3.19.61 \(\int (1-2 x)^{3/2} (2+3 x)^2 (3+5 x) \, dx\) [1861]

Optimal. Leaf size=53 \[ -\frac {539}{40} (1-2 x)^{5/2}+\frac {101}{8} (1-2 x)^{7/2}-\frac {103}{24} (1-2 x)^{9/2}+\frac {45}{88} (1-2 x)^{11/2} \]

[Out]

-539/40*(1-2*x)^(5/2)+101/8*(1-2*x)^(7/2)-103/24*(1-2*x)^(9/2)+45/88*(1-2*x)^(11/2)

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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 = {78} \begin {gather*} \frac {45}{88} (1-2 x)^{11/2}-\frac {103}{24} (1-2 x)^{9/2}+\frac {101}{8} (1-2 x)^{7/2}-\frac {539}{40} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-539*(1 - 2*x)^(5/2))/40 + (101*(1 - 2*x)^(7/2))/8 - (103*(1 - 2*x)^(9/2))/24 + (45*(1 - 2*x)^(11/2))/88

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 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x) \, dx &=\int \left (\frac {539}{8} (1-2 x)^{3/2}-\frac {707}{8} (1-2 x)^{5/2}+\frac {309}{8} (1-2 x)^{7/2}-\frac {45}{8} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {539}{40} (1-2 x)^{5/2}+\frac {101}{8} (1-2 x)^{7/2}-\frac {103}{24} (1-2 x)^{9/2}+\frac {45}{88} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {1}{165} (1-2 x)^{5/2} \left (764+1840 x+1820 x^2+675 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/165*((1 - 2*x)^(5/2)*(764 + 1840*x + 1820*x^2 + 675*x^3))

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Maple [A]
time = 0.09, size = 38, normalized size = 0.72

method result size
gosper \(-\frac {\left (675 x^{3}+1820 x^{2}+1840 x +764\right ) \left (1-2 x \right )^{\frac {5}{2}}}{165}\) \(25\)
trager \(\left (-\frac {180}{11} x^{5}-\frac {916}{33} x^{4}-\frac {151}{33} x^{3}+\frac {828}{55} x^{2}+\frac {1216}{165} x -\frac {764}{165}\right ) \sqrt {1-2 x}\) \(34\)
derivativedivides \(-\frac {539 \left (1-2 x \right )^{\frac {5}{2}}}{40}+\frac {101 \left (1-2 x \right )^{\frac {7}{2}}}{8}-\frac {103 \left (1-2 x \right )^{\frac {9}{2}}}{24}+\frac {45 \left (1-2 x \right )^{\frac {11}{2}}}{88}\) \(38\)
default \(-\frac {539 \left (1-2 x \right )^{\frac {5}{2}}}{40}+\frac {101 \left (1-2 x \right )^{\frac {7}{2}}}{8}-\frac {103 \left (1-2 x \right )^{\frac {9}{2}}}{24}+\frac {45 \left (1-2 x \right )^{\frac {11}{2}}}{88}\) \(38\)
risch \(\frac {\left (2700 x^{5}+4580 x^{4}+755 x^{3}-2484 x^{2}-1216 x +764\right ) \left (-1+2 x \right )}{165 \sqrt {1-2 x}}\) \(40\)
meijerg \(-\frac {9 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{2 \sqrt {\pi }}+\frac {\frac {8 \sqrt {\pi }}{5}-\frac {\sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{5}}{\sqrt {\pi }}-\frac {261 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{32 \sqrt {\pi }}+\frac {\frac {6 \sqrt {\pi }}{77}-\frac {3 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{4928}}{\sqrt {\pi }}\) \(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-539/40*(1-2*x)^(5/2)+101/8*(1-2*x)^(7/2)-103/24*(1-2*x)^(9/2)+45/88*(1-2*x)^(11/2)

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Maxima [A]
time = 0.29, size = 37, normalized size = 0.70 \begin {gather*} \frac {45}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {103}{24} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {101}{8} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {539}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

45/88*(-2*x + 1)^(11/2) - 103/24*(-2*x + 1)^(9/2) + 101/8*(-2*x + 1)^(7/2) - 539/40*(-2*x + 1)^(5/2)

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Fricas [A]
time = 1.03, size = 34, normalized size = 0.64 \begin {gather*} -\frac {1}{165} \, {\left (2700 \, x^{5} + 4580 \, x^{4} + 755 \, x^{3} - 2484 \, x^{2} - 1216 \, x + 764\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/165*(2700*x^5 + 4580*x^4 + 755*x^3 - 2484*x^2 - 1216*x + 764)*sqrt(-2*x + 1)

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Sympy [A]
time = 6.13, size = 46, normalized size = 0.87 \begin {gather*} \frac {45 \left (1 - 2 x\right )^{\frac {11}{2}}}{88} - \frac {103 \left (1 - 2 x\right )^{\frac {9}{2}}}{24} + \frac {101 \left (1 - 2 x\right )^{\frac {7}{2}}}{8} - \frac {539 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

45*(1 - 2*x)**(11/2)/88 - 103*(1 - 2*x)**(9/2)/24 + 101*(1 - 2*x)**(7/2)/8 - 539*(1 - 2*x)**(5/2)/40

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Giac [A]
time = 0.96, size = 65, normalized size = 1.23 \begin {gather*} -\frac {45}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {103}{24} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {101}{8} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {539}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/88*(2*x - 1)^5*sqrt(-2*x + 1) - 103/24*(2*x - 1)^4*sqrt(-2*x + 1) - 101/8*(2*x - 1)^3*sqrt(-2*x + 1) - 539
/40*(2*x - 1)^2*sqrt(-2*x + 1)

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Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {101\,{\left (1-2\,x\right )}^{7/2}}{8}-\frac {539\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {103\,{\left (1-2\,x\right )}^{9/2}}{24}+\frac {45\,{\left (1-2\,x\right )}^{11/2}}{88} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(101*(1 - 2*x)^(7/2))/8 - (539*(1 - 2*x)^(5/2))/40 - (103*(1 - 2*x)^(9/2))/24 + (45*(1 - 2*x)^(11/2))/88

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