3.2074 \(\int \frac{(2+3 x)^2 (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{9}{8} (1-2 x)^{5/2}-\frac{103}{8} (1-2 x)^{3/2}+\frac{707}{8} \sqrt{1-2 x}+\frac{539}{8 \sqrt{1-2 x}} \]

[Out]

539/(8*Sqrt[1 - 2*x]) + (707*Sqrt[1 - 2*x])/8 - (103*(1 - 2*x)^(3/2))/8 + (9*(1 - 2*x)^(5/2))/8

________________________________________________________________________________________

Rubi [A]  time = 0.0098294, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {77} \[ \frac{9}{8} (1-2 x)^{5/2}-\frac{103}{8} (1-2 x)^{3/2}+\frac{707}{8} \sqrt{1-2 x}+\frac{539}{8 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

539/(8*Sqrt[1 - 2*x]) + (707*Sqrt[1 - 2*x])/8 - (103*(1 - 2*x)^(3/2))/8 + (9*(1 - 2*x)^(5/2))/8

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(2+3 x)^2 (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{539}{8 (1-2 x)^{3/2}}-\frac{707}{8 \sqrt{1-2 x}}+\frac{309}{8} \sqrt{1-2 x}-\frac{45}{8} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac{539}{8 \sqrt{1-2 x}}+\frac{707}{8} \sqrt{1-2 x}-\frac{103}{8} (1-2 x)^{3/2}+\frac{9}{8} (1-2 x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.010522, size = 25, normalized size = 0.47 \[ \frac{-9 x^3-38 x^2-132 x+144}{\sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(144 - 132*x - 38*x^2 - 9*x^3)/Sqrt[1 - 2*x]

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{(9\,{x}^{3}+38\,{x}^{2}+132\,x-144){\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(9*x^3+38*x^2+132*x-144)/(1-2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.05032, size = 50, normalized size = 0.94 \begin{align*} \frac{9}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{103}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{707}{8} \, \sqrt{-2 \, x + 1} + \frac{539}{8 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/8*(-2*x + 1)^(5/2) - 103/8*(-2*x + 1)^(3/2) + 707/8*sqrt(-2*x + 1) + 539/8/sqrt(-2*x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.63808, size = 77, normalized size = 1.45 \begin{align*} \frac{{\left (9 \, x^{3} + 38 \, x^{2} + 132 \, x - 144\right )} \sqrt{-2 \, x + 1}}{2 \, x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9*x^3 + 38*x^2 + 132*x - 144)*sqrt(-2*x + 1)/(2*x - 1)

________________________________________________________________________________________

Sympy [A]  time = 11.5917, size = 46, normalized size = 0.87 \begin{align*} \frac{9 \left (1 - 2 x\right )^{\frac{5}{2}}}{8} - \frac{103 \left (1 - 2 x\right )^{\frac{3}{2}}}{8} + \frac{707 \sqrt{1 - 2 x}}{8} + \frac{539}{8 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9*(1 - 2*x)**(5/2)/8 - 103*(1 - 2*x)**(3/2)/8 + 707*sqrt(1 - 2*x)/8 + 539/(8*sqrt(1 - 2*x))

________________________________________________________________________________________

Giac [A]  time = 1.70408, size = 59, normalized size = 1.11 \begin{align*} \frac{9}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{103}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{707}{8} \, \sqrt{-2 \, x + 1} + \frac{539}{8 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/8*(2*x - 1)^2*sqrt(-2*x + 1) - 103/8*(-2*x + 1)^(3/2) + 707/8*sqrt(-2*x + 1) + 539/8/sqrt(-2*x + 1)