3.2692 \(\int \frac {x^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx\)

Optimal. Leaf size=58 \[ \frac {4 b x^{-n/2} \sqrt {a+b x^n}}{3 a^2 n}-\frac {2 x^{-3 n/2} \sqrt {a+b x^n}}{3 a n} \]

[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))

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Rubi [A]  time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {271, 264} \[ \frac {4 b x^{-n/2} \sqrt {a+b x^n}}{3 a^2 n}-\frac {2 x^{-3 n/2} \sqrt {a+b x^n}}{3 a n} \]

Antiderivative was successfully verified.

[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 264

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 271

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*} \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 {(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*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 0.62 \[ -\frac {2 x^{-3 n/2} \left (a-2 b x^n\right ) \sqrt {a+b x^n}}{3 a^2 n} \]

Antiderivative was successfully verified.

[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))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{-\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {x^{-\frac {3 n}{2}-1}}{\sqrt {b \,x^{n}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{-\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^{\frac {3\,n}{2}+1}\,\sqrt {a+b\,x^n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 5.96, size = 51, normalized size = 0.88 \[ - \frac {2 \sqrt {b} x^{- n} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a n} + \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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