Integrand size = 20, antiderivative size = 107 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {b e^3 n \sqrt {x}}{2 d^3}-\frac {b e^2 n x}{4 d^2}+\frac {b e n x^{3/2}}{6 d}-\frac {b e^4 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{2 d^4}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^4 n \log (x)}{4 d^4} \] Output:
1/2*b*e^3*n*x^(1/2)/d^3-1/4*b*e^2*n*x/d^2+1/6*b*e*n*x^(3/2)/d-1/2*b*e^4*n* ln(d+e/x^(1/2))/d^4+1/2*x^2*(a+b*ln(c*(d+e/x^(1/2))^n))-1/4*b*e^4*n*ln(x)/ d^4
Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {a x^2}{2}+\frac {1}{2} b x^2 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+\frac {1}{4} b e n \left (\frac {2 e^2 \sqrt {x}}{d^3}-\frac {e x}{d^2}+\frac {2 x^{3/2}}{3 d}-\frac {2 e^3 \log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {e^3 \log (x)}{d^4}\right ) \] Input:
Integrate[x*(a + b*Log[c*(d + e/Sqrt[x])^n]),x]
Output:
(a*x^2)/2 + (b*x^2*Log[c*(d + e/Sqrt[x])^n])/2 + (b*e*n*((2*e^2*Sqrt[x])/d ^3 - (e*x)/d^2 + (2*x^(3/2))/(3*d) - (2*e^3*Log[d + e/Sqrt[x]])/d^4 - (e^3 *Log[x])/d^4))/4
Time = 0.47 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
\(\displaystyle \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle -2 \int x^{5/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle -2 \left (\frac {1}{4} b e n \int \frac {x^2}{d+\frac {e}{\sqrt {x}}}d\frac {1}{\sqrt {x}}-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )\right )\) |
\(\Big \downarrow \) 54 |
\(\displaystyle -2 \left (\frac {1}{4} b e n \int \left (\frac {e^4}{d^4 \left (d+\frac {e}{\sqrt {x}}\right )}-\frac {\sqrt {x} e^3}{d^4}+\frac {x e^2}{d^3}-\frac {x^{3/2} e}{d^2}+\frac {x^2}{d}\right )d\frac {1}{\sqrt {x}}-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (\frac {1}{4} b e n \left (\frac {e^3 \log \left (d+\frac {e}{\sqrt {x}}\right )}{d^4}-\frac {e^3 \log \left (\frac {1}{\sqrt {x}}\right )}{d^4}-\frac {e^2 \sqrt {x}}{d^3}+\frac {e x}{2 d^2}-\frac {x^{3/2}}{3 d}\right )-\frac {1}{4} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )\right )\) |
Input:
Int[x*(a + b*Log[c*(d + e/Sqrt[x])^n]),x]
Output:
-2*(-1/4*(x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])) + (b*e*n*(-((e^2*Sqrt[x])/ d^3) + (e*x)/(2*d^2) - x^(3/2)/(3*d) + (e^3*Log[d + e/Sqrt[x]])/d^4 - (e^3 *Log[1/Sqrt[x]])/d^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])
\[\int x \left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )d x\]
Input:
int(x*(a+b*ln(c*(d+e/x^(1/2))^n)),x)
Output:
int(x*(a+b*ln(c*(d+e/x^(1/2))^n)),x)
Time = 0.13 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.18 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {6 \, b d^{4} x^{2} \log \left (c\right ) - 3 \, b d^{2} e^{2} n x + 6 \, a d^{4} x^{2} - 6 \, b d^{4} n \log \left (\sqrt {x}\right ) + 6 \, {\left (b d^{4} - b e^{4}\right )} n \log \left (d \sqrt {x} + e\right ) + 6 \, {\left (b d^{4} n x^{2} - b d^{4} n\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) + 2 \, {\left (b d^{3} e n x + 3 \, b d e^{3} n\right )} \sqrt {x}}{12 \, d^{4}} \] Input:
integrate(x*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="fricas")
Output:
1/12*(6*b*d^4*x^2*log(c) - 3*b*d^2*e^2*n*x + 6*a*d^4*x^2 - 6*b*d^4*n*log(s qrt(x)) + 6*(b*d^4 - b*e^4)*n*log(d*sqrt(x) + e) + 6*(b*d^4*n*x^2 - b*d^4* n)*log((d*x + e*sqrt(x))/x) + 2*(b*d^3*e*n*x + 3*b*d*e^3*n)*sqrt(x))/d^4
