Optimal. Leaf size=147 \[ -\frac{3 \left (117-47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{26 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{3 \left (117+47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{26 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{3 (4 x+1)^{m+1}}{4 (m+1)} \]
[Out]
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Rubi [A] time = 0.295671, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{3 \left (117-47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{26 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{3 \left (117+47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{26 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{3 (4 x+1)^{m+1}}{4 (m+1)} \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)^2*(1 + 4*x)^m)/(1 - 5*x + 3*x^2),x]
[Out]
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Rubi in Sympy [A] time = 22.4762, size = 114, normalized size = 0.78 \[ - \frac{\left (- \frac{141 \sqrt{13}}{13} + 27\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{\left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} - \frac{\left (27 + \frac{141 \sqrt{13}}{13}\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{\left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{3 \left (4 x + 1\right )^{m + 1}}{4 \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**2*(1+4*x)**m/(3*x**2-5*x+1),x)
[Out]
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Mathematica [A] time = 0.833655, size = 188, normalized size = 1.28 \[ \frac{3^{-m} (4 x+1)^m \left (\left (117+47 \sqrt{13}\right ) 2^{m+1} \left (-\frac{4 x+1}{-6 x+\sqrt{13}+5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{13+2 \sqrt{13}}{2 \left (-6 x+\sqrt{13}+5\right )}\right )-\left (47 \sqrt{13}-117\right ) 2^{m+1} \left (\frac{4 x+1}{6 x+\sqrt{13}-5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{-13+2 \sqrt{13}}{2 \left (6 x+\sqrt{13}-5\right )}\right )+\frac{13\ 3^{m+1} m (4 x+1)}{m+1}\right )}{52 m} \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)^2*(1 + 4*x)^m)/(1 - 5*x + 3*x^2),x]
[Out]
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Maple [F] time = 0.15, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{2} \left ( 1+4\,x \right ) ^{m}}{3\,{x}^{2}-5\,x+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^2*(1+4*x)^m/(3*x^2-5*x+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (4 \, x + 1\right )}^{m}{\left (3 \, x + 2\right )}^{2}}{3 \, x^{2} - 5 \, x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x + 1)^m*(3*x + 2)^2/(3*x^2 - 5*x + 1),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )}{\left (4 \, x + 1\right )}^{m}}{3 \, x^{2} - 5 \, x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x + 1)^m*(3*x + 2)^2/(3*x^2 - 5*x + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{2} \left (4 x + 1\right )^{m}}{3 x^{2} - 5 x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**2*(1+4*x)**m/(3*x**2-5*x+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (4 \, x + 1\right )}^{m}{\left (3 \, x + 2\right )}^{2}}{3 \, x^{2} - 5 \, x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x + 1)^m*(3*x + 2)^2/(3*x^2 - 5*x + 1),x, algorithm="giac")
[Out]