Integrand size = 21, antiderivative size = 94 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=-\frac {4 b n \sqrt {d+e x}}{e^2}+\frac {8 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2} \]
8*b*n*arctanh((e*x+d)^(1/2)/d^(1/2))*d^(1/2)/e^2+2*d*(a+b*ln(c*x^n))/e^2/( e*x+d)^(1/2)-4*b*n*(e*x+d)^(1/2)/e^2+2*(a+b*ln(c*x^n))*(e*x+d)^(1/2)/e^2
Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.88 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=\frac {2 \left (2 a d-2 b d n+a e x-2 b e n x+4 b \sqrt {d} n \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+b (2 d+e x) \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}} \]
(2*(2*a*d - 2*b*d*n + a*e*x - 2*b*e*n*x + 4*b*Sqrt[d]*n*Sqrt[d + e*x]*ArcT anh[Sqrt[d + e*x]/Sqrt[d]] + b*(2*d + e*x)*Log[c*x^n]))/(e^2*Sqrt[d + e*x] )
Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2792, 27, 90, 73, 221}
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.
\(\displaystyle \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 2792 |
\(\displaystyle -b n \int \frac {2 (2 d+e x)}{e^2 x \sqrt {d+e x}}dx+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 b n \int \frac {2 d+e x}{x \sqrt {d+e x}}dx}{e^2}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -\frac {2 b n \left (2 d \int \frac {1}{x \sqrt {d+e x}}dx+2 \sqrt {d+e x}\right )}{e^2}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {2 b n \left (\frac {4 d \int \frac {1}{\frac {d+e x}{e}-\frac {d}{e}}d\sqrt {d+e x}}{e}+2 \sqrt {d+e x}\right )}{e^2}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{e^2 \sqrt {d+e x}}-\frac {2 b n \left (2 \sqrt {d+e x}-4 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{e^2}\) |
(-2*b*n*(2*Sqrt[d + e*x] - 4*Sqrt[d]*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]))/e^2 + (2*d*(a + b*Log[c*x^n]))/(e^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*L og[c*x^n]))/e^2
3.2.53.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x] }, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2] ) || InverseFunctionFreeQ[u, x]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x ] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])
\[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e x +d \right )^{\frac {3}{2}}}d x\]
Time = 0.38 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.37 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=\left [\frac {2 \, {\left (2 \, {\left (b e n x + b d n\right )} \sqrt {d} \log \left (\frac {e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) - {\left (2 \, b d n - 2 \, a d + {\left (2 \, b e n - a e\right )} x - {\left (b e x + 2 \, b d\right )} \log \left (c\right ) - {\left (b e n x + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt {e x + d}\right )}}{e^{3} x + d e^{2}}, -\frac {2 \, {\left (4 \, {\left (b e n x + b d n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) + {\left (2 \, b d n - 2 \, a d + {\left (2 \, b e n - a e\right )} x - {\left (b e x + 2 \, b d\right )} \log \left (c\right ) - {\left (b e n x + 2 \, b d n\right )} \log \left (x\right )\right )} \sqrt {e x + d}\right )}}{e^{3} x + d e^{2}}\right ] \]
[2*(2*(b*e*n*x + b*d*n)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/ x) - (2*b*d*n - 2*a*d + (2*b*e*n - a*e)*x - (b*e*x + 2*b*d)*log(c) - (b*e* n*x + 2*b*d*n)*log(x))*sqrt(e*x + d))/(e^3*x + d*e^2), -2*(4*(b*e*n*x + b* d*n)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*b*d*n - 2*a*d + (2*b*e *n - a*e)*x - (b*e*x + 2*b*d)*log(c) - (b*e*n*x + 2*b*d*n)*log(x))*sqrt(e* x + d))/(e^3*x + d*e^2)]
Time = 91.74 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.88 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=a \left (\begin {cases} \frac {2 d}{e^{2} \sqrt {d + e x}} + \frac {2 \sqrt {d + e x}}{e^{2}} & \text {for}\: e \neq 0 \\\frac {x^{2}}{2 d^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} - \frac {8 \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} \sqrt {x}} \right )}}{e^{2}} + \frac {4 d}{e^{\frac {5}{2}} \sqrt {x} \sqrt {\frac {d}{e x} + 1}} + \frac {4 \sqrt {x}}{e^{\frac {3}{2}} \sqrt {\frac {d}{e x} + 1}} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\frac {x^{2}}{4 d^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {2 d}{e^{2} \sqrt {d + e x}} + \frac {2 \sqrt {d + e x}}{e^{2}} & \text {for}\: e \neq 0 \\\frac {x^{2}}{2 d^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
a*Piecewise((2*d/(e**2*sqrt(d + e*x)) + 2*sqrt(d + e*x)/e**2, Ne(e, 0)), ( x**2/(2*d**(3/2)), True)) - b*n*Piecewise((-8*sqrt(d)*asinh(sqrt(d)/(sqrt( e)*sqrt(x)))/e**2 + 4*d/(e**(5/2)*sqrt(x)*sqrt(d/(e*x) + 1)) + 4*sqrt(x)/( e**(3/2)*sqrt(d/(e*x) + 1)), (e > -oo) & (e < oo) & Ne(e, 0)), (x**2/(4*d* *(3/2)), True)) + b*Piecewise((2*d/(e**2*sqrt(d + e*x)) + 2*sqrt(d + e*x)/ e**2, Ne(e, 0)), (x**2/(2*d**(3/2)), True))*log(c*x**n)
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.19 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=-4 \, b n {\left (\frac {\sqrt {d} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right )}{e^{2}} + \frac {\sqrt {e x + d}}{e^{2}}\right )} + 2 \, b {\left (\frac {\sqrt {e x + d}}{e^{2}} + \frac {d}{\sqrt {e x + d} e^{2}}\right )} \log \left (c x^{n}\right ) + 2 \, a {\left (\frac {\sqrt {e x + d}}{e^{2}} + \frac {d}{\sqrt {e x + d} e^{2}}\right )} \]
-4*b*n*(sqrt(d)*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d)))/e ^2 + sqrt(e*x + d)/e^2) + 2*b*(sqrt(e*x + d)/e^2 + d/(sqrt(e*x + d)*e^2))* log(c*x^n) + 2*a*(sqrt(e*x + d)/e^2 + d/(sqrt(e*x + d)*e^2))
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]