Integrand size = 22, antiderivative size = 139 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {b e^5 n \sqrt {x}}{3 d^5}-\frac {b e^4 n x}{6 d^4}+\frac {b e^3 n x^{3/2}}{9 d^3}-\frac {b e^2 n x^2}{12 d^2}+\frac {b e n x^{5/2}}{15 d}-\frac {b e^6 n \log \left (d+\frac {e}{\sqrt {x}}\right )}{3 d^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-\frac {b e^6 n \log (x)}{6 d^6} \] Output:
1/3*b*e^5*n*x^(1/2)/d^5-1/6*b*e^4*n*x/d^4+1/9*b*e^3*n*x^(3/2)/d^3-1/12*b*e ^2*n*x^2/d^2+1/15*b*e*n*x^(5/2)/d-1/3*b*e^6*n*ln(d+e/x^(1/2))/d^6+1/3*x^3* (a+b*ln(c*(d+e/x^(1/2))^n))-1/6*b*e^6*n*ln(x)/d^6
Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {a x^3}{3}+\frac {1}{3} b x^3 \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+\frac {1}{6} b e n \left (\frac {2 e^4 \sqrt {x}}{d^5}-\frac {e^3 x}{d^4}+\frac {2 e^2 x^{3/2}}{3 d^3}-\frac {e x^2}{2 d^2}+\frac {2 x^{5/2}}{5 d}-\frac {2 e^5 \log \left (d+\frac {e}{\sqrt {x}}\right )}{d^6}-\frac {e^5 \log (x)}{d^6}\right ) \] Input:
Integrate[x^2*(a + b*Log[c*(d + e/Sqrt[x])^n]),x]
Output:
(a*x^3)/3 + (b*x^3*Log[c*(d + e/Sqrt[x])^n])/3 + (b*e*n*((2*e^4*Sqrt[x])/d ^5 - (e^3*x)/d^4 + (2*e^2*x^(3/2))/(3*d^3) - (e*x^2)/(2*d^2) + (2*x^(5/2)) /(5*d) - (2*e^5*Log[d + e/Sqrt[x]])/d^6 - (e^5*Log[x])/d^6))/6
Time = 0.53 (sec) , antiderivative size = 129, 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.
\(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle -2 \int x^{7/2} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle -2 \left (\frac {1}{6} b e n \int \frac {x^3}{d+\frac {e}{\sqrt {x}}}d\frac {1}{\sqrt {x}}-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )\right )\) |
\(\Big \downarrow \) 54 |
\(\displaystyle -2 \left (\frac {1}{6} b e n \int \left (\frac {e^6}{d^6 \left (d+\frac {e}{\sqrt {x}}\right )}-\frac {\sqrt {x} e^5}{d^6}+\frac {x e^4}{d^5}-\frac {x^{3/2} e^3}{d^4}+\frac {x^2 e^2}{d^3}-\frac {x^{5/2} e}{d^2}+\frac {x^3}{d}\right )d\frac {1}{\sqrt {x}}-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (\frac {1}{6} b e n \left (\frac {e^5 \log \left (d+\frac {e}{\sqrt {x}}\right )}{d^6}-\frac {e^5 \log \left (\frac {1}{\sqrt {x}}\right )}{d^6}-\frac {e^4 \sqrt {x}}{d^5}+\frac {e^3 x}{2 d^4}-\frac {e^2 x^{3/2}}{3 d^3}+\frac {e x^2}{4 d^2}-\frac {x^{5/2}}{5 d}\right )-\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )\right )\) |
Input:
Int[x^2*(a + b*Log[c*(d + e/Sqrt[x])^n]),x]
Output:
-2*(-1/6*(x^3*(a + b*Log[c*(d + e/Sqrt[x])^n])) + (b*e*n*(-((e^4*Sqrt[x])/ d^5) + (e^3*x)/(2*d^4) - (e^2*x^(3/2))/(3*d^3) + (e*x^2)/(4*d^2) - x^(5/2) /(5*d) + (e^5*Log[d + e/Sqrt[x]])/d^6 - (e^5*Log[1/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])
\[\int x^{2} \left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )d x\]
