Integrand size = 22, antiderivative size = 70 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx=-\frac {b e n}{d \sqrt {x}}+\frac {b e^2 n \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}-\frac {b e^2 n \log (x)}{2 d^2} \] Output:
-b*e*n/d/x^(1/2)+b*e^2*n*ln(d+e*x^(1/2))/d^2-(a+b*ln(c*(d+e*x^(1/2))^n))/x -1/2*b*e^2*n*ln(x)/d^2
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.99 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx=-\frac {a}{x}-\frac {b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x}+\frac {1}{2} b e n \left (-\frac {2}{d \sqrt {x}}+\frac {2 e \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {e \log (x)}{d^2}\right ) \] Input:
Integrate[(a + b*Log[c*(d + e*Sqrt[x])^n])/x^2,x]
Output:
-(a/x) - (b*Log[c*(d + e*Sqrt[x])^n])/x + (b*e*n*(-2/(d*Sqrt[x]) + (2*e*Lo g[d + e*Sqrt[x]])/d^2 - (e*Log[x])/d^2))/2
Time = 0.42 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle 2 \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^{3/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle 2 \left (\frac {1}{2} b e n \int \frac {1}{\left (d+e \sqrt {x}\right ) x}d\sqrt {x}-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{2 x}\right )\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 2 \left (\frac {1}{2} b e n \int \left (\frac {e^2}{d^2 \left (d+e \sqrt {x}\right )}-\frac {e}{d^2 \sqrt {x}}+\frac {1}{d x}\right )d\sqrt {x}-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{2 x}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {1}{2} b e n \left (\frac {e \log \left (d+e \sqrt {x}\right )}{d^2}-\frac {e \log \left (\sqrt {x}\right )}{d^2}-\frac {1}{d \sqrt {x}}\right )-\frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{2 x}\right )\) |
Input:
Int[(a + b*Log[c*(d + e*Sqrt[x])^n])/x^2,x]
Output:
2*(-1/2*(a + b*Log[c*(d + e*Sqrt[x])^n])/x + (b*e*n*(-(1/(d*Sqrt[x])) + (e *Log[d + e*Sqrt[x]])/d^2 - (e*Log[Sqrt[x]])/d^2))/2)
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])
\[\int \frac {a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )}{x^{2}}d x\]
Input:
int((a+b*ln(c*(d+e*x^(1/2))^n))/x^2,x)
Output:
int((a+b*ln(c*(d+e*x^(1/2))^n))/x^2,x)
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx=-\frac {b e^{2} n x \log \left (\sqrt {x}\right ) + b d e n \sqrt {x} + b d^{2} \log \left (c\right ) + a d^{2} - {\left (b e^{2} n x - b d^{2} n\right )} \log \left (e \sqrt {x} + d\right )}{d^{2} x} \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^2,x, algorithm="fricas")
Output:
-(b*e^2*n*x*log(sqrt(x)) + b*d*e*n*sqrt(x) + b*d^2*log(c) + a*d^2 - (b*e^2 *n*x - b*d^2*n)*log(e*sqrt(x) + d))/(d^2*x)
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (65) = 130\).
Time = 13.69 (sec) , antiderivative size = 442, normalized size of antiderivative = 6.31 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx=\begin {cases} - \frac {a + b \log {\left (0^{n} c \right )}}{x} & \text {for}\: d = 0 \wedge e = 0 \\- \frac {a}{x} - \frac {b n}{2 x} - \frac {b \log {\left (c \left (e \sqrt {x}\right )^{n} \right )}}{x} & \text {for}\: d = 0 \\- \frac {a + b \log {\left (0^{n} c \right )}}{x} & \text {for}\: d = - e \sqrt {x} \\- \frac {2 a d^{3} \sqrt {x}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 a d^{2} e x}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{3} \sqrt {x} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{2} e n x}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d^{2} e x \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {b d e^{2} n x^{\frac {3}{2}} \log {\left (x \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {2 b d e^{2} n x^{\frac {3}{2}}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} + \frac {2 b d e^{2} x^{\frac {3}{2}} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} - \frac {b e^{3} n x^{2} \log {\left (x \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} + \frac {2 b e^{3} x^{2} \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}}{2 d^{3} x^{\frac {3}{2}} + 2 d^{2} e x^{2}} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*ln(c*(d+e*x**(1/2))**n))/x**2,x)
