3.2741 \(\int x^m \sqrt{a+b x^{2+2 m}} \, dx\)

Optimal. Leaf size=72 \[ \frac{x^{m+1} \sqrt{a+b x^{2 (m+1)}}}{2 (m+1)}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{2 \sqrt{b} (m+1)} \]

[Out]

(x^(1 + m)*Sqrt[a + b*x^(2*(1 + m))])/(2*(1 + m)) + (a*ArcTanh[(Sqrt[b]*x^(1 + m
))/Sqrt[a + b*x^(2*(1 + m))]])/(2*Sqrt[b]*(1 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.074197, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{x^{m+1} \sqrt{a+b x^{2 (m+1)}}}{2 (m+1)}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{2 \sqrt{b} (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m*Sqrt[a + b*x^(2 + 2*m)],x]

[Out]

(x^(1 + m)*Sqrt[a + b*x^(2*(1 + m))])/(2*(1 + m)) + (a*ArcTanh[(Sqrt[b]*x^(1 + m
))/Sqrt[a + b*x^(2*(1 + m))]])/(2*Sqrt[b]*(1 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.85381, size = 56, normalized size = 0.78 \[ \frac{x^{m + 1} \sqrt{a + b x^{2 m + 2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2 m + 2}}{a}} \right )}}{\sqrt{1 + \frac{b x^{2 m + 2}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.139838, size = 93, normalized size = 1.29 \[ \frac{a^{3/2} \sqrt{\frac{b x^{2 m+2}}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )+\sqrt{b} x^{m+1} \left (a+b x^{2 m+2}\right )}{2 \sqrt{b} (m+1) \sqrt{a+b x^{2 m+2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*Sqrt[a + b*x^(2 + 2*m)],x]

[Out]

(Sqrt[b]*x^(1 + m)*(a + b*x^(2 + 2*m)) + a^(3/2)*Sqrt[1 + (b*x^(2 + 2*m))/a]*Arc
Sinh[(Sqrt[b]*x^(1 + m))/Sqrt[a]])/(2*Sqrt[b]*(1 + m)*Sqrt[a + b*x^(2 + 2*m)])

_______________________________________________________________________________________

Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{x}^{m}\sqrt{a+b{x}^{2+2\,m}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{2 \, m + 2} + a} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{2 \, m + 2} + a} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x^(2*m + 2) + a)*x^m, x)