Input:
int(x^2*(a+b*ln(c*(d+e/x^(1/2))^n)),x)
Output:
int(x^2*(a+b*ln(c*(d+e/x^(1/2))^n)),x)
Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.10 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {60 \, b d^{6} x^{3} \log \left (c\right ) - 15 \, b d^{4} e^{2} n x^{2} + 60 \, a d^{6} x^{3} - 30 \, b d^{2} e^{4} n x - 60 \, b d^{6} n \log \left (\sqrt {x}\right ) + 60 \, {\left (b d^{6} - b e^{6}\right )} n \log \left (d \sqrt {x} + e\right ) + 60 \, {\left (b d^{6} n x^{3} - b d^{6} n\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) + 4 \, {\left (3 \, b d^{5} e n x^{2} + 5 \, b d^{3} e^{3} n x + 15 \, b d e^{5} n\right )} \sqrt {x}}{180 \, d^{6}} \] Input:
integrate(x^2*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="fricas")
Output:
1/180*(60*b*d^6*x^3*log(c) - 15*b*d^4*e^2*n*x^2 + 60*a*d^6*x^3 - 30*b*d^2* e^4*n*x - 60*b*d^6*n*log(sqrt(x)) + 60*(b*d^6 - b*e^6)*n*log(d*sqrt(x) + e ) + 60*(b*d^6*n*x^3 - b*d^6*n)*log((d*x + e*sqrt(x))/x) + 4*(3*b*d^5*e*n*x ^2 + 5*b*d^3*e^3*n*x + 15*b*d*e^5*n)*sqrt(x))/d^6
Time = 16.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {a x^{3}}{3} + b \left (\frac {e n \left (\frac {2 x^{\frac {5}{2}}}{5 d} - \frac {e x^{2}}{2 d^{2}} + \frac {2 e^{2} x^{\frac {3}{2}}}{3 d^{3}} - \frac {e^{3} x}{d^{4}} - \frac {2 e^{5} \left (\begin {cases} \frac {\sqrt {x}}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d \sqrt {x} + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{5}} + \frac {2 e^{4} \sqrt {x}}{d^{5}}\right )}{6} + \frac {x^{3} \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}}{3}\right ) \] Input:
integrate(x**2*(a+b*ln(c*(d+e/x**(1/2))**n)),x)
Output:
a*x**3/3 + b*(e*n*(2*x**(5/2)/(5*d) - e*x**2/(2*d**2) + 2*e**2*x**(3/2)/(3 *d**3) - e**3*x/d**4 - 2*e**5*Piecewise((sqrt(x)/e, Eq(d, 0)), (log(d*sqrt (x) + e)/d, True))/d**5 + 2*e**4*sqrt(x)/d**5)/6 + x**3*log(c*(d + e/sqrt( x))**n)/3)
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.69 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {1}{3} \, b x^{3} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + \frac {1}{3} \, a x^{3} - \frac {1}{180} \, b e n {\left (\frac {60 \, e^{5} \log \left (d \sqrt {x} + e\right )}{d^{6}} - \frac {12 \, d^{4} x^{\frac {5}{2}} - 15 \, d^{3} e x^{2} + 20 \, d^{2} e^{2} x^{\frac {3}{2}} - 30 \, d e^{3} x + 60 \, e^{4} \sqrt {x}}{d^{5}}\right )} \] Input:
integrate(x^2*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="maxima")
Output:
1/3*b*x^3*log(c*(d + e/sqrt(x))^n) + 1/3*a*x^3 - 1/180*b*e*n*(60*e^5*log(d *sqrt(x) + e)/d^6 - (12*d^4*x^(5/2) - 15*d^3*e*x^2 + 20*d^2*e^2*x^(3/2) - 30*d*e^3*x + 60*e^4*sqrt(x))/d^5)
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (113) = 226\).
