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

Optimal. Leaf size=183 \[ -\frac{3 \left (5499-1631 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{26 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{3 \left (5499+1631 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{26 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{3687 (4 x+1)^{m+1}}{64 (m+1)}+\frac{207 (4 x+1)^{m+2}}{32 (m+2)}+\frac{27 (4 x+1)^{m+3}}{64 (m+3)} \]

[Out]

(3687*(1 + 4*x)^(1 + m))/(64*(1 + m)) + (207*(1 + 4*x)^(2 + m))/(32*(2 + m)) + (
27*(1 + 4*x)^(3 + m))/(64*(3 + m)) - (3*(5499 - 1631*Sqrt[13])*(1 + 4*x)^(1 + m)
*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sqrt[13])])/(26*(13 -
2*Sqrt[13])*(1 + m)) - (3*(5499 + 1631*Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometri
c2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[13])])/(26*(13 + 2*Sqrt[13])*(1
+ m))

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Rubi [A]  time = 0.433676, antiderivative size = 183, 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 \[ -\frac{3 \left (5499-1631 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{26 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{3 \left (5499+1631 \sqrt{13}\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{26 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{3687 (4 x+1)^{m+1}}{64 (m+1)}+\frac{207 (4 x+1)^{m+2}}{32 (m+2)}+\frac{27 (4 x+1)^{m+3}}{64 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

(3687*(1 + 4*x)^(1 + m))/(64*(1 + m)) + (207*(1 + 4*x)^(2 + m))/(32*(2 + m)) + (
27*(1 + 4*x)^(3 + m))/(64*(3 + m)) - (3*(5499 - 1631*Sqrt[13])*(1 + 4*x)^(1 + m)
*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sqrt[13])])/(26*(13 -
2*Sqrt[13])*(1 + m)) - (3*(5499 + 1631*Sqrt[13])*(1 + 4*x)^(1 + m)*Hypergeometri
c2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[13])])/(26*(13 + 2*Sqrt[13])*(1
+ m))

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Rubi in Sympy [A]  time = 26.4807, size = 144, normalized size = 0.79 \[ \frac{27 \left (4 x + 1\right )^{m + 3}}{64 \left (m + 3\right )} + \frac{207 \left (4 x + 1\right )^{m + 2}}{32 \left (m + 2\right )} - \frac{\left (- \frac{4893 \sqrt{13}}{13} + 1269\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{\left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} - \frac{\left (1269 + \frac{4893 \sqrt{13}}{13}\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{\left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{3687 \left (4 x + 1\right )^{m + 1}}{64 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*(4*x + 1)**(m + 3)/(64*(m + 3)) + 207*(4*x + 1)**(m + 2)/(32*(m + 2)) - (-489
3*sqrt(13)/13 + 1269)*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (-12*x - 3)
/(-13 + 2*sqrt(13)))/((-4*sqrt(13) + 26)*(m + 1)) - (1269 + 4893*sqrt(13)/13)*(4
*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (12*x + 3)/(2*sqrt(13) + 13))/((4*s
qrt(13) + 26)*(m + 1)) + 3687*(4*x + 1)**(m + 1)/(64*(m + 1))

