3.2690 \(\int \left (\frac{6 a x^2}{b (4+m) \sqrt{a+b x^{-2+m}}}+\frac{x^m}{\sqrt{a+b x^{-2+m}}}\right ) \, dx\)

Optimal. Leaf size=26 \[ \frac{2 x^3 \sqrt{a+b x^{m-2}}}{b (m+4)} \]

[Out]

(2*x^3*Sqrt[a + b*x^(-2 + m)])/(b*(4 + m))

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Rubi [C]  time = 0.256842, antiderivative size = 160, normalized size of antiderivative = 6.15, number of steps used = 5, number of rules used = 2, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.044 \[ \frac{x^{m+1} \sqrt{\frac{b x^{m-2}}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{m+1}{2-m};\frac{1-2 m}{2-m};-\frac{b x^{m-2}}{a}\right )}{(m+1) \sqrt{a+b x^{m-2}}}+\frac{2 a x^3 \sqrt{\frac{b x^{m-2}}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{3}{2-m};-\frac{m+1}{2-m};-\frac{b x^{m-2}}{a}\right )}{b (m+4) \sqrt{a+b x^{m-2}}} \]

Antiderivative was successfully verified.

[In]  Int[(6*a*x^2)/(b*(4 + m)*Sqrt[a + b*x^(-2 + m)]) + x^m/Sqrt[a + b*x^(-2 + m)],x]

[Out]

(2*a*x^3*Sqrt[1 + (b*x^(-2 + m))/a]*Hypergeometric2F1[1/2, -3/(2 - m), -((1 + m)
/(2 - m)), -((b*x^(-2 + m))/a)])/(b*(4 + m)*Sqrt[a + b*x^(-2 + m)]) + (x^(1 + m)
*Sqrt[1 + (b*x^(-2 + m))/a]*Hypergeometric2F1[1/2, -((1 + m)/(2 - m)), (1 - 2*m)
/(2 - m), -((b*x^(-2 + m))/a)])/((1 + m)*Sqrt[a + b*x^(-2 + m)])

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Rubi in Sympy [A]  time = 18.8859, size = 117, normalized size = 4.5 \[ \frac{2 x^{3} \sqrt{a + b x^{m - 2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{m - 2} \\ \frac{m + 1}{m - 2} \end{matrix}\middle |{- \frac{b x^{m - 2}}{a}} \right )}}{b \sqrt{1 + \frac{b x^{m - 2}}{a}} \left (m + 4\right )} + \frac{x^{m + 1} \sqrt{a + b x^{m - 2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m + 1}{m - 2} \\ \frac{2 m - 1}{m - 2} \end{matrix}\middle |{- \frac{b x^{m - 2}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{m - 2}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(6*a*x**2/b/(4+m)/(a+b*x**(-2+m))**(1/2)+x**m/(a+b*x**(-2+m))**(1/2),x)

[Out]

2*x**3*sqrt(a + b*x**(m - 2))*hyper((1/2, 3/(m - 2)), ((m + 1)/(m - 2),), -b*x**
(m - 2)/a)/(b*sqrt(1 + b*x**(m - 2)/a)*(m + 4)) + x**(m + 1)*sqrt(a + b*x**(m -
2))*hyper((1/2, (m + 1)/(m - 2)), ((2*m - 1)/(m - 2),), -b*x**(m - 2)/a)/(a*sqrt
(1 + b*x**(m - 2)/a)*(m + 1))

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Mathematica [A]  time = 0.0907494, size = 26, normalized size = 1. \[ \frac{2 x^3 \sqrt{a+b x^{m-2}}}{b (m+4)} \]

Antiderivative was successfully verified.

[In]  Integrate[(6*a*x^2)/(b*(4 + m)*Sqrt[a + b*x^(-2 + m)]) + x^m/Sqrt[a + b*x^(-2 + m)],x]

[Out]

(2*x^3*Sqrt[a + b*x^(-2 + m)])/(b*(4 + m))

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Maple [A]  time = 0.053, size = 40, normalized size = 1.5 \[ 2\,{\frac{x \left ( a{x}^{2}+b{x}^{m} \right ) }{ \left ( 4+m \right ) b}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b{x}^{m}}{{x}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(6*a*x^2/b/(4+m)/(a+b*x^(-2+m))^(1/2)+x^m/(a+b*x^(-2+m))^(1/2),x)

[Out]

2*x*(a*x^2+b*x^m)/b/(4+m)/((a*x^2+b*x^m)/x^2)^(1/2)

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Maxima [A]  time = 1.55777, size = 50, normalized size = 1.92 \[ \frac{2 \,{\left (a x^{4} + b x^{2} x^{m}\right )}}{\sqrt{a x^{2} + b x^{m}} b{\left (m + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^(m - 2) + a) + 6*a*x^2/(sqrt(b*x^(m - 2) + a)*b*(m + 4)),x, algorithm="maxima")

[Out]

2*(a*x^4 + b*x^2*x^m)/(sqrt(a*x^2 + b*x^m)*b*(m + 4))

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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^(m - 2) + a) + 6*a*x^2/(sqrt(b*x^(m - 2) + a)*b*(m + 4)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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(6*a*x**2/b/(4+m)/(a+b*x**(-2+m))**(1/2)+x**m/(a+b*x**(-2+m))**(1/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x^{m - 2} + a}} + \frac{6 \, a x^{2}}{\sqrt{b x^{m - 2} + a} b{\left (m + 4\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^(m - 2) + a) + 6*a*x^2/(sqrt(b*x^(m - 2) + a)*b*(m + 4)),x, algorithm="giac")

[Out]

integrate(x^m/sqrt(b*x^(m - 2) + a) + 6*a*x^2/(sqrt(b*x^(m - 2) + a)*b*(m + 4)),
x)