3.2692 \(\int \left (-\frac{b n x^{-1+m+n}}{2 \left (a+b x^n\right )^{3/2}}+\frac{m x^{-1+m}}{\sqrt{a+b x^n}}\right ) \, dx\)

Optimal. Leaf size=15 \[ \frac{x^m}{\sqrt{a+b x^n}} \]

[Out]

x^m/Sqrt[a + b*x^n]

_______________________________________________________________________________________

Rubi [C]  time = 0.190587, antiderivative size = 126, normalized size of antiderivative = 8.4, number of steps used = 5, number of rules used = 2, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{x^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m}{n};\frac{m+n}{n};-\frac{b x^n}{a}\right )}{\sqrt{a+b x^n}}-\frac{b n x^{m+n} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{m+n}{n};\frac{m}{n}+2;-\frac{b x^n}{a}\right )}{2 a (m+n) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.0928, size = 100, normalized size = 6.67 \[ \frac{x^{m} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{n} \\ \frac{m + n}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}}} - \frac{b n x^{m + n} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m + n}{n} \\ \frac{m}{n} + 2 \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{2} \sqrt{1 + \frac{b x^{n}}{a}} \left (m + n\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(-1/2*b*n*x**(-1+m+n)/(a+b*x**n)**(3/2)+m*x**(-1+m)/(a+b*x**n)**(1/2),x)

[Out]

x**m*sqrt(a + b*x**n)*hyper((1/2, m/n), ((m + n)/n,), -b*x**n/a)/(a*sqrt(1 + b*x
**n/a)) - b*n*x**(m + n)*sqrt(a + b*x**n)*hyper((3/2, (m + n)/n), (m/n + 2,), -b
*x**n/a)/(2*a**2*sqrt(1 + b*x**n/a)*(m + n))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0663734, size = 15, normalized size = 1. \[ \frac{x^m}{\sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

x^m/Sqrt[a + b*x^n]

_______________________________________________________________________________________

Maple [F]  time = 0., size = 0, normalized size = 0. \[ \int -{\frac{bn{x}^{-1+m+n}}{2} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}+{m{x}^{-1+m}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.64235, size = 18, normalized size = 1.2 \[ \frac{x^{m}}{\sqrt{b x^{n} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*b*n*x^(m + n - 1)/(b*x^n + a)^(3/2) + m*x^(m - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

x^m/sqrt(b*x^n + a)

_______________________________________________________________________________________

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{b n x^{m + n - 1}}{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}} + \frac{m x^{m - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*b*n*x^(m + n - 1)/(b*x^n + a)^(3/2) + m*x^(m - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(-1/2*b*n*x^(m + n - 1)/(b*x^n + a)^(3/2) + m*x^(m - 1)/sqrt(b*x^n + a)
, x)