Time = 0.17 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.67 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {1}{3} \, b x^{3} \log \left (c\right ) + \frac {1}{3} \, a x^{3} - \frac {{\left (e^{7} {\left (\frac {60 \, \log \left (\frac {{\left | d \sqrt {x} + e \right |}}{\sqrt {{\left | x \right |}}}\right )}{d^{6}} - \frac {60 \, \log \left ({\left | -d + \frac {d \sqrt {x} + e}{\sqrt {x}} \right |}\right )}{d^{6}} + \frac {137 \, d^{5} - \frac {385 \, {\left (d \sqrt {x} + e\right )} d^{4}}{\sqrt {x}} + \frac {470 \, {\left (d \sqrt {x} + e\right )}^{2} d^{3}}{x} - \frac {270 \, {\left (d \sqrt {x} + e\right )}^{3} d^{2}}{x^{\frac {3}{2}}} + \frac {60 \, {\left (d \sqrt {x} + e\right )}^{4} d}{x^{2}}}{{\left (d - \frac {d \sqrt {x} + e}{\sqrt {x}}\right )}^{5} d^{6}}\right )} - \frac {60 \, e^{7} \log \left (-{\left (e - \frac {d}{\frac {d}{e} - \frac {d \sqrt {x} + e}{e \sqrt {x}}}\right )} {\left (\frac {d}{e} - \frac {d \sqrt {x} + e}{e \sqrt {x}}\right )}\right )}{{\left (d - \frac {d \sqrt {x} + e}{\sqrt {x}}\right )}^{6}}\right )} b n}{180 \, e} \] Input:
integrate(x^2*(a+b*log(c*(d+e/x^(1/2))^n)),x, algorithm="giac")
Output:
1/3*b*x^3*log(c) + 1/3*a*x^3 - 1/180*(e^7*(60*log(abs(d*sqrt(x) + e)/sqrt( abs(x)))/d^6 - 60*log(abs(-d + (d*sqrt(x) + e)/sqrt(x)))/d^6 + (137*d^5 - 385*(d*sqrt(x) + e)*d^4/sqrt(x) + 470*(d*sqrt(x) + e)^2*d^3/x - 270*(d*sqr t(x) + e)^3*d^2/x^(3/2) + 60*(d*sqrt(x) + e)^4*d/x^2)/((d - (d*sqrt(x) + e )/sqrt(x))^5*d^6)) - 60*e^7*log(-(e - d/(d/e - (d*sqrt(x) + e)/(e*sqrt(x)) ))*(d/e - (d*sqrt(x) + e)/(e*sqrt(x))))/(d - (d*sqrt(x) + e)/sqrt(x))^6)*b *n/e
Time = 15.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.76 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {a\,x^3}{3}+\frac {b\,\left (60\,d^6\,x^3\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )-120\,e^6\,n\,\mathrm {atanh}\left (\frac {2\,e}{d\,\sqrt {x}}+1\right )-15\,d^4\,e^2\,n\,x^2+20\,d^3\,e^3\,n\,x^{3/2}-30\,d^2\,e^4\,n\,x+60\,d\,e^5\,n\,\sqrt {x}+12\,d^5\,e\,n\,x^{5/2}\right )}{180\,d^6} \] Input:
int(x^2*(a + b*log(c*(d + e/x^(1/2))^n)),x)
Output:
(a*x^3)/3 + (b*(60*d^6*x^3*log(c*(d + e/x^(1/2))^n) - 120*e^6*n*atanh((2*e )/(d*x^(1/2)) + 1) - 15*d^4*e^2*n*x^2 + 20*d^3*e^3*n*x^(3/2) - 30*d^2*e^4* n*x + 60*d*e^5*n*x^(1/2) + 12*d^5*e*n*x^(5/2)))/(180*d^6)
Time = 0.16 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.98 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \, dx=\frac {12 \sqrt {x}\, b \,d^{5} e n \,x^{2}+20 \sqrt {x}\, b \,d^{3} e^{3} n x +60 \sqrt {x}\, b d \,e^{5} n +60 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b \,d^{6} x^{3}-60 \,\mathrm {log}\left (\frac {\left (\sqrt {x}\, d +e \right )^{n} c}{x^{\frac {n}{2}}}\right ) b \,e^{6}-60 \,\mathrm {log}\left (\sqrt {x}\right ) b \,e^{6} n +60 a \,d^{6} x^{3}-15 b \,d^{4} e^{2} n \,x^{2}-30 b \,d^{2} e^{4} n x}{180 d^{6}} \] Input:
int(x^2*(a+b*log(c*(d+e/x^(1/2))^n)),x)
Output:
(12*sqrt(x)*b*d**5*e*n*x**2 + 20*sqrt(x)*b*d**3*e**3*n*x + 60*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(((sq rt(x)*d + e)**n*c)/x**(n/2))*b*e**6 - 60*log(sqrt(x))*b*e**6*n + 60*a*d**6 *x**3 - 15*b*d**4*e**2*n*x**2 - 30*b*d**2*e**4*n*x)/(180*d**6)