Integrand size = 23, antiderivative size = 84 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b n}{3 d e \sqrt {d+e x^2}}-\frac {b n \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{3 d^{3/2} e}-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}} \]
-1/3*b*n*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(3/2)/e+1/3*(-a-b*ln(c*x^n))/e /(e*x^2+d)^(3/2)+1/3*b*n/d/e/(e*x^2+d)^(1/2)
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.15 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {\frac {a}{\left (d+e x^2\right )^{3/2}}-\frac {b n}{d \sqrt {d+e x^2}}-\frac {b n \log (x)}{d^{3/2}}+\frac {b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}}+\frac {b n \log \left (d+\sqrt {d} \sqrt {d+e x^2}\right )}{d^{3/2}}}{3 e} \]
-1/3*(a/(d + e*x^2)^(3/2) - (b*n)/(d*Sqrt[d + e*x^2]) - (b*n*Log[x])/d^(3/ 2) + (b*Log[c*x^n])/(d + e*x^2)^(3/2) + (b*n*Log[d + Sqrt[d]*Sqrt[d + e*x^ 2]])/d^(3/2))/e
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2776, 243, 61, 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.
\(\displaystyle \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 2776 |
\(\displaystyle \frac {b n \int \frac {1}{x \left (e x^2+d\right )^{3/2}}dx}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {b n \int \frac {1}{x^2 \left (e x^2+d\right )^{3/2}}dx^2}{6 e}-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {b n \left (\frac {\int \frac {1}{x^2 \sqrt {e x^2+d}}dx^2}{d}+\frac {2}{d \sqrt {d+e x^2}}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {b n \left (\frac {2 \int \frac {1}{\frac {x^4}{e}-\frac {d}{e}}d\sqrt {e x^2+d}}{d e}+\frac {2}{d \sqrt {d+e x^2}}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {b n \left (\frac {2}{d \sqrt {d+e x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{3 e \left (d+e x^2\right )^{3/2}}\) |
(b*n*(2/(d*Sqrt[d + e*x^2]) - (2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(3/2) ))/(6*e) - (a + b*Log[c*x^n])/(3*e*(d + e*x^2)^(3/2))
3.3.100.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_)^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]
\[\int \frac {x \left (a +b \ln \left (c \,x^{n}\right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Time = 0.34 (sec) , antiderivative size = 267, normalized size of antiderivative = 3.18 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {{\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {d} \log \left (-\frac {e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {d} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (b d e n x^{2} - b d^{2} n \log \left (x\right ) + b d^{2} n - b d^{2} \log \left (c\right ) - a d^{2}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}}, \frac {{\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d}}{\sqrt {e x^{2} + d}}\right ) + {\left (b d e n x^{2} - b d^{2} n \log \left (x\right ) + b d^{2} n - b d^{2} \log \left (c\right ) - a d^{2}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (d^{2} e^{3} x^{4} + 2 \, d^{3} e^{2} x^{2} + d^{4} e\right )}}\right ] \]
[1/6*((b*e^2*n*x^4 + 2*b*d*e*n*x^2 + b*d^2*n)*sqrt(d)*log(-(e*x^2 - 2*sqrt (e*x^2 + d)*sqrt(d) + 2*d)/x^2) + 2*(b*d*e*n*x^2 - b*d^2*n*log(x) + b*d^2* n - b*d^2*log(c) - a*d^2)*sqrt(e*x^2 + d))/(d^2*e^3*x^4 + 2*d^3*e^2*x^2 + d^4*e), 1/3*((b*e^2*n*x^4 + 2*b*d*e*n*x^2 + b*d^2*n)*sqrt(-d)*arctan(sqrt( -d)/sqrt(e*x^2 + d)) + (b*d*e*n*x^2 - b*d^2*n*log(x) + b*d^2*n - b*d^2*log (c) - a*d^2)*sqrt(e*x^2 + d))/(d^2*e^3*x^4 + 2*d^3*e^2*x^2 + d^4*e)]
Time = 13.69 (sec) , antiderivative size = 272, normalized size of antiderivative = 3.24 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=a \left (\begin {cases} - \frac {1}{3 e \left (d + e x^{2}\right )^{\frac {3}{2}}} & \text {for}\: e \neq 0 \\\frac {x^{2}}{2 d^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} - \frac {2 d^{3} \sqrt {1 + \frac {e x^{2}}{d}}}{6 d^{\frac {9}{2}} e + 6 d^{\frac {7}{2}} e^{2} x^{2}} - \frac {d^{3} \log {\left (\frac {e x^{2}}{d} \right )}}{6 d^{\frac {9}{2}} e + 6 d^{\frac {7}{2}} e^{2} x^{2}} + \frac {2 d^{3} \log {\left (\sqrt {1 + \frac {e x^{2}}{d}} + 1 \right )}}{6 d^{\frac {9}{2}} e + 6 d^{\frac {7}{2}} e^{2} x^{2}} - \frac {d^{2} x^{2} \log {\left (\frac {e x^{2}}{d} \right )}}{6 d^{\frac {9}{2}} + 6 d^{\frac {7}{2}} e x^{2}} + \frac {2 d^{2} x^{2} \log {\left (\sqrt {1 + \frac {e x^{2}}{d}} + 1 \right )}}{6 d^{\frac {9}{2}} + 6 d^{\frac {7}{2}} e x^{2}} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\frac {x^{2}}{4 d^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} - \frac {1}{3 e \left (d + e x^{2}\right )^{\frac {3}{2}}} & \text {for}\: e \neq 0 \\\frac {x^{2}}{2 d^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]
a*Piecewise((-1/(3*e*(d + e*x**2)**(3/2)), Ne(e, 0)), (x**2/(2*d**(5/2)), True)) - b*n*Piecewise((-2*d**3*sqrt(1 + e*x**2/d)/(6*d**(9/2)*e + 6*d**(7 /2)*e**2*x**2) - d**3*log(e*x**2/d)/(6*d**(9/2)*e + 6*d**(7/2)*e**2*x**2) + 2*d**3*log(sqrt(1 + e*x**2/d) + 1)/(6*d**(9/2)*e + 6*d**(7/2)*e**2*x**2) - d**2*x**2*log(e*x**2/d)/(6*d**(9/2) + 6*d**(7/2)*e*x**2) + 2*d**2*x**2* log(sqrt(1 + e*x**2/d) + 1)/(6*d**(9/2) + 6*d**(7/2)*e*x**2), (e > -oo) & (e < oo) & Ne(e, 0)), (x**2/(4*d**(5/2)), True)) + b*Piecewise((-1/(3*e*(d + e*x**2)**(3/2)), Ne(e, 0)), (x**2/(2*d**(5/2)), True))*log(c*x**n)
Exception generated. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]