\right )}^{5} b d n}{e^{5} x^{\frac {5}{2}}} - \frac {{\left (d \sqrt {x} + e\right )}^{6} b n}{e^{5} x^{3}}\right )} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right ) + \frac {10 \, {\left (b n - 6 \, b \log \left (c\right ) - 6 \, a\right )} {\left (d \sqrt {x} + e\right )}^{6}}{e^{5} x^{3}} - \frac {72 \, {\left (b d n - 5 \, b d \log \left (c\right ) - 5 \, a d\right )} {\left (d \sqrt {x} + e\right )}^{5}}{e^{5} x^{\frac {5}{2}}} + \frac {225 \, {\left (b d^{2} n - 4 \, b d^{2} \log \left (c\right ) - 4 \, a d^{2}\right )} {\left (d \sqrt {x} + e\right )}^{4}}{e^{5} x^{2}} - \frac {400 \, {\left (b d^{3} n - 3 \, b d^{3} \log \left (c\right ) - 3 \, a d^{3}\right )} {\left (d \sqrt {x} + e\right )}^{3}}{e^{5} x^{\frac {3}{2}}} + \frac {450 \, {\left (b d^{4} n - 2 \, b d^{4} \log \left (c\right ) - 2 \, a d^{4}\right )} {\left (d \sqrt {x} + e\right )}^{2}}{e^{5} x} - \frac {360 \, {\left (b d^{5} n - b d^{5} \log \left (c\right ) - a d^{5}\right )} {\left (d \sqrt {x} + e\right )}}{e^{5} \sqrt {x}}}{180 \, e} \] Input:
integrate((a+b*log(c*(d+e/x^(1/2))^n))/x^4,x, algorithm="giac")
Output:
1/180*(60*(6*(d*sqrt(x) + e)*b*d^5*n/(e^5*sqrt(x)) - 15*(d*sqrt(x) + e)^2* b*d^4*n/(e^5*x) + 20*(d*sqrt(x) + e)^3*b*d^3*n/(e^5*x^(3/2)) - 15*(d*sqrt( x) + e)^4*b*d^2*n/(e^5*x^2) + 6*(d*sqrt(x) + e)^5*b*d*n/(e^5*x^(5/2)) - (d *sqrt(x) + e)^6*b*n/(e^5*x^3))*log((d*sqrt(x) + e)/sqrt(x)) + 10*(b*n - 6* b*log(c) - 6*a)*(d*sqrt(x) + e)^6/(e^5*x^3) - 72*(b*d*n - 5*b*d*log(c) - 5 *a*d)*(d*sqrt(x) + e)^5/(e^5*x^(5/2)) + 225*(b*d^2*n - 4*b*d^2*log(c) - 4* a*d^2)*(d*sqrt(x) + e)^4/(e^5*x^2) - 400*(b*d^3*n - 3*b*d^3*log(c) - 3*a*d ^3)*(d*sqrt(x) + e)^3/(e^5*x^(3/2)) + 450*(b*d^4*n - 2*b*d^4*log(c) - 2*a* d^4)*(d*sqrt(x) + e)^2/(e^5*x) - 360*(b*d^5*n - b*d^5*log(c) - a*d^5)*(d*s qrt(x) + e)/(e^5*sqrt(x)))/e
Time = 14.60 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx=\frac {b\,n}{18\,x^3}-\frac {a}{3\,x^3}-\frac {b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{3\,x^3}-\frac {b\,d\,n}{15\,e\,x^{5/2}}+\frac {b\,d^6\,n\,\ln \left (d+\frac {e}{\sqrt {x}}\right )}{3\,e^6}+\frac {b\,d^2\,n}{12\,e^2\,x^2}+\frac {b\,d^4\,n}{6\,e^4\,x}-\frac {b\,d^3\,n}{9\,e^3\,x^{3/2}}-\frac {b\,d^5\,n}{3\,e^5\,\sqrt {x}} \] Input:
int((a + b*log(c*(d + e/x^(1/2))^n))/x^4,x)
Output:
(b*n)/(18*x^3) - a/(3*x^3) - (b*log(c*(d + e/x^(1/2))^n))/(3*x^3) - (b*d*n )/(15*e*x^(5/2)) + (b*d^6*n*log(d + e/x^(1/2)))/(3*e^6) + (b*d^2*n)/(12*e^ 2*x^2) + (b*d^4*n)/(6*e^4*x) - (b*d^3*n)/(9*e^3*x^(3/2)) - (b*d^5*n)/(3*e^ 5*x^(1/2))
Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{x^4} \, dx=\frac {-60 \sqrt {x}\, b \,d^{5} e n \,x^{2}-20 \sqrt {x}\, b \,d^{3} e^{3} n x -12 \sqrt {x}\, b d \,e^{5} n +60 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b \,d^{6} x^{3}-60 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b \,e^{6}-60 a \,e^{6}+30 b \,d^{4} e^{2} n \,x^{2}+15 b \,d^{2} e^{4} n x +10 b \,e^{6} n}{180 e^{6} x^{3}} \] Input:
int((a+b*log(c*(d+e/x^(1/2))^n))/x^4,x)
Output:
( - 60*sqrt(x)*b*d**5*e*n*x**2 - 20*sqrt(x)*b*d**3*e**3*n*x - 12*sqrt(x)*b *d*e**5*n + 60*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b*d**6*x**3 - 60*log(( (sqrt(x)*d + e)**n*c)/x**(n/2))*b*e**6 - 60*a*e**6 + 30*b*d**4*e**2*n*x**2 + 15*b*d**2*e**4*n*x + 10*b*e**6*n)/(180*e**6*x**3)