3.501 \(\int \frac{x^m}{(1-a x) \left (1-a^2 x^2\right )} \, dx\)

Optimal. Leaf size=70 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{m+2} \]

[Out]

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

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Rubi [A]  time = 0.143016, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right )}{m+1}+\frac{a x^{m+2} \, _2F_1\left (2,\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right )}{m+2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.0719, size = 51, normalized size = 0.73 \[ \frac{a x^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{a^{2} x^{2}} \right )}}{m + 2} + \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{a^{2} x^{2}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**(m + 2)*hyper((2, m/2 + 1), (m/2 + 2,), a**2*x**2)/(m + 2) + x**(m + 1)*hyp
er((2, m/2 + 1/2), (m/2 + 3/2,), a**2*x**2)/(m + 1)

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Mathematica [A]  time = 0.024319, size = 51, normalized size = 0.73 \[ \frac{x^{m+1} (\, _2F_1(1,m+1;m+2;-a x)+\, _2F_1(1,m+1;m+2;a x)+2 \, _2F_1(2,m+1;m+2;a x))}{4 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{ \left ( -ax+1 \right ) \left ( -{a}^{2}{x}^{2}+1 \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**m/((a*x - 1)**2*(a*x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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