3.428 \(\int \frac{1}{\sqrt{a x^j+b x^n}} \, dx\)

Optimal. Leaf size=93 \[ \frac{2 x \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2 (j-n)};\frac{1-\frac{n}{2}}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{(2-n) \sqrt{a x^j+b x^n}} \]

[Out]

(2*x*Sqrt[1 + (a*x^(j - n))/b]*Hypergeometric2F1[1/2, (2 - n)/(2*(j - n)), 1 + (
1 - n/2)/(j - n), -((a*x^(j - n))/b)])/((2 - n)*Sqrt[a*x^j + b*x^n])

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Rubi [A]  time = 0.12887, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 x \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2 (j-n)};\frac{1-\frac{n}{2}}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{(2-n) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a*x^j + b*x^n],x]

[Out]

(2*x*Sqrt[1 + (a*x^(j - n))/b]*Hypergeometric2F1[1/2, (2 - n)/(2*(j - n)), 1 + (
1 - n/2)/(j - n), -((a*x^(j - n))/b)])/((2 - n)*Sqrt[a*x^j + b*x^n])

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Rubi in Sympy [A]  time = 13.1735, size = 78, normalized size = 0.84 \[ \frac{2 x^{- \frac{n}{2}} x^{- \frac{n}{2} + 1} \sqrt{a x^{j} + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - \frac{n - 2}{2 \left (j - n\right )} \\ \frac{j - \frac{3 n}{2} + 1}{j - n} \end{matrix}\middle |{- \frac{a x^{j - n}}{b}} \right )}}{b \left (- n + 2\right ) \sqrt{\frac{a x^{j - n}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(-n/2)*x**(-n/2 + 1)*sqrt(a*x**j + b*x**n)*hyper((1/2, -(n - 2)/(2*(j - n))
), ((j - 3*n/2 + 1)/(j - n),), -a*x**(j - n)/b)/(b*(-n + 2)*sqrt(a*x**(j - n)/b
+ 1))

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Mathematica [A]  time = 0.0750133, size = 88, normalized size = 0.95 \[ -\frac{2 x \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{n-2}{2 (n-j)};\frac{n-2}{2 (n-j)}+1;-\frac{a x^{j-n}}{b}\right )}{(n-2) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a*x^j + b*x^n],x]

[Out]

(-2*x*Sqrt[1 + (a*x^(j - n))/b]*Hypergeometric2F1[1/2, (-2 + n)/(2*(-j + n)), 1
+ (-2 + n)/(2*(-j + n)), -((a*x^(j - n))/b)])/((-2 + n)*Sqrt[a*x^j + b*x^n])

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{a{x}^{j}+b{x}^{n}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x^j+b*x^n)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x^{j} + b x^{n}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(a*x^j + b*x^n), x)

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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(1/sqrt(a*x^j + b*x^n),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x^{j} + b x^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x**j+b*x**n)**(1/2),x)

[Out]

Integral(1/sqrt(a*x**j + b*x**n), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x^{j} + b x^{n}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(a*x^j + b*x^n), x)