3.2075 \(\int \frac{(2+3 x) (3+5 x)}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{5}{4} (1-2 x)^{3/2}+17 \sqrt{1-2 x}+\frac{77}{4 \sqrt{1-2 x}} \]

[Out]

77/(4*Sqrt[1 - 2*x]) + 17*Sqrt[1 - 2*x] - (5*(1 - 2*x)^(3/2))/4

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Rubi [A]  time = 0.0080628, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{5}{4} (1-2 x)^{3/2}+17 \sqrt{1-2 x}+\frac{77}{4 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

77/(4*Sqrt[1 - 2*x]) + 17*Sqrt[1 - 2*x] - (5*(1 - 2*x)^(3/2))/4

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) (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{77}{4 (1-2 x)^{3/2}}-\frac{17}{\sqrt{1-2 x}}+\frac{15}{4} \sqrt{1-2 x}\right ) \, dx\\ &=\frac{77}{4 \sqrt{1-2 x}}+17 \sqrt{1-2 x}-\frac{5}{4} (1-2 x)^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0080641, size = 20, normalized size = 0.53 \[ \frac{-5 x^2-29 x+35}{\sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35 - 29*x - 5*x^2)/Sqrt[1 - 2*x]

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Maple [A]  time = 0.002, size = 20, normalized size = 0.5 \begin{align*} -{(5\,{x}^{2}+29\,x-35){\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(5*x^2+29*x-35)/(1-2*x)^(1/2)

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Maxima [A]  time = 2.5571, size = 38, normalized size = 1. \begin{align*} -\frac{5}{4} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 17 \, \sqrt{-2 \, x + 1} + \frac{77}{4 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/4*(-2*x + 1)^(3/2) + 17*sqrt(-2*x + 1) + 77/4/sqrt(-2*x + 1)

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Fricas [A]  time = 1.64117, size = 62, normalized size = 1.63 \begin{align*} \frac{{\left (5 \, x^{2} + 29 \, x - 35\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)*(3+5*x)/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

(5*x^2 + 29*x - 35)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]  time = 7.50867, size = 32, normalized size = 0.84 \begin{align*} - \frac{5 \left (1 - 2 x\right )^{\frac{3}{2}}}{4} + 17 \sqrt{1 - 2 x} + \frac{77}{4 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*(1 - 2*x)**(3/2)/4 + 17*sqrt(1 - 2*x) + 77/(4*sqrt(1 - 2*x))

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Giac [A]  time = 2.51225, size = 38, normalized size = 1. \begin{align*} -\frac{5}{4} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 17 \, \sqrt{-2 \, x + 1} + \frac{77}{4 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/4*(-2*x + 1)^(3/2) + 17*sqrt(-2*x + 1) + 77/4/sqrt(-2*x + 1)