3.932 \(\int \frac{(2+3 x)^2 (1+4 x)^m}{1-5 x+3 x^2} \, dx\)

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]

(3*(1 + 4*x)^(1 + m))/(4*(1 + m)) - (3*(117 - 47*Sqrt[13])*(1 + 4*x)^(1 + m)*Hyp
ergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sqrt[13])])/(26*(13 - 2*Sq
rt[13])*(1 + m)) - (3*(117 + 47*Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1,
1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[13])])/(26*(13 + 2*Sqrt[13])*(1 + m))

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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]

(3*(1 + 4*x)^(1 + m))/(4*(1 + m)) - (3*(117 - 47*Sqrt[13])*(1 + 4*x)^(1 + m)*Hyp
ergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sqrt[13])])/(26*(13 - 2*Sq
rt[13])*(1 + m)) - (3*(117 + 47*Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1,
1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[13])])/(26*(13 + 2*Sqrt[13])*(1 + m))

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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]

-(-141*sqrt(13)/13 + 27)*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (-12*x -
3)/(-13 + 2*sqrt(13)))/((-4*sqrt(13) + 26)*(m + 1)) - (27 + 141*sqrt(13)/13)*(4
*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (12*x + 3)/(2*sqrt(13) + 13))/((4*s
qrt(13) + 26)*(m + 1)) + 3*(4*x + 1)**(m + 1)/(4*(m + 1))

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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]

((1 + 4*x)^m*((13*3^(1 + m)*m*(1 + 4*x))/(1 + m) + (2^(1 + m)*(117 + 47*Sqrt[13]
)*Hypergeometric2F1[-m, -m, 1 - m, (13 + 2*Sqrt[13])/(2*(5 + Sqrt[13] - 6*x))])/
(-((1 + 4*x)/(5 + Sqrt[13] - 6*x)))^m - (2^(1 + m)*(-117 + 47*Sqrt[13])*Hypergeo
metric2F1[-m, -m, 1 - m, (-13 + 2*Sqrt[13])/(2*(-5 + Sqrt[13] + 6*x))])/((1 + 4*
x)/(-5 + Sqrt[13] + 6*x))^m))/(52*3^m*m)

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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]

int((2+3*x)^2*(1+4*x)^m/(3*x^2-5*x+1),x)

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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]

integrate((4*x + 1)^m*(3*x + 2)^2/(3*x^2 - 5*x + 1), x)

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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]

integral((9*x^2 + 12*x + 4)*(4*x + 1)^m/(3*x^2 - 5*x + 1), x)

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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]

Integral((3*x + 2)**2*(4*x + 1)**m/(3*x**2 - 5*x + 1), x)

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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]

integrate((4*x + 1)^m*(3*x + 2)^2/(3*x^2 - 5*x + 1), x)