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 = 0.0511551, 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, Rules used = {2011, 365, 364} \[ \frac{2 x \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2 (j-n)};\frac{1-\frac{n}{2}}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{(2-n) \sqrt{a x^j+b x^n}} \]

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])

Rule 2011

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a x^j+b x^n}} \, dx &=\frac{\left (x^{n/2} \sqrt{b+a x^{j-n}}\right ) \int \frac{x^{-n/2}}{\sqrt{b+a x^{j-n}}} \, dx}{\sqrt{a x^j+b x^n}}\\ &=\frac{\left (x^{n/2} \sqrt{1+\frac{a x^{j-n}}{b}}\right ) \int \frac{x^{-n/2}}{\sqrt{1+\frac{a x^{j-n}}{b}}} \, dx}{\sqrt{a x^j+b x^n}}\\ &=\frac{2 x \sqrt{1+\frac{a x^{j-n}}{b}} \, _2F_1\left (\frac{1}{2},\frac{2-n}{2 (j-n)};1+\frac{1-\frac{n}{2}}{j-n};-\frac{a x^{j-n}}{b}\right )}{(2-n) \sqrt{a x^j+b x^n}}\\ \end{align*}

Mathematica [A]  time = 0.0510112, 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 = 0.357, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a{x}^{j}+b{x}^{n}}}}\, dx \end{align*}

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. \begin{align*} \int \frac{1}{\sqrt{a x^{j} + b x^{n}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x^j+b*x^n)^(1/2),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. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*x^j + b*x^n), x)