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

Optimal. Leaf size=177 \[ -\frac{2 \left (2 \left (2+\sqrt{13}\right ) m+9\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{13 \sqrt{13} \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{2 \left (2 \left (2-\sqrt{13}\right ) m+9\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{13 \sqrt{13} \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(7-6 x) (4 x+1)^{m+1}}{39 \left (3 x^2-5 x+1\right )} \]

[Out]

((7 - 6*x)*(1 + 4*x)^(1 + m))/(39*(1 - 5*x + 3*x^2)) - (2*(9 + 2*(2 + Sqrt[13])*m)*(1 + 4*x)^(1 + m)*Hypergeom
etric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sqrt[13])])/(13*Sqrt[13]*(13 - 2*Sqrt[13])*(1 + m)) + (2*(9 +
2*(2 - Sqrt[13])*m)*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[13])])/(13
*Sqrt[13]*(13 + 2*Sqrt[13])*(1 + m))

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Rubi [A]  time = 0.199295, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {740, 830, 68} \[ -\frac{2 \left (2 \left (2+\sqrt{13}\right ) m+9\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13-2 \sqrt{13}}\right )}{13 \sqrt{13} \left (13-2 \sqrt{13}\right ) (m+1)}+\frac{2 \left (2 \left (2-\sqrt{13}\right ) m+9\right ) (4 x+1)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{3 (4 x+1)}{13+2 \sqrt{13}}\right )}{13 \sqrt{13} \left (13+2 \sqrt{13}\right ) (m+1)}+\frac{(7-6 x) (4 x+1)^{m+1}}{39 \left (3 x^2-5 x+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7 - 6*x)*(1 + 4*x)^(1 + m))/(39*(1 - 5*x + 3*x^2)) - (2*(9 + 2*(2 + Sqrt[13])*m)*(1 + 4*x)^(1 + m)*Hypergeom
etric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 - 2*Sqrt[13])])/(13*Sqrt[13]*(13 - 2*Sqrt[13])*(1 + m)) + (2*(9 +
2*(2 - Sqrt[13])*m)*(1 + 4*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (3*(1 + 4*x))/(13 + 2*Sqrt[13])])/(13
*Sqrt[13]*(13 + 2*Sqrt[13])*(1 + m))

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 830

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m, (f + g*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !RationalQ[m]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{(1+4 x)^m}{\left (1-5 x+3 x^2\right )^2} \, dx &=\frac{(7-6 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{1}{507} \int \frac{(1+4 x)^m (26 (9+14 m)-312 m x)}{1-5 x+3 x^2} \, dx\\ &=\frac{(7-6 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{1}{507} \int \left (\frac{\left (-312 m+12 \sqrt{13} (9+4 m)\right ) (1+4 x)^m}{-5-\sqrt{13}+6 x}+\frac{\left (-312 m-12 \sqrt{13} (9+4 m)\right ) (1+4 x)^m}{-5+\sqrt{13}+6 x}\right ) \, dx\\ &=\frac{(7-6 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{\left (4 \left (9+2 \left (2-\sqrt{13}\right ) m\right )\right ) \int \frac{(1+4 x)^m}{-5-\sqrt{13}+6 x} \, dx}{13 \sqrt{13}}+\frac{\left (4 \left (9+2 \left (2+\sqrt{13}\right ) m\right )\right ) \int \frac{(1+4 x)^m}{-5+\sqrt{13}+6 x} \, dx}{13 \sqrt{13}}\\ &=\frac{(7-6 x) (1+4 x)^{1+m}}{39 \left (1-5 x+3 x^2\right )}-\frac{2 \left (9+2 \left (2+\sqrt{13}\right ) m\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{3 (1+4 x)}{13-2 \sqrt{13}}\right )}{13 \sqrt{13} \left (13-2 \sqrt{13}\right ) (1+m)}+\frac{2 \left (9+2 \left (2-\sqrt{13}\right ) m\right ) (1+4 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{3 (1+4 x)}{13+2 \sqrt{13}}\right )}{13 \sqrt{13} \left (13+2 \sqrt{13}\right ) (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.162479, size = 150, normalized size = 0.85 \[ \frac{1}{507} (4 x+1)^{m+1} \left (\frac{6 \left (26 m+\sqrt{13} (4 m+9)\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13-2 \sqrt{13}}\right )}{\left (2 \sqrt{13}-13\right ) (m+1)}+\frac{6 \sqrt{13} \left (9-2 \left (\sqrt{13}-2\right ) m\right ) \, _2F_1\left (1,m+1;m+2;\frac{12 x+3}{13+2 \sqrt{13}}\right )}{\left (13+2 \sqrt{13}\right ) (m+1)}+\frac{91-78 x}{3 x^2-5 x+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 4*x)^(1 + m)*((91 - 78*x)/(1 - 5*x + 3*x^2) + (6*(26*m + Sqrt[13]*(9 + 4*m))*Hypergeometric2F1[1, 1 + m,
2 + m, (3 + 12*x)/(13 - 2*Sqrt[13])])/((-13 + 2*Sqrt[13])*(1 + m)) + (6*Sqrt[13]*(9 - 2*(-2 + Sqrt[13])*m)*Hy
pergeometric2F1[1, 1 + m, 2 + m, (3 + 12*x)/(13 + 2*Sqrt[13])])/((13 + 2*Sqrt[13])*(1 + m))))/507

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Maple [F]  time = 1.174, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 4\,x+1 \right ) ^{m}}{ \left ( 3\,{x}^{2}-5\,x+1 \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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