Integrand size = 22, antiderivative size = 98 \[ \int c^2 x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx=\frac {2 a c^2 x \sqrt {\frac {a}{x^2}+b x^n}}{2+n}+\frac {2 c^2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (2+n)}-\frac {2 a^{3/2} c^2 \text {arctanh}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{2+n} \] Output:
2*a*c^2*x*(a/x^2+b*x^n)^(1/2)/(2+n)+2*c^2*x^3*(a/x^2+b*x^n)^(3/2)/(6+3*n)- 2*a^(3/2)*c^2*arctanh(a^(1/2)/x/(a/x^2+b*x^n)^(1/2))/(2+n)
Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96 \[ \int c^2 x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx=\frac {2 c^2 x \sqrt {\frac {a}{x^2}+b x^n} \left (\sqrt {a+b x^{2+n}} \left (4 a+b x^{2+n}\right )-3 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^{2+n}}}{\sqrt {a}}\right )\right )}{3 (2+n) \sqrt {a+b x^{2+n}}} \] Input:
Integrate[c^2*x^2*(a/x^2 + b*x^n)^(3/2),x]
Output:
(2*c^2*x*Sqrt[a/x^2 + b*x^n]*(Sqrt[a + b*x^(2 + n)]*(4*a + b*x^(2 + n)) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^(2 + n)]/Sqrt[a]]))/(3*(2 + n)*Sqrt[a + b*x ^(2 + n)])
Time = 0.42 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {27, 1934, 1913, 1935, 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.
\(\displaystyle \int c^2 x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^2 \int x^2 \left (b x^n+\frac {a}{x^2}\right )^{3/2}dx\) |
\(\Big \downarrow \) 1934 |
\(\displaystyle c^2 \left (a \int \sqrt {b x^n+\frac {a}{x^2}}dx+\frac {2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (n+2)}\right )\) |
\(\Big \downarrow \) 1913 |
\(\displaystyle c^2 \left (a \left (a \int \frac {1}{x^2 \sqrt {b x^n+\frac {a}{x^2}}}dx+\frac {2 x \sqrt {\frac {a}{x^2}+b x^n}}{n+2}\right )+\frac {2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (n+2)}\right )\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle c^2 \left (a \left (\frac {2 x \sqrt {\frac {a}{x^2}+b x^n}}{n+2}-\frac {2 a \int \frac {1}{1-\frac {a}{x^2 \left (b x^n+\frac {a}{x^2}\right )}}d\frac {1}{x \sqrt {b x^n+\frac {a}{x^2}}}}{n+2}\right )+\frac {2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (n+2)}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle c^2 \left (a \left (\frac {2 x \sqrt {\frac {a}{x^2}+b x^n}}{n+2}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{n+2}\right )+\frac {2 x^3 \left (\frac {a}{x^2}+b x^n\right )^{3/2}}{3 (n+2)}\right )\) |
Input:
Int[c^2*x^2*(a/x^2 + b*x^n)^(3/2),x]
Output:
c^2*((2*x^3*(a/x^2 + b*x^n)^(3/2))/(3*(2 + n)) + a*((2*x*Sqrt[a/x^2 + b*x^ n])/(2 + n) - (2*Sqrt[a]*ArcTanh[Sqrt[a]/(x*Sqrt[a/x^2 + b*x^n])])/(2 + n) ))
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_)^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[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[x*((a*x^j + b*x^n)^p/(p*(n - j))), x] + Simp[a Int[x^j*(a*x^j + b*x^n)^(p - 1), x] , x] /; FreeQ[{a, b, j, n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simp lify[j*p + 1], 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*p*(n - j))), x] + Simp[a/c^j Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, j, m , n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
\[\int c^{2} x^{2} \left (\frac {a}{x^{2}}+b \,x^{n}\right )^{\frac {3}{2}}d x\]
Input:
int(c^2*x^2*(a/x^2+b*x^n)^(3/2),x)
Output:
int(c^2*x^2*(a/x^2+b*x^n)^(3/2),x)
Exception generated. \[ \int c^2 x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(c^2*x^2*(a/x^2+b*x^n)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int c^2 x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx=c^{2} \left (\int a \sqrt {\frac {a}{x^{2}} + b x^{n}}\, dx + \int b x^{2} x^{n} \sqrt {\frac {a}{x^{2}} + b x^{n}}\, dx\right ) \] Input:
integrate(c**2*x**2*(a/x**2+b*x**n)**(3/2),x)
Output:
c**2*(Integral(a*sqrt(a/x**2 + b*x**n), x) + Integral(b*x**2*x**n*sqrt(a/x **2 + b*x**n), x))
\[ \int c^2 x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx=\int { {\left (b x^{n} + \frac {a}{x^{2}}\right )}^{\frac {3}{2}} c^{2} x^{2} \,d x } \] Input:
integrate(c^2*x^2*(a/x^2+b*x^n)^(3/2),x, algorithm="maxima")
Output:
c^2*integrate((b*x^n + a/x^2)^(3/2)*x^2, x)
\[ \int c^2 x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx=\int { {\left (b x^{n} + \frac {a}{x^{2}}\right )}^{\frac {3}{2}} c^{2} x^{2} \,d x } \] Input:
integrate(c^2*x^2*(a/x^2+b*x^n)^(3/2),x, algorithm="giac")
Output:
integrate((b*x^n + a/x^2)^(3/2)*c^2*x^2, x)
Timed out. \[ \int c^2 x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx=\int c^2\,x^2\,{\left (b\,x^n+\frac {a}{x^2}\right )}^{3/2} \,d x \] Input:
int(c^2*x^2*(b*x^n + a/x^2)^(3/2),x)
Output:
int(c^2*x^2*(b*x^n + a/x^2)^(3/2), x)
\[ \int c^2 x^2 \left (\frac {a}{x^2}+b x^n\right )^{3/2} \, dx=c^{2} \left (\left (\int \frac {\sqrt {x^{n} b \,x^{2}+a}}{x}d x \right ) a +\left (\int x^{n} \sqrt {x^{n} b \,x^{2}+a}\, x d x \right ) b \right ) \] Input:
int(c^2*x^2*(a/x^2+b*x^n)^(3/2),x)
Output:
c**2*(int(sqrt(x**n*b*x**2 + a)/x,x)*a + int(x**n*sqrt(x**n*b*x**2 + a)*x, x)*b)