Integrand size = 22, antiderivative size = 141 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^4} \, dx=-\frac {b e n}{15 d x^{5/2}}+\frac {b e^2 n}{12 d^2 x^2}-\frac {b e^3 n}{9 d^3 x^{3/2}}+\frac {b e^4 n}{6 d^4 x}-\frac {b e^5 n}{3 d^5 \sqrt {x}}+\frac {b e^6 n \log \left (d+e \sqrt {x}\right )}{3 d^6}-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{3 x^3}-\frac {b e^6 n \log (x)}{6 d^6} \] Output:
-1/15*b*e*n/d/x^(5/2)+1/12*b*e^2*n/d^2/x^2-1/9*b*e^3*n/d^3/x^(3/2)+1/6*b*e ^4*n/d^4/x-1/3*b*e^5*n/d^5/x^(1/2)+1/3*b*e^6*n*ln(d+e*x^(1/2))/d^6-1/3*(a+ b*ln(c*(d+e*x^(1/2))^n))/x^3-1/6*b*e^6*n*ln(x)/d^6
Time = 0.13 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{3 x^3}+\frac {1}{6} b e n \left (-\frac {2}{5 d x^{5/2}}+\frac {e}{2 d^2 x^2}-\frac {2 e^2}{3 d^3 x^{3/2}}+\frac {e^3}{d^4 x}-\frac {2 e^4}{d^5 \sqrt {x}}+\frac {2 e^5 \log \left (d+e \sqrt {x}\right )}{d^6}-\frac {e^5 \log (x)}{d^6}\right ) \] Input:
Integrate[(a + b*Log[c*(d + e*Sqrt[x])^n])/x^4,x]
Output:
-1/3*a/x^3 - (b*Log[c*(d + e*Sqrt[x])^n])/(3*x^3) + (b*e*n*(-2/(5*d*x^(5/2 )) + e/(2*d^2*x^2) - (2*e^2)/(3*d^3*x^(3/2)) + e^3/(d^4*x) - (2*e^4)/(d^5* Sqrt[x]) + (2*e^5*Log[d + e*Sqrt[x]])/d^6 - (e^5*Log[x])/d^6))/6
Time = 0.52 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2904, 2842, 54, 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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle 2 \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^{7/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle 2 \left (\frac {1}{6} b e n \int \frac {1}{\left (d+e \sqrt {x}\right ) x^3}d\sqrt {x}-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{6 x^3}\right )\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 2 \left (\frac {1}{6} b e n \int \left (\frac {e^6}{d^6 \left (d+e \sqrt {x}\right )}-\frac {e^5}{d^6 \sqrt {x}}+\frac {e^4}{d^5 x}-\frac {e^3}{d^4 x^{3/2}}+\frac {e^2}{d^3 x^2}-\frac {e}{d^2 x^{5/2}}+\frac {1}{d x^3}\right )d\sqrt {x}-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{6 x^3}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {1}{6} b e n \left (\frac {e^5 \log \left (d+e \sqrt {x}\right )}{d^6}-\frac {e^5 \log \left (\sqrt {x}\right )}{d^6}-\frac {e^4}{d^5 \sqrt {x}}+\frac {e^3}{2 d^4 x}-\frac {e^2}{3 d^3 x^{3/2}}+\frac {e}{4 d^2 x^2}-\frac {1}{5 d x^{5/2}}\right )-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{6 x^3}\right )\) |
Input:
Int[(a + b*Log[c*(d + e*Sqrt[x])^n])/x^4,x]
Output:
2*(-1/6*(a + b*Log[c*(d + e*Sqrt[x])^n])/x^3 + (b*e*n*(-1/5*1/(d*x^(5/2)) + e/(4*d^2*x^2) - e^2/(3*d^3*x^(3/2)) + e^3/(2*d^4*x) - e^4/(d^5*Sqrt[x]) + (e^5*Log[d + e*Sqrt[x]])/d^6 - (e^5*Log[Sqrt[x]])/d^6))/6)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[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])
