3.481 \(\int \frac{1+3 x}{(1-8 x+2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{1-2 x}{6 \left (2 x^2-8 x+1\right )^{3/2}}-\frac{2 (2-x)}{21 \sqrt{2 x^2-8 x+1}} \]

[Out]

(1 - 2*x)/(6*(1 - 8*x + 2*x^2)^(3/2)) - (2*(2 - x))/(21*Sqrt[1 - 8*x + 2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0089818, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {638, 613} \[ \frac{1-2 x}{6 \left (2 x^2-8 x+1\right )^{3/2}}-\frac{2 (2-x)}{21 \sqrt{2 x^2-8 x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 3*x)/(1 - 8*x + 2*x^2)^(5/2),x]

[Out]

(1 - 2*x)/(6*(1 - 8*x + 2*x^2)^(3/2)) - (2*(2 - x))/(21*Sqrt[1 - 8*x + 2*x^2])

Rule 638

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

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1+3 x}{\left (1-8 x+2 x^2\right )^{5/2}} \, dx &=\frac{1-2 x}{6 \left (1-8 x+2 x^2\right )^{3/2}}-\frac{2}{3} \int \frac{1}{\left (1-8 x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{1-2 x}{6 \left (1-8 x+2 x^2\right )^{3/2}}-\frac{2 (2-x)}{21 \sqrt{1-8 x+2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0468753, size = 33, normalized size = 0.7 \[ \frac{8 x^3-48 x^2+54 x-1}{42 \left (2 x^2-8 x+1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 3*x)/(1 - 8*x + 2*x^2)^(5/2),x]

[Out]

(-1 + 54*x - 48*x^2 + 8*x^3)/(42*(1 - 8*x + 2*x^2)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 30, normalized size = 0.6 \begin{align*}{\frac{8\,{x}^{3}-48\,{x}^{2}+54\,x-1}{42} \left ( 2\,{x}^{2}-8\,x+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+3*x)/(2*x^2-8*x+1)^(5/2),x)

[Out]

1/42*(8*x^3-48*x^2+54*x-1)/(2*x^2-8*x+1)^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 0.937834, size = 80, normalized size = 1.7 \begin{align*} \frac{2 \, x}{21 \, \sqrt{2 \, x^{2} - 8 \, x + 1}} - \frac{4}{21 \, \sqrt{2 \, x^{2} - 8 \, x + 1}} - \frac{x}{3 \,{\left (2 \, x^{2} - 8 \, x + 1\right )}^{\frac{3}{2}}} + \frac{1}{6 \,{\left (2 \, x^{2} - 8 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+3*x)/(2*x^2-8*x+1)^(5/2),x, algorithm="maxima")

[Out]

2/21*x/sqrt(2*x^2 - 8*x + 1) - 4/21/sqrt(2*x^2 - 8*x + 1) - 1/3*x/(2*x^2 - 8*x + 1)^(3/2) + 1/6/(2*x^2 - 8*x +
1)^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 2.01195, size = 180, normalized size = 3.83 \begin{align*} -\frac{4 \, x^{4} - 32 \, x^{3} + 68 \, x^{2} -{\left (8 \, x^{3} - 48 \, x^{2} + 54 \, x - 1\right )} \sqrt{2 \, x^{2} - 8 \, x + 1} - 16 \, x + 1}{42 \,{\left (4 \, x^{4} - 32 \, x^{3} + 68 \, x^{2} - 16 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+3*x)/(2*x^2-8*x+1)^(5/2),x, algorithm="fricas")

[Out]

-1/42*(4*x^4 - 32*x^3 + 68*x^2 - (8*x^3 - 48*x^2 + 54*x - 1)*sqrt(2*x^2 - 8*x + 1) - 16*x + 1)/(4*x^4 - 32*x^3
+ 68*x^2 - 16*x + 1)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3 x + 1}{\left (2 x^{2} - 8 x + 1\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+3*x)/(2*x**2-8*x+1)**(5/2),x)

[Out]

Integral((3*x + 1)/(2*x**2 - 8*x + 1)**(5/2), x)

________________________________________________________________________________________

Giac [A]  time = 1.08805, size = 36, normalized size = 0.77 \begin{align*} \frac{2 \,{\left (4 \,{\left (x - 6\right )} x + 27\right )} x - 1}{42 \,{\left (2 \, x^{2} - 8 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+3*x)/(2*x^2-8*x+1)^(5/2),x, algorithm="giac")

[Out]

1/42*(2*(4*(x - 6)*x + 27)*x - 1)/(2*x^2 - 8*x + 1)^(3/2)