Integrand size = 23, antiderivative size = 91 \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {x^{-n} (c x)^{3 n/2} \sqrt {a+b x^n}}{b c n}-\frac {a x^{-3 n/2} (c x)^{3 n/2} \text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} c n} \] Output:
(c*x)^(3/2*n)*(a+b*x^n)^(1/2)/b/c/n/(x^n)-a*(c*x)^(3/2*n)*arctanh(b^(1/2)* x^(1/2*n)/(a+b*x^n)^(1/2))/b^(3/2)/c/n/(x^(3/2*n))
Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {a x^{1-\frac {3 n}{2}} (c x)^{-1+\frac {3 n}{2}} \sqrt {1+\frac {b x^n}{a}} \left (\sqrt {b} x^{n/2} \sqrt {\frac {a+b x^n}{a}}-\sqrt {a} \text {arcsinh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )\right )}{b^{3/2} n \sqrt {a+b x^n}} \] Input:
Integrate[(c*x)^(-1 + (3*n)/2)/Sqrt[a + b*x^n],x]
Output:
(a*x^(1 - (3*n)/2)*(c*x)^(-1 + (3*n)/2)*Sqrt[1 + (b*x^n)/a]*(Sqrt[b]*x^(n/ 2)*Sqrt[(a + b*x^n)/a] - Sqrt[a]*ArcSinh[(Sqrt[b]*x^(n/2))/Sqrt[a]]))/(b^( 3/2)*n*Sqrt[a + b*x^n])
Time = 0.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {883, 880, 252, 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.
\(\displaystyle \int \frac {(c x)^{\frac {3 n}{2}-1}}{\sqrt {a+b x^n}} \, dx\) |
\(\Big \downarrow \) 883 |
\(\displaystyle \frac {x^{-3 n/2} (c x)^{3 n/2} \int \frac {x^{\frac {3 n}{2}-1}}{\sqrt {b x^n+a}}dx}{c}\) |
\(\Big \downarrow \) 880 |
\(\displaystyle \frac {2 a x^{-3 n/2} (c x)^{3 n/2} \int \frac {x^n}{\left (b x^n+a\right ) \left (1-\frac {b x^n}{b x^n+a}\right )^2}d\frac {x^{n/2}}{\sqrt {b x^n+a}}}{c n}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {2 a x^{-3 n/2} (c x)^{3 n/2} \left (\frac {x^{n/2}}{2 b \sqrt {a+b x^n} \left (1-\frac {b x^n}{a+b x^n}\right )}-\frac {\int \frac {1}{1-\frac {b x^n}{b x^n+a}}d\frac {x^{n/2}}{\sqrt {b x^n+a}}}{2 b}\right )}{c n}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 a x^{-3 n/2} (c x)^{3 n/2} \left (\frac {x^{n/2}}{2 b \sqrt {a+b x^n} \left (1-\frac {b x^n}{a+b x^n}\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{2 b^{3/2}}\right )}{c n}\) |
Input:
Int[(c*x)^(-1 + (3*n)/2)/Sqrt[a + b*x^n],x]
Output:
(2*a*(c*x)^((3*n)/2)*(x^(n/2)/(2*b*Sqrt[a + b*x^n]*(1 - (b*x^n)/(a + b*x^n ))) - ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]]/(2*b^(3/2))))/(c*n*x^((3* n)/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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[p]}, Simp[k*(a^(p + Simplify[(m + 1)/n])/n) Subst[Int[x^(k*Simplify [(m + 1)/n] - 1)/(1 - b*x^k)^(p + Simplify[(m + 1)/n] + 1), x], x, x^(n/k)/ (a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[p + Simpli fy[(m + 1)/n]] && LtQ[-1, p, 0]
Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^Int Part[m]*((c*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a + b*x^n)^p, x], x] / ; FreeQ[{a, b, c, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n + p]]
\[\int \frac {\left (c x \right )^{-1+\frac {3 n}{2}}}{\sqrt {a +b \,x^{n}}}d x\]
Input:
int((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x)
Output:
int((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x)
Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.62 \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\left [\frac {2 \, \sqrt {b x^{n} + a} b c^{\frac {3}{2} \, n - 1} x^{\frac {1}{2} \, n} + a \sqrt {b} c^{\frac {3}{2} \, n - 1} \log \left (2 \, \sqrt {b x^{n} + a} \sqrt {b} x^{\frac {1}{2} \, n} - 2 \, b x^{n} - a\right )}{2 \, b^{2} n}, \frac {\sqrt {b x^{n} + a} b c^{\frac {3}{2} \, n - 1} x^{\frac {1}{2} \, n} + a \sqrt {-b} c^{\frac {3}{2} \, n - 1} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-b}}{b x^{\frac {1}{2} \, n}}\right )}{b^{2} n}\right ] \] Input:
integrate((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")
Output:
[1/2*(2*sqrt(b*x^n + a)*b*c^(3/2*n - 1)*x^(1/2*n) + a*sqrt(b)*c^(3/2*n - 1 )*log(2*sqrt(b*x^n + a)*sqrt(b)*x^(1/2*n) - 2*b*x^n - a))/(b^2*n), (sqrt(b *x^n + a)*b*c^(3/2*n - 1)*x^(1/2*n) + a*sqrt(-b)*c^(3/2*n - 1)*arctan(sqrt (b*x^n + a)*sqrt(-b)/(b*x^(1/2*n))))/(b^2*n)]
Time = 1.65 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {\sqrt {a} c^{\frac {3 n}{2} - 1} x^{\frac {n}{2}} \sqrt {1 + \frac {b x^{n}}{a}}}{b n} - \frac {a c^{\frac {3 n}{2} - 1} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}} n} \] Input:
integrate((c*x)**(-1+3/2*n)/(a+b*x**n)**(1/2),x)
Output:
sqrt(a)*c**(3*n/2 - 1)*x**(n/2)*sqrt(1 + b*x**n/a)/(b*n) - a*c**(3*n/2 - 1 )*asinh(sqrt(b)*x**(n/2)/sqrt(a))/(b**(3/2)*n)
\[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:
integrate((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")
Output:
integrate((c*x)^(3/2*n - 1)/sqrt(b*x^n + a), x)
\[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:
integrate((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x, algorithm="giac")
Output:
integrate((c*x)^(3/2*n - 1)/sqrt(b*x^n + a), x)
Timed out. \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int \frac {{\left (c\,x\right )}^{\frac {3\,n}{2}-1}}{\sqrt {a+b\,x^n}} \,d x \] Input:
int((c*x)^((3*n)/2 - 1)/(a + b*x^n)^(1/2),x)
Output:
int((c*x)^((3*n)/2 - 1)/(a + b*x^n)^(1/2), x)
\[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {c^{\frac {3 n}{2}} \left (\int \frac {x^{\frac {3 n}{2}} \sqrt {x^{n} b +a}}{x^{n} b x +a x}d x \right )}{c} \] Input:
int((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x)
Output:
(c**((3*n)/2)*int((x**((3*n)/2)*sqrt(x**n*b + a))/(x**n*b*x + a*x),x))/c