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

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

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Rubi [A]  time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {1}{105} (1-2 x)^{3/2} \left (525 x^3+1530 x^2+1788 x+1016\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 49, normalized size = 0.92 \begin {gather*} \frac {1}{840} \left (525 (1-2 x)^{9/2}-4635 (1-2 x)^{7/2}+14847 (1-2 x)^{5/2}-18865 (1-2 x)^{3/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-18865*(1 - 2*x)^(3/2) + 14847*(1 - 2*x)^(5/2) - 4635*(1 - 2*x)^(7/2) + 525*(1 - 2*x)^(9/2))/840

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fricas [A]  time = 1.21, size = 29, normalized size = 0.55 \begin {gather*} \frac {1}{105} \, {\left (1050 \, x^{4} + 2535 \, x^{3} + 2046 \, x^{2} + 244 \, x - 1016\right )} \sqrt {-2 \, x + 1} \end {gather*}

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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giac [A]  time = 1.19, size = 58, normalized size = 1.09 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.00, size = 25, normalized size = 0.47 \begin {gather*} -\frac {\left (525 x^{3}+1530 x^{2}+1788 x +1016\right ) \left (-2 x +1\right )^{\frac {3}{2}}}{105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.51, size = 37, normalized size = 0.70 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {707\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {539\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {309\,{\left (1-2\,x\right )}^{7/2}}{56}+\frac {5\,{\left (1-2\,x\right )}^{9/2}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.44, size = 46, normalized size = 0.87 \begin {gather*} \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 {gather*}

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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