3.2785 \(\int \frac {(c x)^{-1-\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx\)

Optimal. Leaf size=31 \[ -\frac {2 (c x)^{-n/2} \sqrt {a+b x^n}}{a c n} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {264} \[ -\frac {2 (c x)^{-n/2} \sqrt {a+b x^n}}{a c n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*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]

Rubi steps

\begin {align*} \int \frac {(c x)^{-1-\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx &=-\frac {2 (c x)^{-n/2} \sqrt {a+b x^n}}{a c n}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 31, normalized size = 1.00 \[ -\frac {2 x (c x)^{-\frac {n}{2}-1} \sqrt {a+b x^n}}{a n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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((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 (ha
s polynomial part)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 1.21, size = 29, normalized size = 0.94 \[ -\frac {2\,x\,\sqrt {a+b\,x^n}}{a\,n\,{\left (c\,x\right )}^{\frac {n}{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 5.45, size = 29, normalized size = 0.94 \[ - \frac {2 \sqrt {b} c^{- \frac {n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{a c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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