Integrand size = 25, antiderivative size = 138 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {b e^2 n \sqrt {d+e x^2}}{5 d x}-\frac {b e n \left (d+e x^2\right )^{3/2}}{15 d x^3}-\frac {b n \left (d+e x^2\right )^{5/2}}{25 d x^5}+\frac {b e^{5/2} n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5} \]
-1/15*b*e*n*(e*x^2+d)^(3/2)/d/x^3-1/25*b*n*(e*x^2+d)^(5/2)/d/x^5+1/5*b*e^( 5/2)*n*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/d-1/5*(e*x^2+d)^(5/2)*(a+b*ln(c* x^n))/d/x^5-1/5*b*e^2*n*(e*x^2+d)^(1/2)/d/x
Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=-\frac {\sqrt {d+e x^2} \left (15 a \left (d+e x^2\right )^2+b n \left (3 d^2+11 d e x^2+23 e^2 x^4\right )\right )+15 b \left (d+e x^2\right )^{5/2} \log \left (c x^n\right )-15 b e^{5/2} n x^5 \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{75 d x^5} \]
-1/75*(Sqrt[d + e*x^2]*(15*a*(d + e*x^2)^2 + b*n*(3*d^2 + 11*d*e*x^2 + 23* e^2*x^4)) + 15*b*(d + e*x^2)^(5/2)*Log[c*x^n] - 15*b*e^(5/2)*n*x^5*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(d*x^5)
Time = 0.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2773, 247, 247, 247, 224, 219}
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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx\) |
\(\Big \downarrow \) 2773 |
\(\displaystyle \frac {b n \int \frac {\left (e x^2+d\right )^{5/2}}{x^6}dx}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {b n \left (e \int \frac {\left (e x^2+d\right )^{3/2}}{x^4}dx-\frac {\left (d+e x^2\right )^{5/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {b n \left (e \left (e \int \frac {\sqrt {e x^2+d}}{x^2}dx-\frac {\left (d+e x^2\right )^{3/2}}{3 x^3}\right )-\frac {\left (d+e x^2\right )^{5/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {b n \left (e \left (e \left (e \int \frac {1}{\sqrt {e x^2+d}}dx-\frac {\sqrt {d+e x^2}}{x}\right )-\frac {\left (d+e x^2\right )^{3/2}}{3 x^3}\right )-\frac {\left (d+e x^2\right )^{5/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {b n \left (e \left (e \left (e \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}-\frac {\sqrt {d+e x^2}}{x}\right )-\frac {\left (d+e x^2\right )^{3/2}}{3 x^3}\right )-\frac {\left (d+e x^2\right )^{5/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {b n \left (e \left (e \left (\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {d+e x^2}}{x}\right )-\frac {\left (d+e x^2\right )^{3/2}}{3 x^3}\right )-\frac {\left (d+e x^2\right )^{5/2}}{5 x^5}\right )}{5 d}-\frac {\left (d+e x^2\right )^{5/2} \left (a+b \log \left (c x^n\right )\right )}{5 d x^5}\) |
(b*n*(-1/5*(d + e*x^2)^(5/2)/x^5 + e*(-1/3*(d + e*x^2)^(3/2)/x^3 + e*(-(Sq rt[d + e*x^2]/x) + Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]))))/(5*d) - ((d + e*x^2)^(5/2)*(a + b*Log[c*x^n]))/(5*d*x^5)
3.3.72.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Simp[b*(n/(d*(m + 1))) Int[(f*x)^m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && Eq Q[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{6}}d x\]
Time = 0.36 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.28 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\left [\frac {15 \, b e^{\frac {5}{2}} n x^{5} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - 2 \, {\left ({\left (23 \, b e^{2} n + 15 \, a e^{2}\right )} x^{4} + 3 \, b d^{2} n + 15 \, a d^{2} + {\left (11 \, b d e n + 30 \, a d e\right )} x^{2} + 15 \, {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \log \left (c\right ) + 15 \, {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{150 \, d x^{5}}, -\frac {15 \, b \sqrt {-e} e^{2} n x^{5} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left ({\left (23 \, b e^{2} n + 15 \, a e^{2}\right )} x^{4} + 3 \, b d^{2} n + 15 \, a d^{2} + {\left (11 \, b d e n + 30 \, a d e\right )} x^{2} + 15 \, {\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \log \left (c\right ) + 15 \, {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \log \left (x\right )\right )} \sqrt {e x^{2} + d}}{75 \, d x^{5}}\right ] \]
[1/150*(15*b*e^(5/2)*n*x^5*log(-2*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) - 2*((23*b*e^2*n + 15*a*e^2)*x^4 + 3*b*d^2*n + 15*a*d^2 + (11*b*d*e*n + 3 0*a*d*e)*x^2 + 15*(b*e^2*x^4 + 2*b*d*e*x^2 + b*d^2)*log(c) + 15*(b*e^2*n*x ^4 + 2*b*d*e*n*x^2 + b*d^2*n)*log(x))*sqrt(e*x^2 + d))/(d*x^5), -1/75*(15* b*sqrt(-e)*e^2*n*x^5*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + ((23*b*e^2*n + 1 5*a*e^2)*x^4 + 3*b*d^2*n + 15*a*d^2 + (11*b*d*e*n + 30*a*d*e)*x^2 + 15*(b* e^2*x^4 + 2*b*d*e*x^2 + b*d^2)*log(c) + 15*(b*e^2*n*x^4 + 2*b*d*e*n*x^2 + b*d^2*n)*log(x))*sqrt(e*x^2 + d))/(d*x^5)]
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^6} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^6} \,d x \]