3.937 \(\int \frac{(a+b x)^m}{a^2-b^2 x^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^m \, _2F_1\left (1,m;m+1;\frac{a+b x}{2 a}\right )}{2 a b m} \]

[Out]

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

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Rubi [A]  time = 0.0540778, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(a+b x)^m \, _2F_1\left (1,m;m+1;\frac{a+b x}{2 a}\right )}{2 a b m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.5145, size = 26, normalized size = 0.68 \[ \frac{\left (a + b x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} 1, m \\ m + 1 \end{matrix}\middle |{\frac{\frac{a}{2} + \frac{b x}{2}}{a}} \right )}}{2 a b m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0546006, size = 59, normalized size = 1.55 \[ \frac{(a+b x)^m \left (m (a+b x) \, _2F_1\left (1,m+1;m+2;\frac{a+b x}{2 a}\right )+2 a (m+1)\right )}{4 a^2 b m (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{-{b}^{2}{x}^{2}+{a}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m/(-b^2*x^2+a^2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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