\(\int \frac {(c x)^{-1+\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx\) [2784]

Optimal result

Integrand size = 23, antiderivative size = 54 \[ \int \frac {(c x)^{-1+\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {2 x^{-n/2} (c x)^{n/2} \text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{\sqrt {b} c n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, 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 = {354, 352, 223, 212} \[ \int \frac {(c x)^{-1+\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {2 x^{-n/2} (c x)^{n/2} \text {arctanh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a+b x^n}}\right )}{\sqrt {b} c n} \]

[In]

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

[Out]

(2*(c*x)^(n/2)*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/(Sqrt[b]*c*n*x^(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 223

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]

Rule 352

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m +
1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

Rule 354

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, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !In
tegerQ[n]

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int \frac {(c x)^{-1+\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {2 \sqrt {a} x^{-n/2} (c x)^{n/2} \sqrt {1+\frac {b x^n}{a}} \text {arcsinh}\left (\frac {\sqrt {b} x^{n/2}}{\sqrt {a}}\right )}{\sqrt {b} c n \sqrt {a+b x^n}} \]

[In]

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

[Out]

(2*Sqrt[a]*(c*x)^(n/2)*Sqrt[1 + (b*x^n)/a]*ArcSinh[(Sqrt[b]*x^(n/2))/Sqrt[a]])/(Sqrt[b]*c*n*x^(n/2)*Sqrt[a + b
*x^n])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {(c x)^{-1+\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.57 \[ \int \frac {(c x)^{-1+\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {2 c^{\frac {n}{2} - 1} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{\sqrt {b} n} \]

[In]

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

[Out]

2*c**(n/2 - 1)*asinh(sqrt(b)*x**(n/2)/sqrt(a))/(sqrt(b)*n)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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