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

Optimal. Leaf size=202 \[ -\frac{\left (13689-\sqrt{13} \left (-1570 \sqrt{13} m+4474 m+297\right )\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{169 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{\left (\sqrt{13} \left (1570 \sqrt{13} m+4474 m+297\right )+13689\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{169 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(844-2355 x) (4 x+1)^{m+1}}{39 \left (3 x^2-5 x+1\right )}+\frac{9 (4 x+1)^{m+1}}{4 (m+1)} \]

[Out]

(9*(1 + 4*x)^(1 + m))/(4*(1 + m)) + ((844 - 2355*x)*(1 + 4*x)^(1 + m))/(39*(1 -
5*x + 3*x^2)) - ((13689 - Sqrt[13]*(297 + 4474*m - 1570*Sqrt[13]*m))*(1 + 4*x)^(
1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sqrt[13])])/(169
*(13 - 2*Sqrt[13])*(1 + m)) - ((13689 + Sqrt[13]*(297 + 4474*m + 1570*Sqrt[13]*m
))*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sq
rt[13])])/(169*(13 + 2*Sqrt[13])*(1 + m))

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Rubi [A]  time = 0.554124, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\left (13689-\sqrt{13} \left (-1570 \sqrt{13} m+4474 m+297\right )\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{169 \left (13-2 \sqrt{13}\right ) (m+1)}-\frac{\left (\sqrt{13} \left (1570 \sqrt{13} m+4474 m+297\right )+13689\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{169 \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(844-2355 x) (4 x+1)^{m+1}}{39 \left (3 x^2-5 x+1\right )}+\frac{9 (4 x+1)^{m+1}}{4 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(9*(1 + 4*x)^(1 + m))/(4*(1 + m)) + ((844 - 2355*x)*(1 + 4*x)^(1 + m))/(39*(1 -
5*x + 3*x^2)) - ((13689 - Sqrt[13]*(297 + 4474*m - 1570*Sqrt[13]*m))*(1 + 4*x)^(
1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sqrt[13])])/(169
*(13 - 2*Sqrt[13])*(1 + m)) - ((13689 + Sqrt[13]*(297 + 4474*m + 1570*Sqrt[13]*m
))*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sq
rt[13])])/(169*(13 + 2*Sqrt[13])*(1 + m))

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Rubi in Sympy [A]  time = 58.545, size = 255, normalized size = 1.26 \[ \frac{\left (- 30615 x + 10972\right ) \left (4 x + 1\right )^{m + 1}}{507 \left (3 x^{2} - 5 x + 1\right )} - \frac{2 \left (20410 m - \sqrt{13} \left (- 4474 m + 14679\right )\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{169 \left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} - \frac{2 \left (20410 m + \sqrt{13} \left (- 4474 m + 14679\right )\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{169 \left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} - \frac{3 \left (- \frac{768 \sqrt{13}}{13} + 54\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{- 12 x - 3}{-13 + 2 \sqrt{13}}} \right )}}{\left (- 4 \sqrt{13} + 26\right ) \left (m + 1\right )} - \frac{3 \left (54 + \frac{768 \sqrt{13}}{13}\right ) \left (4 x + 1\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{12 x + 3}{2 \sqrt{13} + 13}} \right )}}{\left (4 \sqrt{13} + 26\right ) \left (m + 1\right )} + \frac{9 \left (4 x + 1\right )^{m + 1}}{4 \left (m + 1\right )} \]

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)**2,x)

[Out]

(-30615*x + 10972)*(4*x + 1)**(m + 1)/(507*(3*x**2 - 5*x + 1)) - 2*(20410*m - sq
rt(13)*(-4474*m + 14679))*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (12*x +
3)/(2*sqrt(13) + 13))/(169*(4*sqrt(13) + 26)*(m + 1)) - 2*(20410*m + sqrt(13)*(
-4474*m + 14679))*(4*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (-12*x - 3)/(-1
3 + 2*sqrt(13)))/(169*(-4*sqrt(13) + 26)*(m + 1)) - 3*(-768*sqrt(13)/13 + 54)*(4
*x + 1)**(m + 1)*hyper((1, m + 1), (m + 2,), (-12*x - 3)/(-13 + 2*sqrt(13)))/((-
4*sqrt(13) + 26)*(m + 1)) - 3*(54 + 768*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)) +
9*(4*x + 1)**(m + 1)/(4*(m + 1))

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Mathematica [A]  time = 0.20912, size = 0, normalized size = 0. \[ \int \frac{(2+3 x)^4 (1+4 x)^m}{\left (1-5 x+3 x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.17, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{4} \left ( 1+4\,x \right ) ^{m}}{ \left ( 3\,{x}^{2}-5\,x+1 \right ) ^{2}}}\, dx \]

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)^2,x)

[Out]

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

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

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}{\left (4 \, x + 1\right )}^{m}}{9 \, x^{4} - 30 \, x^{3} + 31 \, x^{2} - 10 \, x + 1}, x\right ) \]

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)^2,x, algorithm="fricas")

[Out]

integral((81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*(4*x + 1)^m/(9*x^4 - 30*x^3 +
31*x^2 - 10*x + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**2,x)

[Out]

Timed out

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

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)^2,x, algorithm="giac")

[Out]

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