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

Optimal. Leaf size=376 \[ \frac{162 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{24565 (m+1)}-\frac{\left (2 \left (211+65 \sqrt{13}\right ) m+423\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{3757 \sqrt{13} \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{9 \left (117+64 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{63869 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{\left (\left (422-130 \sqrt{13}\right ) m+423\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{3757 \sqrt{13} \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{9 \left (117-64 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{63869 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{36 (4 x+1)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{7225 (m+1)}+\frac{(268-195 x) (4 x+1)^{m+1}}{11271 \left (3 x^2-5 x+1\right )} \]

[Out]

((268 - 195*x)*(1 + 4*x)^(1 + m))/(11271*(1 - 5*x + 3*x^2)) + (162*(1 + 4*x)^(1
+ m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(24565*(1 + m)) + (9*
(117 + 64*Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 +
4*x))/(13 - 2*Sqrt[13])])/(63869*(13 - 2*Sqrt[13])*(1 + m)) - ((423 + 2*(211 +
65*Sqrt[13])*m)*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x
))/(13 - 2*Sqrt[13])])/(3757*Sqrt[13]*(13 - 2*Sqrt[13])*(1 + m)) + (9*(117 - 64*
Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13
+ 2*Sqrt[13])])/(63869*(13 + 2*Sqrt[13])*(1 + m)) + ((423 + (422 - 130*Sqrt[13]
)*m)*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*
Sqrt[13])])/(3757*Sqrt[13]*(13 + 2*Sqrt[13])*(1 + m)) + (36*(1 + 4*x)^(1 + m)*Hy
pergeometric2F1[2, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(7225*(1 + m))

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Rubi [A]  time = 1.07929, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{162 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{24565 (m+1)}-\frac{\left (2 \left (211+65 \sqrt{13}\right ) m+423\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{3757 \sqrt{13} \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{9 \left (117+64 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{63869 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{\left (\left (422-130 \sqrt{13}\right ) m+423\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{3757 \sqrt{13} \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{9 \left (117-64 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{63869 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{36 (4 x+1)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{7225 (m+1)}+\frac{(268-195 x) (4 x+1)^{m+1}}{11271 \left (3 x^2-5 x+1\right )} \]

Antiderivative was successfully verified.

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

[Out]

((268 - 195*x)*(1 + 4*x)^(1 + m))/(11271*(1 - 5*x + 3*x^2)) + (162*(1 + 4*x)^(1
+ m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(24565*(1 + m)) + (9*
(117 + 64*Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 +
4*x))/(13 - 2*Sqrt[13])])/(63869*(13 - 2*Sqrt[13])*(1 + m)) - ((423 + 2*(211 +
65*Sqrt[13])*m)*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x
))/(13 - 2*Sqrt[13])])/(3757*Sqrt[13]*(13 - 2*Sqrt[13])*(1 + m)) + (9*(117 - 64*
Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13
+ 2*Sqrt[13])])/(63869*(13 + 2*Sqrt[13])*(1 + m)) + ((423 + (422 - 130*Sqrt[13]
)*m)*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*
Sqrt[13])])/(3757*Sqrt[13]*(13 + 2*Sqrt[13])*(1 + m)) + (36*(1 + 4*x)^(1 + m)*Hy
pergeometric2F1[2, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(7225*(1 + m))

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Rubi in Sympy [A]  time = 64.4264, size = 304, normalized size = 0.81 \[ \frac{\left (- 2535 x + 3484\right ) \left (4 x + 1\right )^{m + 1}}{146523 \left (3 x^{2} - 5 x + 1\right )} - \frac{2 \left (1690 m - \sqrt{13} \left (422 m + 423\right )\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{48841 \left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} - \frac{2 \left (1690 m + \sqrt{13} \left (422 m + 423\right )\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{48841 \left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{3 \left (54 + \frac{384 \sqrt{13}}{13}\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{4913 \left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{3 \left (- \frac{384 \sqrt{13}}{13} + 54\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{4913 \left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{162 \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{12 x}{5} - \frac{3}{5}} \right )}}{24565 \left (m + 1\right )} + \frac{36 \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{12 x}{5} - \frac{3}{5}} \right )}}{7225 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2535*x + 3484)*(4*x + 1)**(m + 1)/(146523*(3*x**2 - 5*x + 1)) - 2*(1690*m - sq
rt(13)*(422*m + 423))*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (12*x + 3)/
(2*sqrt(13) + 13))/(48841*(4*sqrt(13) + 26)*(m + 1)) - 2*(1690*m + sqrt(13)*(422
*m + 423))*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (-12*x - 3)/(-13 + 2*s
qrt(13)))/(48841*(-4*sqrt(13) + 26)*(m + 1)) + 3*(54 + 384*sqrt(13)/13)*(4*x + 1
)**(m + 1)*hyper((1, m + 1), (m + 2,), (-12*x - 3)/(-13 + 2*sqrt(13)))/(4913*(-4
*sqrt(13) + 26)*(m + 1)) + 3*(-384*sqrt(13)/13 + 54)*(4*x + 1)**(m + 1)*hyper((1
, m + 1), (m + 2,), (12*x + 3)/(2*sqrt(13) + 13))/(4913*(4*sqrt(13) + 26)*(m + 1
)) + 162*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), -12*x/5 - 3/5)/(24565*(m
+ 1)) + 36*(4*x + 1)**(m + 1)*hyper((2, m + 1), (m + 2,), -12*x/5 - 3/5)/(7225*
(m + 1))

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Mathematica [A]  time = 0.142896, size = 0, normalized size = 0. \[ \int \frac{(1+4 x)^m}{(2+3 x)^2 \left (1-5 x+3 x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.18, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 1+4\,x \right ) ^{m}}{ \left ( 2+3\,x \right ) ^{2} \left ( 3\,{x}^{2}-5\,x+1 \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x + 1\right )}^{m}}{81 \, x^{6} - 162 \, x^{5} - 45 \, x^{4} + 162 \, x^{3} + 13 \, x^{2} - 28 \, x + 4}, x\right ) \]

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)^2),x, algorithm="fricas")

[Out]

integral((4*x + 1)^m/(81*x^6 - 162*x^5 - 45*x^4 + 162*x^3 + 13*x^2 - 28*x + 4),
x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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