3.2005 \(\int \frac{(2+3 x) (3+5 x)}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{3}{4} (1-2 x)^{5/2}+\frac{17}{3} (1-2 x)^{3/2}-\frac{77}{4} \sqrt{1-2 x} \]

[Out]

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

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Rubi [A]  time = 0.0076623, antiderivative size = 40, 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} \[ -\frac{3}{4} (1-2 x)^{5/2}+\frac{17}{3} (1-2 x)^{3/2}-\frac{77}{4} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0081149, size = 23, normalized size = 0.57 \[ -\frac{1}{3} \sqrt{1-2 x} \left (9 x^2+25 x+43\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(43 + 25*x + 9*x^2))/3

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Maple [A]  time = 0.002, size = 20, normalized size = 0.5 \begin{align*} -{\frac{9\,{x}^{2}+25\,x+43}{3}\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)^(1/2),x)

[Out]

-1/3*(9*x^2+25*x+43)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.33422, size = 38, normalized size = 0.95 \begin{align*} -\frac{3}{4} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{17}{3} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.35554, size = 55, normalized size = 1.38 \begin{align*} -\frac{1}{3} \,{\left (9 \, x^{2} + 25 \, x + 43\right )} \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)^(1/2),x, algorithm="fricas")

[Out]

-1/3*(9*x^2 + 25*x + 43)*sqrt(-2*x + 1)

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Sympy [A]  time = 11.9852, size = 34, normalized size = 0.85 \begin{align*} - \frac{3 \left (1 - 2 x\right )^{\frac{5}{2}}}{4} + \frac{17 \left (1 - 2 x\right )^{\frac{3}{2}}}{3} - \frac{77 \sqrt{1 - 2 x}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.58397, size = 47, normalized size = 1.18 \begin{align*} -\frac{3}{4} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{17}{3} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \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)^(1/2),x, algorithm="giac")

[Out]

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