Time = 5.76 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.82 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {a x^{2}}{2} + b \left (\frac {e n \left (\frac {2 x^{\frac {3}{2}}}{3 d} - \frac {e x}{d^{2}} - \frac {2 e^{3} \left (\begin {cases} \frac {\sqrt {x}}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d \sqrt {x} + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{3}} + \frac {2 e^{2} \sqrt {x}}{d^{3}}\right )}{4} + \frac {x^{2} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{2}\right ) \] Input:
integrate(x*(a+b*ln(c*(d+e/x**(1/2))**n)),x)
Output:
a*x**2/2 + b*(e*n*(2*x**(3/2)/(3*d) - e*x/d**2 - 2*e**3*Piecewise((sqrt(x) /e, Eq(d, 0)), (log(d*sqrt(x) + e)/d, True))/d**3 + 2*e**2*sqrt(x)/d**3)/4 + x**2*log(c*(d + e/sqrt(x))**n)/2)
Time = 0.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.69 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=-\frac {1}{12} \, b e n {\left (\frac {6 \, e^{3} \log \left (d \sqrt {x} + e\right )}{d^{4}} - \frac {2 \, d^{2} x^{\frac {3}{2}} - 3 \, d e x + 6 \, e^{2} \sqrt {x}}{d^{3}}\right )} + \frac {1}{2} \, b x^{2} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + \frac {1}{2} \, a x^{2} \] Input:
integrate(x*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="maxima")
Output:
-1/12*b*e*n*(6*e^3*log(d*sqrt(x) + e)/d^4 - (2*d^2*x^(3/2) - 3*d*e*x + 6*e ^2*sqrt(x))/d^3) + 1/2*b*x^2*log(c*(d + e/sqrt(x))^n) + 1/2*a*x^2
Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {1}{2} \, b x^{2} \log \left (c\right ) + \frac {1}{12} \, {\left (6 \, x^{2} \log \left (d + \frac {e}{\sqrt {x}}\right ) - e {\left (\frac {6 \, e^{3} \log \left ({\left | d \sqrt {x} + e \right |}\right )}{d^{4}} - \frac {2 \, d^{2} x^{\frac {3}{2}} - 3 \, d e x + 6 \, e^{2} \sqrt {x}}{d^{3}}\right )}\right )} b n + \frac {1}{2} \, a x^{2} \] Input:
integrate(x*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="giac")
Output:
1/2*b*x^2*log(c) + 1/12*(6*x^2*log(d + e/sqrt(x)) - e*(6*e^3*log(abs(d*sqr t(x) + e))/d^4 - (2*d^2*x^(3/2) - 3*d*e*x + 6*e^2*sqrt(x))/d^3))*b*n + 1/2 *a*x^2
Time = 15.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {x^{3/2}\,\left (\frac {b\,e\,n}{3\,d}-\frac {b\,e^2\,n}{2\,d^2\,\sqrt {x}}+\frac {b\,e^3\,n}{d^3\,x}\right )}{2}+\frac {a\,x^2}{2}+\frac {b\,x^2\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}{2}-\frac {b\,e^4\,n\,\mathrm {atanh}\left (\frac {2\,e}{d\,\sqrt {x}}+1\right )}{d^4} \] Input:
int(x*(a + b*log(c*(d + e/x^(1/2))^n)),x)
Output:
(x^(3/2)*((b*e*n)/(3*d) - (b*e^2*n)/(2*d^2*x^(1/2)) + (b*e^3*n)/(d^3*x)))/ 2 + (a*x^2)/2 + (b*x^2*log(c*(d + e/x^(1/2))^n))/2 - (b*e^4*n*atanh((2*e)/ (d*x^(1/2)) + 1))/d^4
Time = 0.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.01 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {2 \sqrt {x}\, b \,d^{3} e n x +6 \sqrt {x}\, b d \,e^{3} n +6 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b \,d^{4} x^{2}-6 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b \,e^{4}-6 \,\mathrm {log}\left (\sqrt {x}\right ) b \,e^{4} n +6 a \,d^{4} x^{2}-3 b \,d^{2} e^{2} n x}{12 d^{4}} \] Input:
int(x*(a+b*log(c*(d+e/x^(1/2))^n)),x)
Output:
(2*sqrt(x)*b*d**3*e*n*x + 6*sqrt(x)*b*d*e**3*n + 6*log(((sqrt(x)*d + e)**n *c)/x**(n/2))*b*d**4*x**2 - 6*log(((sqrt(x)*d + e)**n*c)/x**(n/2))*b*e**4 - 6*log(sqrt(x))*b*e**4*n + 6*a*d**4*x**2 - 3*b*d**2*e**2*n*x)/(12*d**4)