\(\int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx\) [2783]

Optimal result

Integrand size = 23, antiderivative size = 91 \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {x^{-n} (c x)^{3 n/2} \sqrt {a+b x^n}}{b c n}-\frac {a x^{-3 n/2} (c x)^{3 n/2} \text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} c n} \]

[Out]

-a*(c*x)^(3/2*n)*arctanh(x^(1/2*n)*b^(1/2)/(a+b*x^n)^(1/2))/b^(3/2)/c/n/(x^(3/2*n))+(c*x)^(3/2*n)*(a+b*x^n)^(1
/2)/b/c/n/(x^n)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {364, 362, 294, 212} \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {x^{-n} (c x)^{3 n/2} \sqrt {a+b x^n}}{b c n}-\frac {a x^{-3 n/2} (c x)^{3 n/2} \text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} c n} \]

[In]

Int[(c*x)^(-1 + (3*n)/2)/Sqrt[a + b*x^n],x]

[Out]

((c*x)^((3*n)/2)*Sqrt[a + b*x^n])/(b*c*n*x^n) - (a*(c*x)^((3*n)/2)*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])
/(b^(3/2)*c*n*x^((3*n)/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 362

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[p]}, Dist[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}, x] && IntegerQ[p + Simplify[(m + 1)/n]] && LtQ[-1, p, 0]

Rule 364

Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[c^IntPart[m]*((c*x)^FracPart[m]/x^FracPa
rt[m]), Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IntegerQ[p + Simplify[(m + 1)/n]] && LtQ
[-1, p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-3 n/2} (c x)^{3 n/2}\right ) \int \frac {x^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx}{c} \\ & = \frac {\left (2 a x^{-3 n/2} (c x)^{3 n/2}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1-b x^2\right )^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{c n} \\ & = \frac {x^{-n} (c x)^{3 n/2} \sqrt {a+b x^n}}{b c n}-\frac {\left (a x^{-3 n/2} (c x)^{3 n/2}\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{n/2}}{\sqrt {a+b x^n}}\right )}{b c n} \\ & = \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} \tanh ^{-1}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{b^{3/2} c n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {a x^{1-\frac {3 n}{2}} (c x)^{-1+\frac {3 n}{2}} \sqrt {1+\frac {b x^n}{a}} \left (\sqrt {b} x^{n/2} \sqrt {\frac {a+b x^n}{a}}-\sqrt {a} \text {arcsinh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )\right )}{b^{3/2} n \sqrt {a+b x^n}} \]

[In]

Integrate[(c*x)^(-1 + (3*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(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]*Arc
Sinh[(Sqrt[b]*x^(n/2))/Sqrt[a]]))/(b^(3/2)*n*Sqrt[a + b*x^n])

Maple [F]

\[\int \frac {\left (c x \right )^{-1+\frac {3 n}{2}}}{\sqrt {a +b \,x^{n}}}d x\]

[In]

int((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.56 \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\left [\frac {2 \, \sqrt {b x^{n} + a} b c^{\frac {3}{2} \, n - 1} x^{\frac {1}{2} \, n} + a \sqrt {b} c^{\frac {3}{2} \, n - 1} \log \left (2 \, \sqrt {b x^{n} + a} \sqrt {b} x^{\frac {1}{2} \, n} - 2 \, b x^{n} - a\right )}{2 \, b^{2} n}, \frac {\sqrt {b x^{n} + a} b c^{\frac {3}{2} \, n - 1} x^{\frac {1}{2} \, n} + a \sqrt {-b} c^{\frac {3}{2} \, n - 1} \arctan \left (\frac {\sqrt {-b} x^{\frac {1}{2} \, n}}{\sqrt {b x^{n} + a}}\right )}{b^{2} n}\right ] \]

[In]

integrate((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

[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(sq
rt(-b)*x^(1/2*n)/sqrt(b*x^n + a)))/(b^2*n)]

Sympy [A] (verification not implemented)

Time = 1.40 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {\sqrt {a} c^{\frac {3 n}{2} - 1} x^{\frac {n}{2}} \sqrt {1 + \frac {b x^{n}}{a}}}{b n} - \frac {a c^{\frac {3 n}{2} - 1} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{b^{\frac {3}{2}} n} \]

[In]

integrate((c*x)**(-1+3/2*n)/(a+b*x**n)**(1/2),x)

[Out]

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)

Maxima [F]

\[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \]

[In]

integrate((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate((c*x)^(3/2*n - 1)/sqrt(b*x^n + a), x)

Giac [F]

\[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \]

[In]

integrate((c*x)^(-1+3/2*n)/(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate((c*x)^(3/2*n - 1)/sqrt(b*x^n + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c x)^{-1+\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int \frac {{\left (c\,x\right )}^{\frac {3\,n}{2}-1}}{\sqrt {a+b\,x^n}} \,d x \]

[In]

int((c*x)^((3*n)/2 - 1)/(a + b*x^n)^(1/2),x)

[Out]

int((c*x)^((3*n)/2 - 1)/(a + b*x^n)^(1/2), x)