Integrand size = 23, antiderivative size = 57 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {b n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d} e}-\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}} \] Output:
-b*n*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(1/2)/e-(a+b*ln(c*x^n))/e/(e*x^2+d )^(1/2)
Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.35 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {\frac {a}{\sqrt {d+e x^2}}-\frac {b n \log (x)}{\sqrt {d}}+\frac {b \log \left (c x^n\right )}{\sqrt {d+e x^2}}+\frac {b n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{\sqrt {d}}}{e} \] Input:
Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x^2)^(3/2),x]
Output:
-((a/Sqrt[d + e*x^2] - (b*n*Log[x])/Sqrt[d] + (b*Log[c*x^n])/Sqrt[d + e*x^ 2] + (b*n*Log[d + Sqrt[d]*Sqrt[d + e*x^2]])/Sqrt[d])/e)
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2776, 243, 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.
\(\displaystyle \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2776 |
\(\displaystyle \frac {b n \int \frac {1}{x \sqrt {e x^2+d}}dx}{e}-\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {b n \int \frac {1}{x^2 \sqrt {e x^2+d}}dx^2}{2 e}-\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {b n \int \frac {1}{\frac {x^4}{e}-\frac {d}{e}}d\sqrt {e x^2+d}}{e^2}-\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {a+b \log \left (c x^n\right )}{e \sqrt {d+e x^2}}-\frac {b n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d} e}\) |
Input:
Int[(x*(a + b*Log[c*x^n]))/(d + e*x^2)^(3/2),x]
Output:
-((b*n*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(Sqrt[d]*e)) - (a + b*Log[c*x^n]) /(e*Sqrt[d + e*x^2])
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_)^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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :> Simp[f^m*(d + e*x^r)^(q + 1)*((a + b*L og[c*x^n])^p/(e*r*(q + 1))), x] - Simp[b*f^m*n*(p/(e*r*(q + 1))) Int[(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d , e, f, m, n, q, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || G tQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]
\[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
Input:
int(x*(a+b*ln(c*x^n))/(e*x^2+d)^(3/2),x)
Output:
int(x*(a+b*ln(c*x^n))/(e*x^2+d)^(3/2),x)
Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 3.02 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\left [\frac {{\left (b e n x^{2} + b d n\right )} \sqrt {d} \log \left (-\frac {e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \sqrt {e x^{2} + d}}{2 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac {{\left (b e n x^{2} + b d n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {e x^{2} + d} \sqrt {-d}}{d}\right ) - {\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \sqrt {e x^{2} + d}}{d e^{2} x^{2} + d^{2} e}\right ] \] Input:
integrate(x*(a+b*log(c*x^n))/(e*x^2+d)^(3/2),x, algorithm="fricas")
Output:
[1/2*((b*e*n*x^2 + b*d*n)*sqrt(d)*log(-(e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(d) + 2*d)/x^2) - 2*(b*d*n*log(x) + b*d*log(c) + a*d)*sqrt(e*x^2 + d))/(d*e^2* x^2 + d^2*e), ((b*e*n*x^2 + b*d*n)*sqrt(-d)*arctan(sqrt(e*x^2 + d)*sqrt(-d )/d) - (b*d*n*log(x) + b*d*log(c) + a*d)*sqrt(e*x^2 + d))/(d*e^2*x^2 + d^2 *e)]
Time = 5.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.65 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=a \left (\begin {cases} - \frac {1}{e \sqrt {d + e x^{2}}} & \text {for}\: e \neq 0 \\\frac {x^{2}}{2 d^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} \frac {x^{2}}{4 d^{\frac {3}{2}}} & \text {for}\: e = 0 \\\frac {\operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} x} \right )}}{\sqrt {d} e} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {x^{2}}{2 d^{\frac {3}{2}}} & \text {for}\: e = 0 \\- \frac {1}{e \sqrt {d + e x^{2}}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \] Input:
integrate(x*(a+b*ln(c*x**n))/(e*x**2+d)**(3/2),x)
Output:
a*Piecewise((-1/(e*sqrt(d + e*x**2)), Ne(e, 0)), (x**2/(2*d**(3/2)), True) ) - b*n*Piecewise((x**2/(4*d**(3/2)), Eq(e, 0)), (asinh(sqrt(d)/(sqrt(e)*x ))/(sqrt(d)*e), True)) + b*Piecewise((x**2/(2*d**(3/2)), Eq(e, 0)), (-1/(e *sqrt(d + e*x**2)), True))*log(c*x**n)
Exception generated. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x*(a+b*log(c*x^n))/(e*x^2+d)^(3/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(x*(a+b*log(c*x^n))/(e*x^2+d)^(3/2),x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)*x/(e*x^2 + d)^(3/2), x)
Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:
int((x*(a + b*log(c*x^n)))/(d + e*x^2)^(3/2),x)
Output:
int((x*(a + b*log(c*x^n)))/(d + e*x^2)^(3/2), x)
Time = 0.16 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.82 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, \mathrm {log}\left (\frac {d^{\frac {n}{2}} \left (\sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +e \,x^{2}\right )^{n} c}{e^{\frac {n}{2}} \left (\sqrt {d}\, \sqrt {e \,x^{2}+d}+\sqrt {e}\, \sqrt {d}\, x \right )^{n}}\right ) b d -\sqrt {e \,x^{2}+d}\, a d +\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d n +\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b e n \,x^{2}-\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d n -\sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b e n \,x^{2}}{d e \left (e \,x^{2}+d \right )} \] Input:
int(x*(a+b*log(c*x^n))/(e*x^2+d)^(3/2),x)
Output:
( - sqrt(d + e*x**2)*log((d**(n/2)*(sqrt(e)*sqrt(d + e*x**2)*x + e*x**2)** n*c)/(e**(n/2)*(sqrt(d)*sqrt(d + e*x**2) + sqrt(e)*sqrt(d)*x)**n))*b*d - s qrt(d + e*x**2)*a*d + sqrt(d)*log((sqrt(d + e*x**2) - sqrt(d) + sqrt(e)*x) /sqrt(d))*b*d*n + sqrt(d)*log((sqrt(d + e*x**2) - sqrt(d) + sqrt(e)*x)/sqr t(d))*b*e*n*x**2 - sqrt(d)*log((sqrt(d + e*x**2) + sqrt(d) + sqrt(e)*x)/sq rt(d))*b*d*n - sqrt(d)*log((sqrt(d + e*x**2) + sqrt(d) + sqrt(e)*x)/sqrt(d ))*b*e*n*x**2)/(d*e*(d + e*x**2))