Output:
Piecewise((-(a + b*log(0**n*c))/x, Eq(d, 0) & Eq(e, 0)), (-a/x - b*n/(2*x) - b*log(c*(e*sqrt(x))**n)/x, Eq(d, 0)), (-(a + b*log(0**n*c))/x, Eq(d, -e *sqrt(x))), (-2*a*d**3*sqrt(x)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - 2*a*d** 2*e*x/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - 2*b*d**3*sqrt(x)*log(c*(d + e*sq rt(x))**n)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) - 2*b*d**2*e*n*x/(2*d**3*x**( 3/2) + 2*d**2*e*x**2) - 2*b*d**2*e*x*log(c*(d + e*sqrt(x))**n)/(2*d**3*x** (3/2) + 2*d**2*e*x**2) - b*d*e**2*n*x**(3/2)*log(x)/(2*d**3*x**(3/2) + 2*d **2*e*x**2) - 2*b*d*e**2*n*x**(3/2)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) + 2* b*d*e**2*x**(3/2)*log(c*(d + e*sqrt(x))**n)/(2*d**3*x**(3/2) + 2*d**2*e*x* *2) - b*e**3*n*x**2*log(x)/(2*d**3*x**(3/2) + 2*d**2*e*x**2) + 2*b*e**3*x* *2*log(c*(d + e*sqrt(x))**n)/(2*d**3*x**(3/2) + 2*d**2*e*x**2), True))
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx=\frac {1}{2} \, b e n {\left (\frac {2 \, e \log \left (e \sqrt {x} + d\right )}{d^{2}} - \frac {e \log \left (x\right )}{d^{2}} - \frac {2}{d \sqrt {x}}\right )} - \frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )}{x} - \frac {a}{x} \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^2,x, algorithm="maxima")
Output:
1/2*b*e*n*(2*e*log(e*sqrt(x) + d)/d^2 - e*log(x)/d^2 - 2/(d*sqrt(x))) - b* log((e*sqrt(x) + d)^n*c)/x - a/x
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (62) = 124\).
Time = 0.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.06 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx=-\frac {\frac {b e^{3} n \log \left (e \sqrt {x} + d\right )}{{\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d + d^{2}} - \frac {b e^{3} n \log \left (e \sqrt {x} + d\right )}{d^{2}} + \frac {b e^{3} n \log \left (e \sqrt {x}\right )}{d^{2}} + \frac {{\left (e \sqrt {x} + d\right )} b e^{3} n - b d e^{3} n + b d e^{3} \log \left (c\right ) + a d e^{3}}{{\left (e \sqrt {x} + d\right )}^{2} d - 2 \, {\left (e \sqrt {x} + d\right )} d^{2} + d^{3}}}{e} \] Input:
integrate((a+b*log(c*(d+e*x^(1/2))^n))/x^2,x, algorithm="giac")
Output:
-(b*e^3*n*log(e*sqrt(x) + d)/((e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)*d + d^ 2) - b*e^3*n*log(e*sqrt(x) + d)/d^2 + b*e^3*n*log(e*sqrt(x))/d^2 + ((e*sqr t(x) + d)*b*e^3*n - b*d*e^3*n + b*d*e^3*log(c) + a*d*e^3)/((e*sqrt(x) + d) ^2*d - 2*(e*sqrt(x) + d)*d^2 + d^3))/e
Time = 15.33 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx=\frac {2\,b\,e^2\,n\,\mathrm {atanh}\left (\frac {2\,e\,\sqrt {x}}{d}+1\right )}{d^2}-\frac {b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{x}-\frac {b\,e\,n}{d\,\sqrt {x}}-\frac {a}{x} \] Input:
int((a + b*log(c*(d + e*x^(1/2))^n))/x^2,x)
Output:
(2*b*e^2*n*atanh((2*e*x^(1/2))/d + 1))/d^2 - (b*log(c*(d + e*x^(1/2))^n))/ x - (b*e*n)/(d*x^(1/2)) - a/x
Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{x^2} \, dx=\frac {-\sqrt {x}\, b d e n -\mathrm {log}\left (\sqrt {x}\right ) b \,e^{2} n x -\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b \,d^{2}+\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b \,e^{2} x -a \,d^{2}}{d^{2} x} \] Input:
int((a+b*log(c*(d+e*x^(1/2))^n))/x^2,x)
Output:
( - sqrt(x)*b*d*e*n - log(sqrt(x))*b*e**2*n*x - log((sqrt(x)*e + d)**n*c)* b*d**2 + log((sqrt(x)*e + d)**n*c)*b*e**2*x - a*d**2)/(d**2*x)