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

Optimal. Leaf size=36 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{a x^2}{b}\right )}{m+1} \]

[Out]

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

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Rubi [A]  time = 0.060137, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{a x^2}{b}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.94023, size = 27, normalized size = 0.75 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{a x^{2}}{b}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0188131, size = 38, normalized size = 1.06 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{a x^2}{b}\right )}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.185, size = 0, normalized size = 0. \[ \int{{x}^{m} \left ( 1-{x\sqrt{a}{\frac{1}{\sqrt{-b}}}} \right ) ^{-2} \left ( 1+{x\sqrt{a}{\frac{1}{\sqrt{-b}}}} \right ) ^{-2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 33.8287, size = 541, normalized size = 15.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*b**2*m**2*x**m*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, -m/2 + 3/2)*gamma(-m/2
+ 3/2)/(x*(8*a**3*x**2*gamma(-m/2 + 5/2) + 8*a**2*b*gamma(-m/2 + 5/2))) - 4*a*b*
*2*m*x**m*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, -m/2 + 3/2)*gamma(-m/2 + 3/2)/
(x*(8*a**3*x**2*gamma(-m/2 + 5/2) + 8*a**2*b*gamma(-m/2 + 5/2))) + 2*a*b**2*m*x*
*m*gamma(-m/2 + 3/2)/(x*(8*a**3*x**2*gamma(-m/2 + 5/2) + 8*a**2*b*gamma(-m/2 + 5
/2))) + 3*a*b**2*x**m*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, -m/2 + 3/2)*gamma(
-m/2 + 3/2)/(x*(8*a**3*x**2*gamma(-m/2 + 5/2) + 8*a**2*b*gamma(-m/2 + 5/2))) - 6
*a*b**2*x**m*gamma(-m/2 + 3/2)/(x*(8*a**3*x**2*gamma(-m/2 + 5/2) + 8*a**2*b*gamm
a(-m/2 + 5/2))) + b**3*m**2*x**m*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, -m/2 +
3/2)*gamma(-m/2 + 3/2)/(x**3*(8*a**3*x**2*gamma(-m/2 + 5/2) + 8*a**2*b*gamma(-m/
2 + 5/2))) - 4*b**3*m*x**m*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, -m/2 + 3/2)*g
amma(-m/2 + 3/2)/(x**3*(8*a**3*x**2*gamma(-m/2 + 5/2) + 8*a**2*b*gamma(-m/2 + 5/
2))) + 3*b**3*x**m*lerchphi(b*exp_polar(I*pi)/(a*x**2), 1, -m/2 + 3/2)*gamma(-m/
2 + 3/2)/(x**3*(8*a**3*x**2*gamma(-m/2 + 5/2) + 8*a**2*b*gamma(-m/2 + 5/2)))

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GIAC/XCAS [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(a)*x/sqrt(-b) + 1)^2*(sqrt(a)*x/sqrt(-b) - 1)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError