Integrand size = 18, antiderivative size = 100 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^3} \, dx=-\frac {b p}{6 a x^{3/2}}+\frac {b^2 p}{4 a^2 x}-\frac {b^3 p}{2 a^3 \sqrt {x}}+\frac {b^4 p \log \left (a+b \sqrt {x}\right )}{2 a^4}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{2 x^2}-\frac {b^4 p \log (x)}{4 a^4} \] Output:
-1/6*b*p/a/x^(3/2)+1/4*b^2*p/a^2/x-1/2*b^3*p/a^3/x^(1/2)+1/2*b^4*p*ln(a+b* x^(1/2))/a^4-1/2*ln(c*(a+b*x^(1/2))^p)/x^2-1/4*b^4*p*ln(x)/a^4
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.90 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^3} \, dx=-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{2 x^2}+\frac {1}{4} b p \left (-\frac {2}{3 a x^{3/2}}+\frac {b}{a^2 x}-\frac {2 b^2}{a^3 \sqrt {x}}+\frac {2 b^3 \log \left (a+b \sqrt {x}\right )}{a^4}-\frac {b^3 \log (x)}{a^4}\right ) \] Input:
Integrate[Log[c*(a + b*Sqrt[x])^p]/x^3,x]
Output:
-1/2*Log[c*(a + b*Sqrt[x])^p]/x^2 + (b*p*(-2/(3*a*x^(3/2)) + b/(a^2*x) - ( 2*b^2)/(a^3*Sqrt[x]) + (2*b^3*Log[a + b*Sqrt[x]])/a^4 - (b^3*Log[x])/a^4)) /4
Time = 0.45 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, 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.
\(\displaystyle \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle 2 \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^{5/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle 2 \left (\frac {1}{4} b p \int \frac {1}{\left (a+b \sqrt {x}\right ) x^2}d\sqrt {x}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{4 x^2}\right )\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 2 \left (\frac {1}{4} b p \int \left (\frac {b^4}{a^4 \left (a+b \sqrt {x}\right )}-\frac {b^3}{a^4 \sqrt {x}}+\frac {b^2}{a^3 x}-\frac {b}{a^2 x^{3/2}}+\frac {1}{a x^2}\right )d\sqrt {x}-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{4 x^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {1}{4} b p \left (\frac {b^3 \log \left (a+b \sqrt {x}\right )}{a^4}-\frac {b^3 \log \left (\sqrt {x}\right )}{a^4}-\frac {b^2}{a^3 \sqrt {x}}+\frac {b}{2 a^2 x}-\frac {1}{3 a x^{3/2}}\right )-\frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{4 x^2}\right )\) |
Input:
Int[Log[c*(a + b*Sqrt[x])^p]/x^3,x]
Output:
2*(-1/4*Log[c*(a + b*Sqrt[x])^p]/x^2 + (b*p*(-1/3*1/(a*x^(3/2)) + b/(2*a^2 *x) - b^2/(a^3*Sqrt[x]) + (b^3*Log[a + b*Sqrt[x]])/a^4 - (b^3*Log[Sqrt[x]] )/a^4))/4)
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])
Time = 0.43 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.77
method | result | size |
parts | \(-\frac {\ln \left (c \left (a +b \sqrt {x}\right )^{p}\right )}{2 x^{2}}+\frac {p b \left (\frac {2 b^{3} \ln \left (a +b \sqrt {x}\right )}{a^{4}}-\frac {2}{3 a \,x^{\frac {3}{2}}}-\frac {2 b^{2}}{a^{3} \sqrt {x}}+\frac {b}{a^{2} x}-\frac {b^{3} \ln \left (x \right )}{a^{4}}\right )}{4}\) | \(77\) |
Input:
int(ln(c*(a+b*x^(1/2))^p)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*ln(c*(a+b*x^(1/2))^p)/x^2+1/4*p*b*(2*b^3/a^4*ln(a+b*x^(1/2))-2/3/a/x^ (3/2)-2*b^2/a^3/x^(1/2)+b/a^2/x-b^3/a^4*ln(x))
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.84 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^3} \, dx=-\frac {6 \, b^{4} p x^{2} \log \left (\sqrt {x}\right ) - 3 \, a^{2} b^{2} p x + 6 \, a^{4} \log \left (c\right ) - 6 \, {\left (b^{4} p x^{2} - a^{4} p\right )} \log \left (b \sqrt {x} + a\right ) + 2 \, {\left (3 \, a b^{3} p x + a^{3} b p\right )} \sqrt {x}}{12 \, a^{4} x^{2}} \] Input:
integrate(log(c*(a+b*x^(1/2))^p)/x^3,x, algorithm="fricas")
Output:
-1/12*(6*b^4*p*x^2*log(sqrt(x)) - 3*a^2*b^2*p*x + 6*a^4*log(c) - 6*(b^4*p* x^2 - a^4*p)*log(b*sqrt(x) + a) + 2*(3*a*b^3*p*x + a^3*b*p)*sqrt(x))/(a^4* x^2)
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (90) = 180\).
