3.2778 \(\int \frac{(c x)^{-1-\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.033722, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{2 (c x)^{-n/2} \sqrt{a+b x^n}}{a c n} \]

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))

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Rubi in Sympy [A]  time = 3.78073, size = 24, normalized size = 0.77 \[ - \frac{2 \left (c x\right )^{- \frac{n}{2}} \sqrt{a + b x^{n}}}{a c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(-1-1/2*n)/(a+b*x**n)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0386011, size = 31, normalized size = 1. \[ -\frac{2 (c x)^{-n/2} \sqrt{a+b x^n}}{a c n} \]

Antiderivative was successfully verified.

[In]  Integrate[(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))

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-1-{\frac{n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 1.44208, size = 39, normalized size = 1.26 \[ -\frac{2 \, \sqrt{b x^{n} + a} c^{-\frac{1}{2} \, n - 1} x^{-\frac{1}{2} \, n}}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

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)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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