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

Optimal. Leaf size=165 \[ -\frac{3 \left (416-135 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{13 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{3 \left (416+135 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{13 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{123 (4 x+1)^{m+1}}{16 (m+1)}+\frac{9 (4 x+1)^{m+2}}{16 (m+2)} \]

[Out]

(123*(1 + 4*x)^(1 + m))/(16*(1 + m)) + (9*(1 + 4*x)^(2 + m))/(16*(2 + m)) - (3*(
416 - 135*Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 +
4*x))/(13 - 2*Sqrt[13])])/(13*(13 - 2*Sqrt[13])*(1 + m)) - (3*(416 + 135*Sqrt[1
3])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*S
qrt[13])])/(13*(13 + 2*Sqrt[13])*(1 + m))

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Rubi [A]  time = 0.304733, antiderivative size = 165, 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 \[ -\frac{3 \left (416-135 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{13 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{3 \left (416+135 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{13 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{123 (4 x+1)^{m+1}}{16 (m+1)}+\frac{9 (4 x+1)^{m+2}}{16 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

(123*(1 + 4*x)^(1 + m))/(16*(1 + m)) + (9*(1 + 4*x)^(2 + m))/(16*(2 + m)) - (3*(
416 - 135*Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 +
4*x))/(13 - 2*Sqrt[13])])/(13*(13 - 2*Sqrt[13])*(1 + m)) - (3*(416 + 135*Sqrt[1
3])*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*S
qrt[13])])/(13*(13 + 2*Sqrt[13])*(1 + m))

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Rubi in Sympy [A]  time = 24.1704, size = 129, normalized size = 0.78 \[ \frac{9 \left (4 x + 1\right )^{m + 2}}{16 \left (m + 2\right )} - \frac{\left (- \frac{810 \sqrt{13}}{13} + 192\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{\left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} - \frac{\left (192 + \frac{810 \sqrt{13}}{13}\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{\left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{123 \left (4 x + 1\right )^{m + 1}}{16 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*(4*x + 1)**(m + 2)/(16*(m + 2)) - (-810*sqrt(13)/13 + 192)*(4*x + 1)**(m + 1)*
hyper((1, m + 1), (m + 2,), (-12*x - 3)/(-13 + 2*sqrt(13)))/((-4*sqrt(13) + 26)*
(m + 1)) - (192 + 810*sqrt(13)/13)*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,)
, (12*x + 3)/(2*sqrt(13) + 13))/((4*sqrt(13) + 26)*(m + 1)) + 123*(4*x + 1)**(m
+ 1)/(16*(m + 1))

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Mathematica [B]  time = 0.506035, size = 410, normalized size = 2.48 \[ \frac{(4 x+1)^m \left (13\ 2^{m+9} 3^{-m} (m+1) (m+2) \left (-\frac{4 x+1}{-6 x+\sqrt{13}+5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{13+2 \sqrt{13}}{2 \left (-6 x+\sqrt{13}+5\right )}\right )+5 \sqrt{13} 2^{m+4} 3^{3-m} (m+1) (m+2) \left (-\frac{4 x+1}{-6 x+\sqrt{13}+5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{13+2 \sqrt{13}}{2 \left (-6 x+\sqrt{13}+5\right )}\right )+13\ 2^{m+9} 3^{-m} (m+1) (m+2) \left (\frac{4 x+1}{6 x+\sqrt{13}-5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{-13+2 \sqrt{13}}{2 \left (6 x+\sqrt{13}-5\right )}\right )-5 \sqrt{13} 2^{m+4} 3^{3-m} (m+1) (m+2) \left (\frac{4 x+1}{6 x+\sqrt{13}-5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{-13+2 \sqrt{13}}{2 \left (6 x+\sqrt{13}-5\right )}\right )+1872 m^2 x^2+1716 m^2 x+312 m^2+1872 m x^2+325 m \left (-\frac{12 x}{5}-\frac{3}{5}\right )^{-m}+2496 m x+5616 m (m+2) x+507 m+1404 m (m+2)\right )}{208 m (m+1) (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

((1 + 4*x)^m*(507*m + 312*m^2 + 1404*m*(2 + m) + (325*m)/(-3/5 - (12*x)/5)^m + 2
496*m*x + 1716*m^2*x + 5616*m*(2 + m)*x + 1872*m*x^2 + 1872*m^2*x^2 + (13*2^(9 +
m)*(1 + m)*(2 + m)*Hypergeometric2F1[-m, -m, 1 - m, (13 + 2*Sqrt[13])/(2*(5 + S
qrt[13] - 6*x))])/(3^m*(-((1 + 4*x)/(5 + Sqrt[13] - 6*x)))^m) + (5*2^(4 + m)*3^(
3 - m)*Sqrt[13]*(1 + m)*(2 + 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 + (13*2^(9 +
m)*(1 + m)*(2 + m)*Hypergeometric2F1[-m, -m, 1 - m, (-13 + 2*Sqrt[13])/(2*(-5 +
Sqrt[13] + 6*x))])/(3^m*((1 + 4*x)/(-5 + Sqrt[13] + 6*x))^m) - (5*2^(4 + m)*3^(
3 - m)*Sqrt[13]*(1 + m)*(2 + m)*Hypergeometric2F1[-m, -m, 1 - m, (-13 + 2*Sqrt[1
3])/(2*(-5 + Sqrt[13] + 6*x))])/((1 + 4*x)/(-5 + Sqrt[13] + 6*x))^m))/(208*m*(1
+ m)*(2 + m))

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Maple [F]  time = 0.163, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{3} \left ( 1+4\,x \right ) ^{m}}{3\,{x}^{2}-5\,x+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}{\left (4 \, x + 1\right )}^{m}}{3 \, x^{2} - 5 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)*(4*x + 1)^m/(3*x^2 - 5*x + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)**3*(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. \[ \int \frac{{\left (4 \, x + 1\right )}^{m}{\left (3 \, x + 2\right )}^{3}}{3 \, x^{2} - 5 \, x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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