3.2668 \(\int \frac{x^m}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=55 \[ \frac{x^{m+1} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{m+1}{n}+\frac{1}{2};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0670956, antiderivative size = 64, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{(m+1) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x^m/Sqrt[a + b*x^n],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.08685, size = 51, normalized size = 0.93 \[ \frac{x^{m + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(a+b*x**n)**(1/2),x)

[Out]

x**(m + 1)*sqrt(a + b*x**n)*hyper((1/2, (m + 1)/n), ((m + n + 1)/n,), -b*x**n/a)
/(a*sqrt(1 + b*x**n/a)*(m + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0509487, size = 66, normalized size = 1.2 \[ \frac{x^{m+1} \sqrt{\frac{a+b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+1}{n}+1;-\frac{b x^n}{a}\right )}{(m+1) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/Sqrt[a + b*x^n],x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{{x}^{m}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(a+b*x^n)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

integrate(x^m/sqrt(b*x^n + a), x)

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^m/sqrt(b*x^n + a), x)