3.21.74 \(\int \frac {(2+3 x)^2 (3+5 x)}{(1-2 x)^{3/2}} \, dx\) [2074]

Optimal. Leaf size=53 \[ \frac {539}{8 \sqrt {1-2 x}}+\frac {707}{8} \sqrt {1-2 x}-\frac {103}{8} (1-2 x)^{3/2}+\frac {9}{8} (1-2 x)^{5/2} \]

[Out]

-103/8*(1-2*x)^(3/2)+9/8*(1-2*x)^(5/2)+539/8/(1-2*x)^(1/2)+707/8*(1-2*x)^(1/2)

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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 {9}{8} (1-2 x)^{5/2}-\frac {103}{8} (1-2 x)^{3/2}+\frac {707}{8} \sqrt {1-2 x}+\frac {539}{8 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

539/(8*Sqrt[1 - 2*x]) + (707*Sqrt[1 - 2*x])/8 - (103*(1 - 2*x)^(3/2))/8 + (9*(1 - 2*x)^(5/2))/8

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

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Mathematica [A]
time = 0.02, size = 25, normalized size = 0.47 \begin {gather*} \frac {144-132 x-38 x^2-9 x^3}{\sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(144 - 132*x - 38*x^2 - 9*x^3)/Sqrt[1 - 2*x]

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Maple [A]
time = 0.16, size = 38, normalized size = 0.72

method	result	size
gosper	\(-\frac {9 x^{3}+38 x^{2}+132 x -144}{\sqrt {1-2 x}}\)	\(25\)
risch	\(-\frac {9 x^{3}+38 x^{2}+132 x -144}{\sqrt {1-2 x}}\)	\(25\)
trager	\(\frac {\left (9 x^{3}+38 x^{2}+132 x -144\right ) \sqrt {1-2 x}}{-1+2 x}\)	\(31\)
derivativedivides	\(-\frac {103 \left (1-2 x \right )^{\frac {3}{2}}}{8}+\frac {9 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {539}{8 \sqrt {1-2 x}}+\frac {707 \sqrt {1-2 x}}{8}\)	\(38\)
default	\(-\frac {103 \left (1-2 x \right )^{\frac {3}{2}}}{8}+\frac {9 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {539}{8 \sqrt {1-2 x}}+\frac {707 \sqrt {1-2 x}}{8}\)	\(38\)
meijerg	\(-\frac {12 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-56 \sqrt {\pi }+\frac {7 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {87 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{4 \sqrt {\pi }}+\frac {-18 \sqrt {\pi }+\frac {9 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{64 \sqrt {1-2 x}}}{\sqrt {\pi }}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-103/8*(1-2*x)^(3/2)+9/8*(1-2*x)^(5/2)+539/8/(1-2*x)^(1/2)+707/8*(1-2*x)^(1/2)

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Maxima [A]
time = 0.30, size = 37, normalized size = 0.70 \begin {gather*} \frac {9}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {103}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {707}{8} \, \sqrt {-2 \, x + 1} + \frac {539}{8 \, \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)^(3/2),x, algorithm="maxima")

[Out]

9/8*(-2*x + 1)^(5/2) - 103/8*(-2*x + 1)^(3/2) + 707/8*sqrt(-2*x + 1) + 539/8/sqrt(-2*x + 1)

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Fricas [A]
time = 1.20, size = 30, normalized size = 0.57 \begin {gather*} \frac {{\left (9 \, x^{3} + 38 \, x^{2} + 132 \, x - 144\right )} \sqrt {-2 \, x + 1}}{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)^(3/2),x, algorithm="fricas")

[Out]

(9*x^3 + 38*x^2 + 132*x - 144)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]
time = 8.63, size = 46, normalized size = 0.87 \begin {gather*} \frac {9 \left (1 - 2 x\right )^{\frac {5}{2}}}{8} - \frac {103 \left (1 - 2 x\right )^{\frac {3}{2}}}{8} + \frac {707 \sqrt {1 - 2 x}}{8} + \frac {539}{8 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9*(1 - 2*x)**(5/2)/8 - 103*(1 - 2*x)**(3/2)/8 + 707*sqrt(1 - 2*x)/8 + 539/(8*sqrt(1 - 2*x))

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Giac [A]
time = 1.31, size = 44, normalized size = 0.83 \begin {gather*} \frac {9}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {103}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {707}{8} \, \sqrt {-2 \, x + 1} + \frac {539}{8 \, \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)^(3/2),x, algorithm="giac")

[Out]

9/8*(2*x - 1)^2*sqrt(-2*x + 1) - 103/8*(-2*x + 1)^(3/2) + 707/8*sqrt(-2*x + 1) + 539/8/sqrt(-2*x + 1)

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Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {539}{8\,\sqrt {1-2\,x}}+\frac {707\,\sqrt {1-2\,x}}{8}-\frac {103\,{\left (1-2\,x\right )}^{3/2}}{8}+\frac {9\,{\left (1-2\,x\right )}^{5/2}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

539/(8*(1 - 2*x)^(1/2)) + (707*(1 - 2*x)^(1/2))/8 - (103*(1 - 2*x)^(3/2))/8 + (9*(1 - 2*x)^(5/2))/8

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