3.5.28 \(\int \frac {1}{\sqrt {a x^j+b x^n}} \, dx\) [428]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2036, 372, 371} \begin {gather*} \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}} \end {gather*}

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 2036

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]

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*}

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Mathematica [A]
time = 0.06, size = 88, normalized size = 0.95 \begin {gather*} -\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 {-2+n}{2 (-j+n)};-\frac {a x^{j-n}}{b}\right )}{(-2+n) \sqrt {a x^j+b x^n}} \end {gather*}

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 = 0.09, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a \,x^{j}+b \,x^{n}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x^{j} + b x^{n}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 5.27, size = 83, normalized size = 0.89 \begin {gather*} -\frac {x\,\sqrt {\frac {b\,x^{n-j}}{a}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {\frac {j}{2}-1}{j-n};\ \frac {\frac {j}{2}-1}{j-n}+1;\ -\frac {b\,x^{n-j}}{a}\right )}{\left (\frac {j}{2}-1\right )\,\sqrt {a\,x^j+b\,x^n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^j + b*x^n)^(1/2),x)

[Out]

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

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