3.1794 \(\int \sqrt{1-2 x} (2+3 x)^2 (3+5 x) \, dx\)

Optimal. Leaf size=53 \[ \frac{5}{8} (1-2 x)^{9/2}-\frac{309}{56} (1-2 x)^{7/2}+\frac{707}{40} (1-2 x)^{5/2}-\frac{539}{24} (1-2 x)^{3/2} \]

[Out]

(-539*(1 - 2*x)^(3/2))/24 + (707*(1 - 2*x)^(5/2))/40 - (309*(1 - 2*x)^(7/2))/56 + (5*(1 - 2*x)^(9/2))/8

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Rubi [A]  time = 0.0108028, 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} \[ \frac{5}{8} (1-2 x)^{9/2}-\frac{309}{56} (1-2 x)^{7/2}+\frac{707}{40} (1-2 x)^{5/2}-\frac{539}{24} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-539*(1 - 2*x)^(3/2))/24 + (707*(1 - 2*x)^(5/2))/40 - (309*(1 - 2*x)^(7/2))/56 + (5*(1 - 2*x)^(9/2))/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 \sqrt{1-2 x} (2+3 x)^2 (3+5 x) \, dx &=\int \left (\frac{539}{8} \sqrt{1-2 x}-\frac{707}{8} (1-2 x)^{3/2}+\frac{309}{8} (1-2 x)^{5/2}-\frac{45}{8} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac{539}{24} (1-2 x)^{3/2}+\frac{707}{40} (1-2 x)^{5/2}-\frac{309}{56} (1-2 x)^{7/2}+\frac{5}{8} (1-2 x)^{9/2}\\ \end{align*}

Mathematica [A]  time = 0.0118286, size = 28, normalized size = 0.53 \[ -\frac{1}{105} (1-2 x)^{3/2} \left (525 x^3+1530 x^2+1788 x+1016\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(3/2)*(1016 + 1788*x + 1530*x^2 + 525*x^3))/105

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{525\,{x}^{3}+1530\,{x}^{2}+1788\,x+1016}{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)^2*(3+5*x)*(1-2*x)^(1/2),x)

[Out]

-1/105*(525*x^3+1530*x^2+1788*x+1016)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.053, size = 50, normalized size = 0.94 \begin{align*} \frac{5}{8} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{309}{56} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{707}{40} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{539}{24} \,{\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)^2*(3+5*x)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

5/8*(-2*x + 1)^(9/2) - 309/56*(-2*x + 1)^(7/2) + 707/40*(-2*x + 1)^(5/2) - 539/24*(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.36828, size = 95, normalized size = 1.79 \begin{align*} \frac{1}{105} \,{\left (1050 \, x^{4} + 2535 \, x^{3} + 2046 \, x^{2} + 244 \, x - 1016\right )} \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)^(1/2),x, algorithm="fricas")

[Out]

1/105*(1050*x^4 + 2535*x^3 + 2046*x^2 + 244*x - 1016)*sqrt(-2*x + 1)

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Sympy [A]  time = 1.73982, size = 46, normalized size = 0.87 \begin{align*} \frac{5 \left (1 - 2 x\right )^{\frac{9}{2}}}{8} - \frac{309 \left (1 - 2 x\right )^{\frac{7}{2}}}{56} + \frac{707 \left (1 - 2 x\right )^{\frac{5}{2}}}{40} - \frac{539 \left (1 - 2 x\right )^{\frac{3}{2}}}{24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*(1 - 2*x)**(9/2)/8 - 309*(1 - 2*x)**(7/2)/56 + 707*(1 - 2*x)**(5/2)/40 - 539*(1 - 2*x)**(3/2)/24

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Giac [A]  time = 2.67413, size = 78, normalized size = 1.47 \begin{align*} \frac{5}{8} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{309}{56} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{707}{40} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{539}{24} \,{\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)^2*(3+5*x)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

5/8*(2*x - 1)^4*sqrt(-2*x + 1) + 309/56*(2*x - 1)^3*sqrt(-2*x + 1) + 707/40*(2*x - 1)^2*sqrt(-2*x + 1) - 539/2
4*(-2*x + 1)^(3/2)