\(\int \frac {x^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx\) [2692]

Optimal result

Integrand size = 21, antiderivative size = 58 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {2 x^{-3 n/2} \sqrt {a+b x^n}}{3 a n}+\frac {4 b x^{-n/2} \sqrt {a+b x^n}}{3 a^2 n} \]

[Out]

-2/3*(a+b*x^n)^(1/2)/a/n/(x^(3/2*n))+4/3*b*(a+b*x^n)^(1/2)/a^2/n/(x^(1/2*n))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {277, 270} \[ \int \frac {x^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\frac {4 b x^{-n/2} \sqrt {a+b x^n}}{3 a^2 n}-\frac {2 x^{-3 n/2} \sqrt {a+b x^n}}{3 a n} \]

[In]

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

[Out]

(-2*Sqrt[a + b*x^n])/(3*a*n*x^((3*n)/2)) + (4*b*Sqrt[a + b*x^n])/(3*a^2*n*x^(n/2))

Rule 270

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

Rule 277

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^{-3 n/2} \sqrt {a+b x^n}}{3 a n}-\frac {(2 b) \int \frac {x^{-1-\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx}{3 a} \\ & = -\frac {2 x^{-3 n/2} \sqrt {a+b x^n}}{3 a n}+\frac {4 b x^{-n/2} \sqrt {a+b x^n}}{3 a^2 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.62 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {2 x^{-3 n/2} \left (a-2 b x^n\right ) \sqrt {a+b x^n}}{3 a^2 n} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.88 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx=- \frac {2 \sqrt {b} x^{- n} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a n} + \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a^{2} n} \]

[In]

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

[Out]

-2*sqrt(b)*sqrt(a/(b*x**n) + 1)/(3*a*n*x**n) + 4*b**(3/2)*sqrt(a/(b*x**n) + 1)/(3*a**2*n)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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