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

Optimal. Leaf size=340 \[ \frac{9 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{1445 (m+1)}-\frac{\left (\left (62+22 \sqrt{13}\right ) m+81\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{221 \sqrt{13} \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{9 \left (13+9 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{7514 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{\left (\left (62-22 \sqrt{13}\right ) m+81\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{221 \sqrt{13} \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{9 \left (13-9 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{7514 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(43-33 x) (4 x+1)^{m+1}}{663 \left (3 x^2-5 x+1\right )} \]

[Out]

((43 - 33*x)*(1 + 4*x)^(1 + m))/(663*(1 - 5*x + 3*x^2)) + (9*(1 + 4*x)^(1 + m)*H
ypergeometric2F1[1, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(1445*(1 + m)) + (9*(13 + 9
*Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(1
3 - 2*Sqrt[13])])/(7514*(13 - 2*Sqrt[13])*(1 + m)) - ((81 + (62 + 22*Sqrt[13])*m
)*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sqr
t[13])])/(221*Sqrt[13]*(13 - 2*Sqrt[13])*(1 + m)) + (9*(13 - 9*Sqrt[13])*(1 + 4*
x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[13])])/
(7514*(13 + 2*Sqrt[13])*(1 + m)) + ((81 + (62 - 22*Sqrt[13])*m)*(1 + 4*x)^(1 + m
)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[13])])/(221*Sqrt
[13]*(13 + 2*Sqrt[13])*(1 + m))

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Rubi [A]  time = 1.00982, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{9 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{1445 (m+1)}-\frac{\left (\left (62+22 \sqrt{13}\right ) m+81\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{221 \sqrt{13} \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{9 \left (13+9 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{7514 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{\left (\left (62-22 \sqrt{13}\right ) m+81\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{221 \sqrt{13} \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{9 \left (13-9 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{7514 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(43-33 x) (4 x+1)^{m+1}}{663 \left (3 x^2-5 x+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

((43 - 33*x)*(1 + 4*x)^(1 + m))/(663*(1 - 5*x + 3*x^2)) + (9*(1 + 4*x)^(1 + m)*H
ypergeometric2F1[1, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(1445*(1 + m)) + (9*(13 + 9
*Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(1
3 - 2*Sqrt[13])])/(7514*(13 - 2*Sqrt[13])*(1 + m)) - ((81 + (62 + 22*Sqrt[13])*m
)*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sqr
t[13])])/(221*Sqrt[13]*(13 - 2*Sqrt[13])*(1 + m)) + (9*(13 - 9*Sqrt[13])*(1 + 4*
x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[13])])/
(7514*(13 + 2*Sqrt[13])*(1 + m)) + ((81 + (62 - 22*Sqrt[13])*m)*(1 + 4*x)^(1 + m
)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[13])])/(221*Sqrt
[13]*(13 + 2*Sqrt[13])*(1 + m))

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Rubi in Sympy [A]  time = 62.158, size = 274, normalized size = 0.81 \[ \frac{\left (- 429 x + 559\right ) \left (4 x + 1\right )^{m + 1}}{8619 \left (3 x^{2} - 5 x + 1\right )} - \frac{2 \left (286 m - \sqrt{13} \left (62 m + 81\right )\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{2873 \left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} - \frac{2 \left (286 m + \sqrt{13} \left (62 m + 81\right )\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{2873 \left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{3 \left (3 + \frac{27 \sqrt{13}}{13}\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{289 \left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{3 \left (- \frac{27 \sqrt{13}}{13} + 3\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{289 \left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{9 \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{12 x}{5} - \frac{3}{5}} \right )}}{1445 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-429*x + 559)*(4*x + 1)**(m + 1)/(8619*(3*x**2 - 5*x + 1)) - 2*(286*m - sqrt(13
)*(62*m + 81))*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (12*x + 3)/(2*sqrt
(13) + 13))/(2873*(4*sqrt(13) + 26)*(m + 1)) - 2*(286*m + sqrt(13)*(62*m + 81))*
(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (-12*x - 3)/(-13 + 2*sqrt(13)))/(
2873*(-4*sqrt(13) + 26)*(m + 1)) + 3*(3 + 27*sqrt(13)/13)*(4*x + 1)**(m + 1)*hyp
er((1, m + 1), (m + 2,), (-12*x - 3)/(-13 + 2*sqrt(13)))/(289*(-4*sqrt(13) + 26)
*(m + 1)) + 3*(-27*sqrt(13)/13 + 3)*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,
), (12*x + 3)/(2*sqrt(13) + 13))/(289*(4*sqrt(13) + 26)*(m + 1)) + 9*(4*x + 1)**
(m + 1)*hyper((1, m + 1), (m + 2,), -12*x/5 - 3/5)/(1445*(m + 1))

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Mathematica [A]  time = 0.139731, size = 0, normalized size = 0. \[ \int \frac{(1+4 x)^m}{(2+3 x) \left (1-5 x+3 x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.177, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 1+4\,x \right ) ^{m}}{ \left ( 2+3\,x \right ) \left ( 3\,{x}^{2}-5\,x+1 \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x + 1\right )}^{m}}{27 \, x^{5} - 72 \, x^{4} + 33 \, x^{3} + 32 \, x^{2} - 17 \, x + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x + 1)^m/(27*x^5 - 72*x^4 + 33*x^3 + 32*x^2 - 17*x + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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