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

Optimal. Leaf size=199 \[ \frac{27 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{1445 (m+1)}+\frac{3 \left (117+47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{7514 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{3 \left (117-47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{7514 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{12 (4 x+1)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{425 (m+1)} \]

[Out]

(27*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(144
5*(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])])/(7514*(13 - 2*Sqrt[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])])/(7514*(13 + 2*Sqrt[13])*(1 + m)) + (12*(1 + 4*x)^(1
+ m)*Hypergeometric2F1[2, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(425*(1 + m))

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Rubi [A]  time = 0.488883, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{27 (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{1445 (m+1)}+\frac{3 \left (117+47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{7514 \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{3 \left (117-47 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{7514 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{12 (4 x+1)^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac{3}{5} (4 x+1)\right )}{425 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(27*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(144
5*(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])])/(7514*(13 - 2*Sqrt[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])])/(7514*(13 + 2*Sqrt[13])*(1 + m)) + (12*(1 + 4*x)^(1
+ m)*Hypergeometric2F1[2, 1 + m, 2 + m, (-3*(1 + 4*x))/5])/(425*(1 + m))

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Rubi in Sympy [A]  time = 35.5173, size = 163, normalized size = 0.82 \[ \frac{\left (27 + \frac{141 \sqrt{13}}{13}\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{289 \left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{\left (- \frac{141 \sqrt{13}}{13} + 27\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{289 \left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{27 \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{12 x}{5} - \frac{3}{5}} \right )}}{1445 \left (m + 1\right )} + \frac{12 \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{12 x}{5} - \frac{3}{5}} \right )}}{425 \left (m + 1\right )} \]

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),x)

[Out]

(27 + 141*sqrt(13)/13)*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (-12*x - 3
)/(-13 + 2*sqrt(13)))/(289*(-4*sqrt(13) + 26)*(m + 1)) + (-141*sqrt(13)/13 + 27)
*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (12*x + 3)/(2*sqrt(13) + 13))/(2
89*(4*sqrt(13) + 26)*(m + 1)) + 27*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,)
, -12*x/5 - 3/5)/(1445*(m + 1)) + 12*(4*x + 1)**(m + 1)*hyper((2, m + 1), (m + 2
,), -12*x/5 - 3/5)/(425*(m + 1))

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Mathematica [A]  time = 3.96925, size = 284, normalized size = 1.43 \[ \frac{(4 x+1)^m \left (\frac{\left (\frac{12 x+3}{6 x+4}\right )^{-m} \left (\left (47 \sqrt{13}-117\right ) \left (\frac{4 x+1}{3 x+2}\right )^m \left (-\frac{4 x+1}{-6 x+\sqrt{13}+5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{13+2 \sqrt{13}}{2 \left (-6 x+\sqrt{13}+5\right )}\right )-\left (117+47 \sqrt{13}\right ) \left (\frac{4 x+1}{3 x+2}\right )^m \left (\frac{4 x+1}{6 x+\sqrt{13}-5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{-13+2 \sqrt{13}}{2 \left (6 x+\sqrt{13}-5\right )}\right )+117\ 2^{m+1} \, _2F_1\left (-m,-m;1-m;\frac{5}{12 x+8}\right )\right )}{m}+\frac{442 \left (1-\frac{5}{12 x+8}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{5}{12 x+8}\right )}{(m-1) (3 x+2)}\right )}{7514} \]

Antiderivative was successfully verified.

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

[Out]

((1 + 4*x)^m*((442*Hypergeometric2F1[1 - m, -m, 2 - m, 5/(8 + 12*x)])/((-1 + m)*
(2 + 3*x)*(1 - 5/(8 + 12*x))^m) + (((-117 + 47*Sqrt[13])*((1 + 4*x)/(2 + 3*x))^m
*Hypergeometric2F1[-m, -m, 1 - m, (13 + 2*Sqrt[13])/(2*(5 + Sqrt[13] - 6*x))])/(
-((1 + 4*x)/(5 + Sqrt[13] - 6*x)))^m - ((117 + 47*Sqrt[13])*((1 + 4*x)/(2 + 3*x)
)^m*Hypergeometric2F1[-m, -m, 1 - m, (-13 + 2*Sqrt[13])/(2*(-5 + Sqrt[13] + 6*x)
)])/((1 + 4*x)/(-5 + Sqrt[13] + 6*x))^m + 117*2^(1 + m)*Hypergeometric2F1[-m, -m
, 1 - m, 5/(8 + 12*x)])/(m*((3 + 12*x)/(4 + 6*x))^m)))/7514

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Maple [F]  time = 0.171, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 1+4\,x \right ) ^{m}}{ \left ( 2+3\,x \right ) ^{2} \left ( 3\,{x}^{2}-5\,x+1 \right ) }}\, dx \]

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x + 1\right )}^{m}}{27 \, x^{4} - 9 \, x^{3} - 39 \, x^{2} - 8 \, x + 4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((4*x + 1)^m/(27*x^4 - 9*x^3 - 39*x^2 - 8*x + 4), x)

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

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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