Optimal. Leaf size=81 \[ \frac{x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )}{3 a^4 b (m+1)}+\frac{x^{m+1} (A b-a B)}{3 a b (a+b x)^3} \]
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Rubi [A] time = 0.0981282, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )}{3 a^4 b (m+1)}+\frac{x^{m+1} (A b-a B)}{3 a b (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Int[(x^m*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
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Rubi in Sympy [A] time = 22.8603, size = 63, normalized size = 0.78 \[ \frac{x^{m + 1} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{3}} + \frac{x^{m + 1} \left (A b \left (- m + 2\right ) + B a \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{3 a^{4} b \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.0951725, size = 60, normalized size = 0.74 \[ \frac{x^{m+1} \left ((A b-a B) \, _2F_1\left (4,m+1;m+2;-\frac{b x}{a}\right )+a B \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )\right )}{a^4 b (m+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(x^m*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Maple [F] time = 0.189, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( Bx+A \right ) }{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} x^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^m/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} x^{m}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^m/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m} \left (A + B x\right )}{\left (a + b x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} x^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^m/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="giac")
[Out]