3.18.93 \(\int \sqrt {1-2 x} (2+3 x)^3 (3+5 x) \, dx\) [1793]

Optimal. Leaf size=66 \[ -\frac {3773}{48} (1-2 x)^{3/2}+\frac {3283}{40} (1-2 x)^{5/2}-\frac {153}{4} (1-2 x)^{7/2}+\frac {69}{8} (1-2 x)^{9/2}-\frac {135}{176} (1-2 x)^{11/2} \]

[Out]

-3773/48*(1-2*x)^(3/2)+3283/40*(1-2*x)^(5/2)-153/4*(1-2*x)^(7/2)+69/8*(1-2*x)^(9/2)-135/176*(1-2*x)^(11/2)

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Rubi [A]
time = 0.01, antiderivative size = 66, 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 = {78} \begin {gather*} -\frac {135}{176} (1-2 x)^{11/2}+\frac {69}{8} (1-2 x)^{9/2}-\frac {153}{4} (1-2 x)^{7/2}+\frac {3283}{40} (1-2 x)^{5/2}-\frac {3773}{48} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3773*(1 - 2*x)^(3/2))/48 + (3283*(1 - 2*x)^(5/2))/40 - (153*(1 - 2*x)^(7/2))/4 + (69*(1 - 2*x)^(9/2))/8 - (1
35*(1 - 2*x)^(11/2))/176

Rule 78

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 (3+5 x) \, dx &=\int \left (\frac {3773}{16} \sqrt {1-2 x}-\frac {3283}{8} (1-2 x)^{3/2}+\frac {1071}{4} (1-2 x)^{5/2}-\frac {621}{8} (1-2 x)^{7/2}+\frac {135}{16} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {3773}{48} (1-2 x)^{3/2}+\frac {3283}{40} (1-2 x)^{5/2}-\frac {153}{4} (1-2 x)^{7/2}+\frac {69}{8} (1-2 x)^{9/2}-\frac {135}{176} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {1}{165} (1-2 x)^{3/2} \left (4442+9366 x+11205 x^2+7335 x^3+2025 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/165*((1 - 2*x)^(3/2)*(4442 + 9366*x + 11205*x^2 + 7335*x^3 + 2025*x^4))

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Maple [A]
time = 0.10, size = 47, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (2025 x^{4}+7335 x^{3}+11205 x^{2}+9366 x +4442\right ) \left (1-2 x \right )^{\frac {3}{2}}}{165}\)	\(30\)
trager	\(\left (\frac {270}{11} x^{5}+\frac {843}{11} x^{4}+\frac {1005}{11} x^{3}+\frac {2509}{55} x^{2}-\frac {482}{165} x -\frac {4442}{165}\right ) \sqrt {1-2 x}\)	\(34\)
risch	\(-\frac {\left (4050 x^{5}+12645 x^{4}+15075 x^{3}+7527 x^{2}-482 x -4442\right ) \left (-1+2 x \right )}{165 \sqrt {1-2 x}}\)	\(40\)
derivativedivides	\(-\frac {3773 \left (1-2 x \right )^{\frac {3}{2}}}{48}+\frac {3283 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {153 \left (1-2 x \right )^{\frac {7}{2}}}{4}+\frac {69 \left (1-2 x \right )^{\frac {9}{2}}}{8}-\frac {135 \left (1-2 x \right )^{\frac {11}{2}}}{176}\)	\(47\)
default	\(-\frac {3773 \left (1-2 x \right )^{\frac {3}{2}}}{48}+\frac {3283 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {153 \left (1-2 x \right )^{\frac {7}{2}}}{4}+\frac {69 \left (1-2 x \right )^{\frac {9}{2}}}{8}-\frac {135 \left (1-2 x \right )^{\frac {11}{2}}}{176}\)	\(47\)
meijerg	\(\frac {8 \sqrt {\pi }-4 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {37 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{2 \sqrt {\pi }}+\frac {\frac {228 \sqrt {\pi }}{35}-\frac {57 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{70}}{\sqrt {\pi }}-\frac {351 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{32 \sqrt {\pi }}+\frac {\frac {24 \sqrt {\pi }}{77}-\frac {3 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{1232}}{\sqrt {\pi }}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3773/48*(1-2*x)^(3/2)+3283/40*(1-2*x)^(5/2)-153/4*(1-2*x)^(7/2)+69/8*(1-2*x)^(9/2)-135/176*(1-2*x)^(11/2)

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Maxima [A]
time = 0.29, size = 46, normalized size = 0.70 \begin {gather*} -\frac {135}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {69}{8} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {153}{4} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {3283}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {3773}{48} \, {\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)^3*(3+5*x)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-135/176*(-2*x + 1)^(11/2) + 69/8*(-2*x + 1)^(9/2) - 153/4*(-2*x + 1)^(7/2) + 3283/40*(-2*x + 1)^(5/2) - 3773/
48*(-2*x + 1)^(3/2)

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Fricas [A]
time = 1.26, size = 34, normalized size = 0.52 \begin {gather*} \frac {1}{165} \, {\left (4050 \, x^{5} + 12645 \, x^{4} + 15075 \, x^{3} + 7527 \, x^{2} - 482 \, x - 4442\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/165*(4050*x^5 + 12645*x^4 + 15075*x^3 + 7527*x^2 - 482*x - 4442)*sqrt(-2*x + 1)

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Sympy [A]
time = 1.77, size = 58, normalized size = 0.88 \begin {gather*} - \frac {135 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} + \frac {69 \left (1 - 2 x\right )^{\frac {9}{2}}}{8} - \frac {153 \left (1 - 2 x\right )^{\frac {7}{2}}}{4} + \frac {3283 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} - \frac {3773 \left (1 - 2 x\right )^{\frac {3}{2}}}{48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-135*(1 - 2*x)**(11/2)/176 + 69*(1 - 2*x)**(9/2)/8 - 153*(1 - 2*x)**(7/2)/4 + 3283*(1 - 2*x)**(5/2)/40 - 3773*
(1 - 2*x)**(3/2)/48

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Giac [A]
time = 0.55, size = 74, normalized size = 1.12 \begin {gather*} \frac {135}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {69}{8} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {153}{4} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {3283}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {3773}{48} \, {\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)^3*(3+5*x)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

135/176*(2*x - 1)^5*sqrt(-2*x + 1) + 69/8*(2*x - 1)^4*sqrt(-2*x + 1) + 153/4*(2*x - 1)^3*sqrt(-2*x + 1) + 3283
/40*(2*x - 1)^2*sqrt(-2*x + 1) - 3773/48*(-2*x + 1)^(3/2)

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Mupad [B]
time = 0.03, size = 46, normalized size = 0.70 \begin {gather*} \frac {3283\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {3773\,{\left (1-2\,x\right )}^{3/2}}{48}-\frac {153\,{\left (1-2\,x\right )}^{7/2}}{4}+\frac {69\,{\left (1-2\,x\right )}^{9/2}}{8}-\frac {135\,{\left (1-2\,x\right )}^{11/2}}{176} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3283*(1 - 2*x)^(5/2))/40 - (3773*(1 - 2*x)^(3/2))/48 - (153*(1 - 2*x)^(7/2))/4 + (69*(1 - 2*x)^(9/2))/8 - (13
5*(1 - 2*x)^(11/2))/176

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