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