3.21.73 \(\int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx\) [2073]

Optimal. Leaf size=66 \[ \frac {3773}{16 \sqrt {1-2 x}}+\frac {3283}{8} \sqrt {1-2 x}-\frac {357}{4} (1-2 x)^{3/2}+\frac {621}{40} (1-2 x)^{5/2}-\frac {135}{112} (1-2 x)^{7/2} \]

[Out]

-357/4*(1-2*x)^(3/2)+621/40*(1-2*x)^(5/2)-135/112*(1-2*x)^(7/2)+3773/16/(1-2*x)^(1/2)+3283/8*(1-2*x)^(1/2)

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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}{112} (1-2 x)^{7/2}+\frac {621}{40} (1-2 x)^{5/2}-\frac {357}{4} (1-2 x)^{3/2}+\frac {3283}{8} \sqrt {1-2 x}+\frac {3773}{16 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3773/(16*Sqrt[1 - 2*x]) + (3283*Sqrt[1 - 2*x])/8 - (357*(1 - 2*x)^(3/2))/4 + (621*(1 - 2*x)^(5/2))/40 - (135*(
1 - 2*x)^(7/2))/112

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)^3 (3+5 x)}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {3773}{16 (1-2 x)^{3/2}}-\frac {3283}{8 \sqrt {1-2 x}}+\frac {1071}{4} \sqrt {1-2 x}-\frac {621}{8} (1-2 x)^{3/2}+\frac {135}{16} (1-2 x)^{5/2}\right ) \, dx\\ &=\frac {3773}{16 \sqrt {1-2 x}}+\frac {3283}{8} \sqrt {1-2 x}-\frac {357}{4} (1-2 x)^{3/2}+\frac {621}{40} (1-2 x)^{5/2}-\frac {135}{112} (1-2 x)^{7/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} \frac {19994-19154 x-6987 x^2-2997 x^3-675 x^4}{35 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(19994 - 19154*x - 6987*x^2 - 2997*x^3 - 675*x^4)/(35*Sqrt[1 - 2*x])

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Maple [A]
time = 0.12, size = 47, normalized size = 0.71

method	result	size
gosper	\(-\frac {675 x^{4}+2997 x^{3}+6987 x^{2}+19154 x -19994}{35 \sqrt {1-2 x}}\)	\(30\)
risch	\(-\frac {675 x^{4}+2997 x^{3}+6987 x^{2}+19154 x -19994}{35 \sqrt {1-2 x}}\)	\(30\)
trager	\(\frac {\left (675 x^{4}+2997 x^{3}+6987 x^{2}+19154 x -19994\right ) \sqrt {1-2 x}}{-35+70 x}\)	\(37\)
derivativedivides	\(-\frac {357 \left (1-2 x \right )^{\frac {3}{2}}}{4}+\frac {621 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {135 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {3773}{16 \sqrt {1-2 x}}+\frac {3283 \sqrt {1-2 x}}{8}\)	\(47\)
default	\(-\frac {357 \left (1-2 x \right )^{\frac {3}{2}}}{4}+\frac {621 \left (1-2 x \right )^{\frac {5}{2}}}{40}-\frac {135 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {3773}{16 \sqrt {1-2 x}}+\frac {3283 \sqrt {1-2 x}}{8}\)	\(47\)
meijerg	\(-\frac {24 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-148 \sqrt {\pi }+\frac {37 \sqrt {\pi }\, \left (-8 x +8\right )}{2 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {171 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{2 \sqrt {\pi }}+\frac {-\frac {702 \sqrt {\pi }}{5}+\frac {351 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{320 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {135 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-357/4*(1-2*x)^(3/2)+621/40*(1-2*x)^(5/2)-135/112*(1-2*x)^(7/2)+3773/16/(1-2*x)^(1/2)+3283/8*(1-2*x)^(1/2)

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Maxima [A]
time = 0.28, size = 46, normalized size = 0.70 \begin {gather*} -\frac {135}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {621}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {357}{4} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {3283}{8} \, \sqrt {-2 \, x + 1} + \frac {3773}{16 \, \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)^(3/2),x, algorithm="maxima")

[Out]

-135/112*(-2*x + 1)^(7/2) + 621/40*(-2*x + 1)^(5/2) - 357/4*(-2*x + 1)^(3/2) + 3283/8*sqrt(-2*x + 1) + 3773/16
/sqrt(-2*x + 1)

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Fricas [A]
time = 0.97, size = 36, normalized size = 0.55 \begin {gather*} \frac {{\left (675 \, x^{4} + 2997 \, x^{3} + 6987 \, x^{2} + 19154 \, x - 19994\right )} \sqrt {-2 \, x + 1}}{35 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(675*x^4 + 2997*x^3 + 6987*x^2 + 19154*x - 19994)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]
time = 13.43, size = 58, normalized size = 0.88 \begin {gather*} - \frac {135 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} + \frac {621 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} - \frac {357 \left (1 - 2 x\right )^{\frac {3}{2}}}{4} + \frac {3283 \sqrt {1 - 2 x}}{8} + \frac {3773}{16 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-135*(1 - 2*x)**(7/2)/112 + 621*(1 - 2*x)**(5/2)/40 - 357*(1 - 2*x)**(3/2)/4 + 3283*sqrt(1 - 2*x)/8 + 3773/(16
*sqrt(1 - 2*x))

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Giac [A]
time = 1.37, size = 60, normalized size = 0.91 \begin {gather*} \frac {135}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {621}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {357}{4} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {3283}{8} \, \sqrt {-2 \, x + 1} + \frac {3773}{16 \, \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)^(3/2),x, algorithm="giac")

[Out]

135/112*(2*x - 1)^3*sqrt(-2*x + 1) + 621/40*(2*x - 1)^2*sqrt(-2*x + 1) - 357/4*(-2*x + 1)^(3/2) + 3283/8*sqrt(
-2*x + 1) + 3773/16/sqrt(-2*x + 1)

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Mupad [B]
time = 0.03, size = 46, normalized size = 0.70 \begin {gather*} \frac {3773}{16\,\sqrt {1-2\,x}}+\frac {3283\,\sqrt {1-2\,x}}{8}-\frac {357\,{\left (1-2\,x\right )}^{3/2}}{4}+\frac {621\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {135\,{\left (1-2\,x\right )}^{7/2}}{112} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3773/(16*(1 - 2*x)^(1/2)) + (3283*(1 - 2*x)^(1/2))/8 - (357*(1 - 2*x)^(3/2))/4 + (621*(1 - 2*x)^(5/2))/40 - (1
35*(1 - 2*x)^(7/2))/112

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