\[\int \frac {a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )}{x^{4}}d x\]
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 = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^4} \, dx=-\frac {60 \, b e^{6} n x^{3} \log \left (\sqrt {x}\right ) - 30 \, b d^{2} e^{4} n x^{2} - 15 \, b d^{4} e^{2} n x + 60 \, b d^{6} \log \left (c\right ) + 60 \, a d^{6} - 60 \, {\left (b e^{6} n x^{3} - b d^{6} n\right )} \log \left (e \sqrt {x} + d\right ) + 4 \, {\left (15 \, b d e^{5} n x^{2} + 5 \, b d^{3} e^{3} n x + 3 \, b d^{5} e n\right )} \sqrt {x}}{180 \, d^{6} x^{3}} \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^4,x, algorithm="fricas")
Output:
-1/180*(60*b*e^6*n*x^3*log(sqrt(x)) - 30*b*d^2*e^4*n*x^2 - 15*b*d^4*e^2*n* x + 60*b*d^6*log(c) + 60*a*d^6 - 60*(b*e^6*n*x^3 - b*d^6*n)*log(e*sqrt(x) + d) + 4*(15*b*d*e^5*n*x^2 + 5*b*d^3*e^3*n*x + 3*b*d^5*e*n)*sqrt(x))/(d^6* x^3)
Timed out. \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^4} \, dx=\text {Timed out} \] Input:
integrate((a+b*ln(c*(d+e*x**(1/2))**n))/x**4,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.75 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^4} \, dx=\frac {1}{180} \, b e n {\left (\frac {60 \, e^{5} \log \left (e \sqrt {x} + d\right )}{d^{6}} - \frac {30 \, e^{5} \log \left (x\right )}{d^{6}} - \frac {60 \, e^{4} x^{2} - 30 \, d e^{3} x^{\frac {3}{2}} + 20 \, d^{2} e^{2} x - 15 \, d^{3} e \sqrt {x} + 12 \, d^{4}}{d^{5} x^{\frac {5}{2}}}\right )} - \frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )}{3 \, x^{3}} - \frac {a}{3 \, x^{3}} \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^4,x, algorithm="maxima")
Output:
1/180*b*e*n*(60*e^5*log(e*sqrt(x) + d)/d^6 - 30*e^5*log(x)/d^6 - (60*e^4*x ^2 - 30*d*e^3*x^(3/2) + 20*d^2*e^2*x - 15*d^3*e*sqrt(x) + 12*d^4)/(d^5*x^( 5/2))) - 1/3*b*log((e*sqrt(x) + d)^n*c)/x^3 - 1/3*a/x^3
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (115) = 230\).
Time = 0.13 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.43 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^4} \, dx=-\frac {\frac {60 \, b e^{7} n \log \left (e \sqrt {x} + d\right )}{{\left (e \sqrt {x} + d\right )}^{6} - 6 \, {\left (e \sqrt {x} + d\right )}^{5} d + 15 \, {\left (e \sqrt {x} + d\right )}^{4} d^{2} - 20 \, {\left (e \sqrt {x} + d\right )}^{3} d^{3} + 15 \, {\left (e \sqrt {x} + d\right )}^{2} d^{4} - 6 \, {\left (e \sqrt {x} + d\right )} d^{5} + d^{6}} - \frac {60 \, b e^{7} n \log \left (e \sqrt {x} + d\right )}{d^{6}} + \frac {60 \, b e^{7} n \log \left (e \sqrt {x}\right )}{d^{6}} + \frac {60 \, {\left (e \sqrt {x} + d\right )}^{5} b e^{7} n - 330 \, {\left (e \sqrt {x} + d\right )}^{4} b d e^{7} n + 740 \, {\left (e \sqrt {x} + d\right )}^{3} b d^{2} e^{7} n - 855 \, {\left (e \sqrt {x} + d\right )}^{2} b d^{3} e^{7} n + 522 \, {\left (e \sqrt {x} + d\right )} b d^{4} e^{7} n - 137 \, b d^{5} e^{7} n + 60 \, b d^{5} e^{7} \log \left (c\right ) + 60 \, a d^{5} e^{7}}{{\left (e \sqrt {x} + d\right )}^{6} d^{5} - 6 \, {\left (e \sqrt {x} + d\right )}^{5} d^{6} + 15 \, {\left (e \sqrt {x} + d\right )}^{4} d^{7} - 20 \, {\left (e \sqrt {x} + d\right )}^{3} d^{8} + 15 \, {\left (e \sqrt {x} + d\right )}^{2} d^{9} - 6 \, {\left (e \sqrt {x} + d\right )} d^{10} + d^{11}}}{180 \, e} \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^4,x, algorithm="giac")
Output:
-1/180*(60*b*e^7*n*log(e*sqrt(x) + d)/((e*sqrt(x) + d)^6 - 6*(e*sqrt(x) + d)^5*d + 15*(e*sqrt(x) + d)^4*d^2 - 20*(e*sqrt(x) + d)^3*d^3 + 15*(e*sqrt( x) + d)^2*d^4 - 6*(e*sqrt(x) + d)*d^5 + d^6) - 60*b*e^7*n*log(e*sqrt(x) + d)/d^6 + 60*b*e^7*n*log(e*sqrt(x))/d^6 + (60*(e*sqrt(x) + d)^5*b*e^7*n - 3 30*(e*sqrt(x) + d)^4*b*d*e^7*n + 740*(e*sqrt(x) + d)^3*b*d^2*e^7*n - 855*( e*sqrt(x) + d)^2*b*d^3*e^7*n + 522*(e*sqrt(x) + d)*b*d^4*e^7*n - 137*b*d^5 *e^7*n + 60*b*d^5*e^7*log(c) + 60*a*d^5*e^7)/((e*sqrt(x) + d)^6*d^5 - 6*(e *sqrt(x) + d)^5*d^6 + 15*(e*sqrt(x) + d)^4*d^7 - 20*(e*sqrt(x) + d)^3*d^8 + 15*(e*sqrt(x) + d)^2*d^9 - 6*(e*sqrt(x) + d)*d^10 + d^11))/e
Time = 15.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.78 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^4} \, dx=\frac {2\,b\,e^6\,n\,\mathrm {atanh}\left (\frac {2\,e\,\sqrt {x}}{d}+1\right )}{3\,d^6}-\frac {\frac {b\,e\,n}{5\,d}+\frac {b\,e^3\,n\,x}{3\,d^3}-\frac {b\,e^2\,n\,\sqrt {x}}{4\,d^2}+\frac {b\,e^5\,n\,x^2}{d^5}-\frac {b\,e^4\,n\,x^{3/2}}{2\,d^4}}{3\,x^{5/2}}-\frac {b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3\,x^3}-\frac {a}{3\,x^3} \] Input:
int((a + b*log(c*(d + e*x^(1/2))^n))/x^4,x)
Output:
(2*b*e^6*n*atanh((2*e*x^(1/2))/d + 1))/(3*d^6) - ((b*e*n)/(5*d) + (b*e^3*n *x)/(3*d^3) - (b*e^2*n*x^(1/2))/(4*d^2) + (b*e^5*n*x^2)/d^5 - (b*e^4*n*x^( 3/2))/(2*d^4))/(3*x^(5/2)) - (b*log(c*(d + e*x^(1/2))^n))/(3*x^3) - a/(3*x ^3)
Time = 0.23 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^4} \, dx=\frac {-12 \sqrt {x}\, b \,d^{5} e n -20 \sqrt {x}\, b \,d^{3} e^{3} n x -60 \sqrt {x}\, b d \,e^{5} n \,x^{2}-60 \,\mathrm {log}\left (\sqrt {x}\right ) b \,e^{6} n \,x^{3}-60 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b \,d^{6}+60 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b \,e^{6} x^{3}-60 a \,d^{6}+15 b \,d^{4} e^{2} n x +30 b \,d^{2} e^{4} n \,x^{2}}{180 d^{6} x^{3}} \] Input:
int((a+b*log(c*(d+e*x^(1/2))^n))/x^4,x)
Output:
( - 12*sqrt(x)*b*d**5*e*n - 20*sqrt(x)*b*d**3*e**3*n*x - 60*sqrt(x)*b*d*e* *5*n*x**2 - 60*log(sqrt(x))*b*e**6*n*x**3 - 60*log((sqrt(x)*e + d)**n*c)*b *d**6 + 60*log((sqrt(x)*e + d)**n*c)*b*e**6*x**3 - 60*a*d**6 + 15*b*d**4*e **2*n*x + 30*b*d**2*e**4*n*x**2)/(180*d**6*x**3)