3.429 \(\int \frac {1}{(a x^j+b x^n)^{3/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac {2 x^{1-n} \sqrt {\frac {a x^{j-n}}{b}+1} \, _2F_1\left (\frac {3}{2},\frac {1-\frac {3 n}{2}}{j-n};\frac {1-\frac {3 n}{2}}{j-n}+1;-\frac {a x^{j-n}}{b}\right )}{b (2-3 n) \sqrt {a x^j+b x^n}} \]

[Out]

2*x^(1-n)*hypergeom([3/2, (1-3/2*n)/(j-n)],[1+(2-3*n)/(2*j-2*n)],-a*x^(j-n)/b)*(1+a*x^(j-n)/b)^(1/2)/b/(2-3*n)
/(a*x^j+b*x^n)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 101, 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 = {2011, 365, 364} \[ \frac {2 x^{1-n} \sqrt {\frac {a x^{j-n}}{b}+1} \, _2F_1\left (\frac {3}{2},\frac {1-\frac {3 n}{2}}{j-n};\frac {1-\frac {3 n}{2}}{j-n}+1;-\frac {a x^{j-n}}{b}\right )}{b (2-3 n) \sqrt {a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^j + b*x^n)^(-3/2),x]

[Out]

(2*x^(1 - n)*Sqrt[1 + (a*x^(j - n))/b]*Hypergeometric2F1[3/2, (1 - (3*n)/2)/(j - n), 1 + (1 - (3*n)/2)/(j - n)
, -((a*x^(j - n))/b)])/(b*(2 - 3*n)*Sqrt[a*x^j + b*x^n])

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])

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 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]

Rubi steps

\begin {align*} \int \frac {1}{\left (a x^j+b x^n\right )^{3/2}} \, dx &=\frac {\left (x^{n/2} \sqrt {b+a x^{j-n}}\right ) \int \frac {x^{-3 n/2}}{\left (b+a x^{j-n}\right )^{3/2}} \, 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^{-3 n/2}}{\left (1+\frac {a x^{j-n}}{b}\right )^{3/2}} \, dx}{b \sqrt {a x^j+b x^n}}\\ &=\frac {2 x^{1-n} \sqrt {1+\frac {a x^{j-n}}{b}} \, _2F_1\left (\frac {3}{2},\frac {1-\frac {3 n}{2}}{j-n};1+\frac {1-\frac {3 n}{2}}{j-n};-\frac {a x^{j-n}}{b}\right )}{b (2-3 n) \sqrt {a x^j+b x^n}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 104, normalized size = 1.03 \[ \frac {2 x^{1-j} \left (\sqrt {\frac {a x^{j-n}}{b}+1} \, _2F_1\left (\frac {1}{2},-\frac {2 j+n-2}{2 (j-n)};\frac {2-3 n}{2 j-2 n};-\frac {a x^{j-n}}{b}\right )-1\right )}{a (j-n) \sqrt {a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^j + b*x^n)^(-3/2),x]

[Out]

(2*x^(1 - j)*(-1 + Sqrt[1 + (a*x^(j - n))/b]*Hypergeometric2F1[1/2, -1/2*(-2 + 2*j + n)/(j - n), (2 - 3*n)/(2*
j - 2*n), -((a*x^(j - n))/b)]))/(a*(j - n)*Sqrt[a*x^j + b*x^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(1/(a*x^j+b*x^n)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x^{j} + b x^{n}\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x^j+b*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate((a*x^j + b*x^n)^(-3/2), x)

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maple [F]  time = 0.77, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \,x^{j}+b \,x^{n}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^j+b*x^n)^(3/2),x)

[Out]

int(1/(a*x^j+b*x^n)^(3/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x^{j} + b x^{n}\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x^j+b*x^n)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*x^j + b*x^n)^(-3/2), x)

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mupad [B]  time = 5.48, size = 83, normalized size = 0.82 \[ -\frac {x\,{\left (\frac {b\,x^{n-j}}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {\frac {3\,j}{2}-1}{j-n};\ \frac {\frac {3\,j}{2}-1}{j-n}+1;\ -\frac {b\,x^{n-j}}{a}\right )}{\left (\frac {3\,j}{2}-1\right )\,{\left (a\,x^j+b\,x^n\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^j + b*x^n)^(3/2),x)

[Out]

-(x*((b*x^(n - j))/a + 1)^(3/2)*hypergeom([3/2, ((3*j)/2 - 1)/(j - n)], ((3*j)/2 - 1)/(j - n) + 1, -(b*x^(n -
j))/a))/(((3*j)/2 - 1)*(a*x^j + b*x^n)^(3/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a x^{j} + b x^{n}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x**j+b*x**n)**(3/2),x)

[Out]

Integral((a*x**j + b*x**n)**(-3/2), x)

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