Integrand size = 22, antiderivative size = 58 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {b n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d \sqrt {e}}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}} \] Output:
-b*n*arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/d/e^(1/2)+x*(a+b*ln(c*x^n))/d/(e*x ^2+d)^(1/2)
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {\frac {a x}{\sqrt {d+e x^2}}+\frac {b x \log \left (c x^n\right )}{\sqrt {d+e x^2}}-\frac {b n \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{\sqrt {e}}}{d} \] Input:
Integrate[(a + b*Log[c*x^n])/(d + e*x^2)^(3/2),x]
Output:
((a*x)/Sqrt[d + e*x^2] + (b*x*Log[c*x^n])/Sqrt[d + e*x^2] - (b*n*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/Sqrt[e])/d
Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2751, 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.
\(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2751 |
\(\displaystyle \frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}}-\frac {b n \int \frac {1}{\sqrt {e x^2+d}}dx}{d}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}}-\frac {b n \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {x \left (a+b \log \left (c x^n\right )\right )}{d \sqrt {d+e x^2}}-\frac {b n \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{d \sqrt {e}}\) |
Input:
Int[(a + b*Log[c*x^n])/(d + e*x^2)^(3/2),x]
Output:
-((b*n*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(d*Sqrt[e])) + (x*(a + b*Log[ c*x^n]))/(d*Sqrt[d + e*x^2])
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[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
Input:
int((a+b*ln(c*x^n))/(e*x^2+d)^(3/2),x)
Output:
int((a+b*ln(c*x^n))/(e*x^2+d)^(3/2),x)
Time = 0.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.97 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\left [\frac {{\left (b e n x^{2} + b d n\right )} \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (b e n x \log \left (x\right ) + b e x \log \left (c\right ) + a e x\right )} \sqrt {e x^{2} + d}}{2 \, {\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac {{\left (b e n x^{2} + b d n\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + {\left (b e n x \log \left (x\right ) + b e x \log \left (c\right ) + a e x\right )} \sqrt {e x^{2} + d}}{d e^{2} x^{2} + d^{2} e}\right ] \] Input:
integrate((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(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e) *x - d) + 2*(b*e*n*x*log(x) + b*e*x*log(c) + a*e*x)*sqrt(e*x^2 + d))/(d*e^ 2*x^2 + d^2*e), ((b*e*n*x^2 + b*d*n)*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + (b*e*n*x*log(x) + b*e*x*log(c) + a*e*x)*sqrt(e*x^2 + d))/(d*e^2*x ^2 + d^2*e)]
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*ln(c*x**n))/(e*x**2+d)**(3/2),x)
Output:
Integral((a + b*log(c*x**n))/(d + e*x**2)**(3/2), x)
Exception generated. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((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
\[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*log(c*x^n))/(e*x^2+d)^(3/2),x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)/(e*x^2 + d)^(3/2), x)
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:
int((a + b*log(c*x^n))/(d + e*x^2)^(3/2),x)
Output:
int((a + b*log(c*x^n))/(d + e*x^2)^(3/2), x)
Time = 0.19 (sec) , antiderivative size = 351, normalized size of antiderivative = 6.05 \[ \int \frac {a+b \log \left (c x^n\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {\sqrt {e \,x^{2}+d}\, \mathrm {log}\left (\frac {\left (2 \sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +2 e \,x^{2}\right )^{n} c}{e^{\frac {n}{2}} \left (\sqrt {e \,x^{2}+d}+\sqrt {e}\, x \right )^{n} 2^{n}}\right ) b e x +\sqrt {e \,x^{2}+d}\, a e x -\sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d n -\sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b e n \,x^{2}-\sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b d n -\sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) b e n \,x^{2}+\sqrt {e}\, \mathrm {log}\left (\frac {\left (2 \sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +2 e \,x^{2}\right )^{n} c}{e^{\frac {n}{2}} \left (\sqrt {e \,x^{2}+d}+\sqrt {e}\, x \right )^{n} 2^{n}}\right ) b d +\sqrt {e}\, \mathrm {log}\left (\frac {\left (2 \sqrt {e}\, \sqrt {e \,x^{2}+d}\, x +2 e \,x^{2}\right )^{n} c}{e^{\frac {n}{2}} \left (\sqrt {e \,x^{2}+d}+\sqrt {e}\, x \right )^{n} 2^{n}}\right ) b e \,x^{2}+\sqrt {e}\, a d +\sqrt {e}\, a e \,x^{2}}{d e \left (e \,x^{2}+d \right )} \] Input:
int((a+b*log(c*x^n))/(e*x^2+d)^(3/2),x)
Output:
(sqrt(d + e*x**2)*log(((2*sqrt(e)*sqrt(d + e*x**2)*x + 2*e*x**2)**n*c)/(e* *(n/2)*(sqrt(d + e*x**2) + sqrt(e)*x)**n*2**n))*b*e*x + sqrt(d + e*x**2)*a *e*x - sqrt(e)*log((sqrt(d + e*x**2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*b*d*n - sqrt(e)*log((sqrt(d + e*x**2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*b*e*n*x** 2 - sqrt(e)*log((sqrt(d + e*x**2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*b*d*n - sqrt(e)*log((sqrt(d + e*x**2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*b*e*n*x**2 + sqrt(e)*log(((2*sqrt(e)*sqrt(d + e*x**2)*x + 2*e*x**2)**n*c)/(e**(n/2)*(s qrt(d + e*x**2) + sqrt(e)*x)**n*2**n))*b*d + sqrt(e)*log(((2*sqrt(e)*sqrt( d + e*x**2)*x + 2*e*x**2)**n*c)/(e**(n/2)*(sqrt(d + e*x**2) + sqrt(e)*x)** n*2**n))*b*e*x**2 + sqrt(e)*a*d + sqrt(e)*a*e*x**2)/(d*e*(d + e*x**2))