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

Optimal. Leaf size=177 \[ -\frac{2 \left (2 \left (2+\sqrt{13}\right ) m+9\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{13 \sqrt{13} \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{2 \left (2 \left (2-\sqrt{13}\right ) m+9\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{13 \sqrt{13} \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(7-6 x) (4 x+1)^{m+1}}{39 \left (3 x^2-5 x+1\right )} \]

[Out]

((7 - 6*x)*(1 + 4*x)^(1 + m))/(39*(1 - 5*x + 3*x^2)) - (2*(9 + 2*(2 + Sqrt[13])*
m)*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sq
rt[13])])/(13*Sqrt[13]*(13 - 2*Sqrt[13])*(1 + m)) + (2*(9 + 2*(2 - Sqrt[13])*m)*
(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[
13])])/(13*Sqrt[13]*(13 + 2*Sqrt[13])*(1 + m))

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Rubi [A]  time = 0.413095, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{2 \left (2 \left (2+\sqrt{13}\right ) m+9\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{13 \sqrt{13} \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{2 \left (2 \left (2-\sqrt{13}\right ) m+9\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{13 \sqrt{13} \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(7-6 x) (4 x+1)^{m+1}}{39 \left (3 x^2-5 x+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 4*x)^m/(1 - 5*x + 3*x^2)^2,x]

[Out]

((7 - 6*x)*(1 + 4*x)^(1 + m))/(39*(1 - 5*x + 3*x^2)) - (2*(9 + 2*(2 + Sqrt[13])*
m)*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sq
rt[13])])/(13*Sqrt[13]*(13 - 2*Sqrt[13])*(1 + m)) + (2*(9 + 2*(2 - Sqrt[13])*m)*
(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[
13])])/(13*Sqrt[13]*(13 + 2*Sqrt[13])*(1 + m))

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Rubi in Sympy [A]  time = 31.8334, size = 136, normalized size = 0.77 \[ \frac{\left (- 78 x + 91\right ) \left (4 x + 1\right )^{m + 1}}{507 \left (3 x^{2} - 5 x + 1\right )} - \frac{4 \left (26 m - \sqrt{13} \left (4 m + 9\right )\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{169 \left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} - \frac{4 \left (26 m + \sqrt{13} \left (4 m + 9\right )\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{169 \left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+4*x)**m/(3*x**2-5*x+1)**2,x)

[Out]

(-78*x + 91)*(4*x + 1)**(m + 1)/(507*(3*x**2 - 5*x + 1)) - 4*(26*m - sqrt(13)*(4
*m + 9))*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (12*x + 3)/(2*sqrt(13) +
13))/(169*(4*sqrt(13) + 26)*(m + 1)) - 4*(26*m + sqrt(13)*(4*m + 9))*(4*x + 1)*
*(m + 1)*hyper((1, m + 1), (m + 2,), (-12*x - 3)/(-13 + 2*sqrt(13)))/(169*(-4*sq
rt(13) + 26)*(m + 1))

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Mathematica [A]  time = 0.0373103, size = 0, normalized size = 0. \[ \int \frac{(1+4 x)^m}{\left (1-5 x+3 x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(1 + 4*x)^m/(1 - 5*x + 3*x^2)^2,x]

[Out]

Integrate[(1 + 4*x)^m/(1 - 5*x + 3*x^2)^2, x]

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Maple [F]  time = 0.155, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 1+4\,x \right ) ^{m}}{ \left ( 3\,{x}^{2}-5\,x+1 \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (4 \, x + 1\right )}^{m}}{{\left (3 \, x^{2} - 5 \, x + 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x + 1)^m/(3*x^2 - 5*x + 1)^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x + 1\right )}^{m}}{9 \, x^{4} - 30 \, x^{3} + 31 \, x^{2} - 10 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x + 1)^m/(3*x^2 - 5*x + 1)^2,x, algorithm="fricas")

[Out]

integral((4*x + 1)^m/(9*x^4 - 30*x^3 + 31*x^2 - 10*x + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (4 x + 1\right )^{m}}{\left (3 x^{2} - 5 x + 1\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+4*x)**m/(3*x**2-5*x+1)**2,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (4 \, x + 1\right )}^{m}}{{\left (3 \, x^{2} - 5 \, x + 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x + 1)^m/(3*x^2 - 5*x + 1)^2,x, algorithm="giac")

[Out]

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