3.345 \(\int \frac{x^m}{\left (a+b x^2\right )^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0303773, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a^2 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 4.45326, size = 31, normalized size = 0.79 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0298717, size = 41, normalized size = 1.05 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right )}{a^2 (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x^2)^2,x]

[Out]

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

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( b{x}^{2}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x^2+a)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^2 + a)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 27.6192, size = 374, normalized size = 9.59 \[ - \frac{a m^{2} x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{8 a^{3} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{2 a m x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{8 a^{3} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{a x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{8 a^{3} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{2 a x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{8 a^{3} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} - \frac{b m^{2} x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{8 a^{3} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{b x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{8 a^{3} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ) + 8 a^{2} b x^{2} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(b*x**2+a)**2,x)

[Out]

-a*m**2*x*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)
/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + 2*a*m*x*x**m*gamma
(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + a*x*x**
m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**3*gamm
a(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + 2*a*x*x**m*gamma(m/2 + 1/2)/(8*
a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) - b*m**2*x**3*x**m*lerch
phi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 +
3/2) + 8*a**2*b*x**2*gamma(m/2 + 3/2)) + b*x**3*x**m*lerchphi(b*x**2*exp_polar(
I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*a**3*gamma(m/2 + 3/2) + 8*a**2*b*x**2
*gamma(m/2 + 3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^2 + a)^2, x)