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Mathematica [B]  time = 1.43798, size = 568, normalized size = 3.1 \[ \frac{17\ 2^{m+6} 3^{-m} (4 x+1)^m \left (\left (\frac{4 x+1}{6 x+\sqrt{13}-5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{-13+2 \sqrt{13}}{2 \left (6 x+\sqrt{13}-5\right )}\right )-\left (-\frac{4 x+1}{-6 x+\sqrt{13}+5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{13+2 \sqrt{13}}{2 \left (-6 x+\sqrt{13}+5\right )}\right )\right )}{\sqrt{13} m}+\frac{47\ 2^{2 m-1} 3^{2-m} \left (2 x+\frac{1}{2}\right )^m \left (\left (9+\sqrt{13}\right ) \left (-\frac{4 x+1}{-6 x+\sqrt{13}+5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{13+2 \sqrt{13}}{2 \left (-6 x+\sqrt{13}+5\right )}\right )+\left (\sqrt{13}-9\right ) \left (\frac{4 x+1}{6 x+\sqrt{13}-5}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{-13+2 \sqrt{13}}{2 \left (6 x+\sqrt{13}-5\right )}\right )\right )}{\sqrt{13} m}+\frac{3^{-m} (4 x+1)^m \left (24 m^2 (3 x+2)^2 (4 x+1) (-12 x-3)^m+1728 x^3 (-12 x-3)^m+3456 x^2 (-12 x-3)^m+12 m \left (216 x^3+402 x^2+223 x+34\right ) (-12 x-3)^m+2304 x (-12 x-3)^m+387 (-12 x-3)^m+5^{m+3}\right ) (-4 x-1)^{-m}}{32 (m+1) (m+2) (m+3)}+\frac{3^{2-m} (4 x+1)^m \left (144 x^2 (-12 x-3)^m+12 m \left (12 x^2+11 x+2\right ) (-12 x-3)^m+192 x (-12 x-3)^m+39 (-12 x-3)^m+5^{m+2}\right ) (-4 x-1)^{-m}}{16 (m+1) (m+2)}+\frac{48 (4 x+1)^{m+1}}{m+1} \]

Antiderivative was successfully verified.

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

[Out]

(48*(1 + 4*x)^(1 + m))/(1 + m) + (3^(2 - m)*(1 + 4*x)^m*(5^(2 + m) + 39*(-3 - 12
*x)^m + 192*(-3 - 12*x)^m*x + 144*(-3 - 12*x)^m*x^2 + 12*m*(-3 - 12*x)^m*(2 + 11
*x + 12*x^2)))/(16*(1 + m)*(2 + m)*(-1 - 4*x)^m) + ((1 + 4*x)^m*(5^(3 + m) + 387
*(-3 - 12*x)^m + 2304*(-3 - 12*x)^m*x + 3456*(-3 - 12*x)^m*x^2 + 1728*(-3 - 12*x
)^m*x^3 + 24*m^2*(-3 - 12*x)^m*(2 + 3*x)^2*(1 + 4*x) + 12*m*(-3 - 12*x)^m*(34 +
223*x + 402*x^2 + 216*x^3)))/(32*3^m*(1 + m)*(2 + m)*(3 + m)*(-1 - 4*x)^m) + (17
*2^(6 + m)*(1 + 4*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) + Hypergeometric2F
1[-m, -m, 1 - m, (-13 + 2*Sqrt[13])/(2*(-5 + Sqrt[13] + 6*x))]/((1 + 4*x)/(-5 +
Sqrt[13] + 6*x))^m))/(3^m*Sqrt[13]*m) + (47*2^(-1 + 2*m)*3^(2 - m)*(1/2 + 2*x)^m
*(((9 + Sqrt[13])*Hypergeometric2F1[-m, -m, 1 - m, (13 + 2*Sqrt[13])/(2*(5 + Sqr
t[13] - 6*x))])/(-((1 + 4*x)/(5 + Sqrt[13] - 6*x)))^m + ((-9 + Sqrt[13])*Hyperge
ometric2F1[-m, -m, 1 - m, (-13 + 2*Sqrt[13])/(2*(-5 + Sqrt[13] + 6*x))])/((1 + 4
*x)/(-5 + Sqrt[13] + 6*x))^m))/(Sqrt[13]*m)

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Maple [F]  time = 0.269, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{4} \left ( 1+4\,x \right ) ^{m}}{3\,{x}^{2}-5\,x+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}{\left (4 \, x + 1\right )}^{m}}{3 \, x^{2} - 5 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*(4*x + 1)^m/(3*x^2 - 5*x + 1),
x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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