3.2508 \(\int \frac {1}{(a+b x^n)^{3/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac {x \, _2F_1\left (1,\frac {1}{n}-\frac {1}{2};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a \sqrt {a+b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {246, 245} \[ \frac {x \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {3}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a \sqrt {a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^(-3/2),x]

[Out]

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

Rule 245

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 51, normalized size = 1.31 \[ \frac {x \sqrt {\frac {b x^n}{a}+1} \, _2F_1\left (\frac {3}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a \sqrt {a+b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^(-3/2),x]

[Out]

(x*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[3/2, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*Sqrt[a + 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+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 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^(-3/2), x)

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maple [F]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{n}+a \right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^(-3/2), x)

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mupad [B]  time = 1.28, size = 43, normalized size = 1.10 \[ \frac {x\,{\left (\frac {b\,x^n}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{{\left (a+b\,x^n\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^n)^(3/2),x)

[Out]

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

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sympy [C]  time = 1.37, size = 39, normalized size = 1.00 \[ \frac {x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{a^{\frac {3}{2}} n \Gamma \left (1 + \frac {1}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*gamma(1/n)*hyper((3/2, 1/n), (1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(a**(3/2)*n*gamma(1 + 1/n))

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