Time = 73.06 (sec) , antiderivative size = 435, normalized size of antiderivative = 4.35 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^3} \, dx=\begin {cases} - \frac {\log {\left (0^{p} c \right )}}{2 x^{2}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {p}{8 x^{2}} - \frac {\log {\left (c \left (b \sqrt {x}\right )^{p} \right )}}{2 x^{2}} & \text {for}\: a = 0 \\- \frac {\log {\left (0^{p} c \right )}}{2 x^{2}} & \text {for}\: a = - b \sqrt {x} \\- \frac {6 a^{5} \sqrt {x} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {2 a^{4} b p x}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {6 a^{4} b x \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} + \frac {a^{3} b^{2} p x^{\frac {3}{2}}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {3 a^{2} b^{3} p x^{2}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {3 a b^{4} p x^{\frac {5}{2}} \log {\left (x \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {6 a b^{4} p x^{\frac {5}{2}}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} + \frac {6 a b^{4} x^{\frac {5}{2}} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} - \frac {3 b^{5} p x^{3} \log {\left (x \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} + \frac {6 b^{5} x^{3} \log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{12 a^{5} x^{\frac {5}{2}} + 12 a^{4} b x^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(ln(c*(a+b*x**(1/2))**p)/x**3,x)
Output:
Piecewise((-log(0**p*c)/(2*x**2), Eq(a, 0) & Eq(b, 0)), (-p/(8*x**2) - log (c*(b*sqrt(x))**p)/(2*x**2), Eq(a, 0)), (-log(0**p*c)/(2*x**2), Eq(a, -b*s qrt(x))), (-6*a**5*sqrt(x)*log(c*(a + b*sqrt(x))**p)/(12*a**5*x**(5/2) + 1 2*a**4*b*x**3) - 2*a**4*b*p*x/(12*a**5*x**(5/2) + 12*a**4*b*x**3) - 6*a**4 *b*x*log(c*(a + b*sqrt(x))**p)/(12*a**5*x**(5/2) + 12*a**4*b*x**3) + a**3* b**2*p*x**(3/2)/(12*a**5*x**(5/2) + 12*a**4*b*x**3) - 3*a**2*b**3*p*x**2/( 12*a**5*x**(5/2) + 12*a**4*b*x**3) - 3*a*b**4*p*x**(5/2)*log(x)/(12*a**5*x **(5/2) + 12*a**4*b*x**3) - 6*a*b**4*p*x**(5/2)/(12*a**5*x**(5/2) + 12*a** 4*b*x**3) + 6*a*b**4*x**(5/2)*log(c*(a + b*sqrt(x))**p)/(12*a**5*x**(5/2) + 12*a**4*b*x**3) - 3*b**5*p*x**3*log(x)/(12*a**5*x**(5/2) + 12*a**4*b*x** 3) + 6*b**5*x**3*log(c*(a + b*sqrt(x))**p)/(12*a**5*x**(5/2) + 12*a**4*b*x **3), True))
Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.76 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^3} \, dx=\frac {1}{12} \, b p {\left (\frac {6 \, b^{3} \log \left (b \sqrt {x} + a\right )}{a^{4}} - \frac {3 \, b^{3} \log \left (x\right )}{a^{4}} - \frac {6 \, b^{2} x - 3 \, a b \sqrt {x} + 2 \, a^{2}}{a^{3} x^{\frac {3}{2}}}\right )} - \frac {\log \left ({\left (b \sqrt {x} + a\right )}^{p} c\right )}{2 \, x^{2}} \] Input:
integrate(log(c*(a+b*x^(1/2))^p)/x^3,x, algorithm="maxima")
Output:
1/12*b*p*(6*b^3*log(b*sqrt(x) + a)/a^4 - 3*b^3*log(x)/a^4 - (6*b^2*x - 3*a *b*sqrt(x) + 2*a^2)/(a^3*x^(3/2))) - 1/2*log((b*sqrt(x) + a)^p*c)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (80) = 160\).
