3.1795 \(\int \sqrt{1-2 x} (2+3 x) (3+5 x) \, dx\)

Optimal. Leaf size=40 \[ -\frac{15}{28} (1-2 x)^{7/2}+\frac{17}{5} (1-2 x)^{5/2}-\frac{77}{12} (1-2 x)^{3/2} \]

[Out]

(-77*(1 - 2*x)^(3/2))/12 + (17*(1 - 2*x)^(5/2))/5 - (15*(1 - 2*x)^(7/2))/28

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Rubi [A]  time = 0.0084678, 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} \[ -\frac{15}{28} (1-2 x)^{7/2}+\frac{17}{5} (1-2 x)^{5/2}-\frac{77}{12} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-77*(1 - 2*x)^(3/2))/12 + (17*(1 - 2*x)^(5/2))/5 - (15*(1 - 2*x)^(7/2))/28

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 \sqrt{1-2 x} (2+3 x) (3+5 x) \, dx &=\int \left (\frac{77}{4} \sqrt{1-2 x}-17 (1-2 x)^{3/2}+\frac{15}{4} (1-2 x)^{5/2}\right ) \, dx\\ &=-\frac{77}{12} (1-2 x)^{3/2}+\frac{17}{5} (1-2 x)^{5/2}-\frac{15}{28} (1-2 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0100952, size = 23, normalized size = 0.57 \[ -\frac{1}{105} (1-2 x)^{3/2} \left (225 x^2+489 x+373\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(3/2)*(373 + 489*x + 225*x^2))/105

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Maple [A]  time = 0.003, size = 20, normalized size = 0.5 \begin{align*} -{\frac{225\,{x}^{2}+489\,x+373}{105} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \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]

-1/105*(225*x^2+489*x+373)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.08022, size = 38, normalized size = 0.95 \begin{align*} -\frac{15}{28} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{17}{5} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{77}{12} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \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]

-15/28*(-2*x + 1)^(7/2) + 17/5*(-2*x + 1)^(5/2) - 77/12*(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.35289, size = 76, normalized size = 1.9 \begin{align*} \frac{1}{105} \,{\left (450 \, x^{3} + 753 \, x^{2} + 257 \, x - 373\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]

1/105*(450*x^3 + 753*x^2 + 257*x - 373)*sqrt(-2*x + 1)

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Sympy [A]  time = 1.57053, size = 34, normalized size = 0.85 \begin{align*} - \frac{15 \left (1 - 2 x\right )^{\frac{7}{2}}}{28} + \frac{17 \left (1 - 2 x\right )^{\frac{5}{2}}}{5} - \frac{77 \left (1 - 2 x\right )^{\frac{3}{2}}}{12} \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]

-15*(1 - 2*x)**(7/2)/28 + 17*(1 - 2*x)**(5/2)/5 - 77*(1 - 2*x)**(3/2)/12

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Giac [A]  time = 1.71598, size = 57, normalized size = 1.42 \begin{align*} \frac{15}{28} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{17}{5} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{77}{12} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \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]

15/28*(2*x - 1)^3*sqrt(-2*x + 1) + 17/5*(2*x - 1)^2*sqrt(-2*x + 1) - 77/12*(-2*x + 1)^(3/2)