3.25.88 \(\int \sqrt {a+b x^n} \, dx\) [2488]

Optimal. Leaf size=39 \[ \frac {x \left (a+b x^n\right )^{3/2} \, _2F_1\left (1,\frac {3}{2}+\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.23, 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 = {252, 251} \begin {gather*} \frac {x \sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{\sqrt {\frac {b x^n}{a}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x^n],x]

[Out]

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

Rule 251

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 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[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 \sqrt {a+b x^n} \, dx &=\frac {\sqrt {a+b x^n} \int \sqrt {1+\frac {b x^n}{a}} \, dx}{\sqrt {1+\frac {b x^n}{a}}}\\ &=\frac {x \sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{\sqrt {1+\frac {b x^n}{a}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 48, normalized size = 1.23 \begin {gather*} \frac {x \sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{\sqrt {1+\frac {b x^n}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x^n],x]

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \sqrt {a +b \,x^{n}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

integrate(sqrt(b*x^n + a), x)

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 41, normalized size = 1.05 \begin {gather*} \frac {\sqrt {a} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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((a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^n + a), x)

________________________________________________________________________________________

Mupad [B]
time = 1.46, size = 43, normalized size = 1.10 \begin {gather*} \frac {x\,\sqrt {a+b\,x^n}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{\sqrt {\frac {b\,x^n}{a}+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^(1/2),x)

[Out]

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

________________________________________________________________________________________