Time = 0.13 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.32 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^3} \, dx=-\frac {\frac {6 \, b^{5} p \log \left (b \sqrt {x} + a\right )}{{\left (b \sqrt {x} + a\right )}^{4} - 4 \, {\left (b \sqrt {x} + a\right )}^{3} a + 6 \, {\left (b \sqrt {x} + a\right )}^{2} a^{2} - 4 \, {\left (b \sqrt {x} + a\right )} a^{3} + a^{4}} - \frac {6 \, b^{5} p \log \left (b \sqrt {x} + a\right )}{a^{4}} + \frac {6 \, b^{5} p \log \left (b \sqrt {x}\right )}{a^{4}} + \frac {6 \, {\left (b \sqrt {x} + a\right )}^{3} b^{5} p - 21 \, {\left (b \sqrt {x} + a\right )}^{2} a b^{5} p + 26 \, {\left (b \sqrt {x} + a\right )} a^{2} b^{5} p - 11 \, a^{3} b^{5} p + 6 \, a^{3} b^{5} \log \left (c\right )}{{\left (b \sqrt {x} + a\right )}^{4} a^{3} - 4 \, {\left (b \sqrt {x} + a\right )}^{3} a^{4} + 6 \, {\left (b \sqrt {x} + a\right )}^{2} a^{5} - 4 \, {\left (b \sqrt {x} + a\right )} a^{6} + a^{7}}}{12 \, b} \] Input:
integrate(log(c*(a+b*x^(1/2))^p)/x^3,x, algorithm="giac")
Output:
-1/12*(6*b^5*p*log(b*sqrt(x) + a)/((b*sqrt(x) + a)^4 - 4*(b*sqrt(x) + a)^3 *a + 6*(b*sqrt(x) + a)^2*a^2 - 4*(b*sqrt(x) + a)*a^3 + a^4) - 6*b^5*p*log( b*sqrt(x) + a)/a^4 + 6*b^5*p*log(b*sqrt(x))/a^4 + (6*(b*sqrt(x) + a)^3*b^5 *p - 21*(b*sqrt(x) + a)^2*a*b^5*p + 26*(b*sqrt(x) + a)*a^2*b^5*p - 11*a^3* b^5*p + 6*a^3*b^5*log(c))/((b*sqrt(x) + a)^4*a^3 - 4*(b*sqrt(x) + a)^3*a^4 + 6*(b*sqrt(x) + a)^2*a^5 - 4*(b*sqrt(x) + a)*a^6 + a^7))/b
Time = 15.50 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.72 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^3} \, dx=\frac {b^4\,p\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^4}-\frac {\ln \left (c\,{\left (a+b\,\sqrt {x}\right )}^p\right )}{2\,x^2}-\frac {\frac {b\,p}{3\,a}-\frac {b^2\,p\,\sqrt {x}}{2\,a^2}+\frac {b^3\,p\,x}{a^3}}{2\,x^{3/2}} \] Input:
int(log(c*(a + b*x^(1/2))^p)/x^3,x)
Output:
(b^4*p*atanh((2*b*x^(1/2))/a + 1))/a^4 - log(c*(a + b*x^(1/2))^p)/(2*x^2) - ((b*p)/(3*a) - (b^2*p*x^(1/2))/(2*a^2) + (b^3*p*x)/a^3)/(2*x^(3/2))
Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x^3} \, dx=\frac {-2 \sqrt {x}\, a^{3} b p -6 \sqrt {x}\, a \,b^{3} p x -6 \,\mathrm {log}\left (\sqrt {x}\right ) b^{4} p \,x^{2}-6 \,\mathrm {log}\left (\left (\sqrt {x}\, b +a \right )^{p} c \right ) a^{4}+6 \,\mathrm {log}\left (\left (\sqrt {x}\, b +a \right )^{p} c \right ) b^{4} x^{2}+3 a^{2} b^{2} p x}{12 a^{4} x^{2}} \] Input:
int(log(c*(a+b*x^(1/2))^p)/x^3,x)
Output:
( - 2*sqrt(x)*a**3*b*p - 6*sqrt(x)*a*b**3*p*x - 6*log(sqrt(x))*b**4*p*x**2 - 6*log((sqrt(x)*b + a)**p*c)*a**4 + 6*log((sqrt(x)*b + a)**p*c)*b**4*x** 2 + 3*a**2*b**2*p*x)/(12*a**4*x**2)