3.22.37 \(\int \frac {(2+3 x)^3 (3+5 x)}{(1-2 x)^{5/2}} \, dx\) [2137]

Optimal. Leaf size=66 \[ \frac {3773}{48 (1-2 x)^{3/2}}-\frac {3283}{8 \sqrt {1-2 x}}-\frac {1071}{4} \sqrt {1-2 x}+\frac {207}{8} (1-2 x)^{3/2}-\frac {27}{16} (1-2 x)^{5/2} \]

[Out]

3773/48/(1-2*x)^(3/2)+207/8*(1-2*x)^(3/2)-27/16*(1-2*x)^(5/2)-3283/8/(1-2*x)^(1/2)-1071/4*(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 {27}{16} (1-2 x)^{5/2}+\frac {207}{8} (1-2 x)^{3/2}-\frac {1071}{4} \sqrt {1-2 x}-\frac {3283}{8 \sqrt {1-2 x}}+\frac {3773}{48 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3773/(48*(1 - 2*x)^(3/2)) - 3283/(8*Sqrt[1 - 2*x]) - (1071*Sqrt[1 - 2*x])/4 + (207*(1 - 2*x)^(3/2))/8 - (27*(1
- 2*x)^(5/2))/16

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)^{5/2}} \, dx &=\int \left (\frac {3773}{16 (1-2 x)^{5/2}}-\frac {3283}{8 (1-2 x)^{3/2}}+\frac {1071}{4 \sqrt {1-2 x}}-\frac {621}{8} \sqrt {1-2 x}+\frac {135}{16} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac {3773}{48 (1-2 x)^{3/2}}-\frac {3283}{8 \sqrt {1-2 x}}-\frac {1071}{4} \sqrt {1-2 x}+\frac {207}{8} (1-2 x)^{3/2}-\frac {27}{16} (1-2 x)^{5/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {1726-5250 x+2403 x^2+459 x^3+81 x^4}{3 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(1726 - 5250*x + 2403*x^2 + 459*x^3 + 81*x^4)/(1 - 2*x)^(3/2)

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Maple [A]
time = 0.11, size = 47, normalized size = 0.71

method	result	size
gosper	\(-\frac {81 x^{4}+459 x^{3}+2403 x^{2}-5250 x +1726}{3 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(30\)
trager	\(-\frac {\left (81 x^{4}+459 x^{3}+2403 x^{2}-5250 x +1726\right ) \sqrt {1-2 x}}{3 \left (-1+2 x \right )^{2}}\)	\(37\)
risch	\(\frac {81 x^{4}+459 x^{3}+2403 x^{2}-5250 x +1726}{3 \left (-1+2 x \right ) \sqrt {1-2 x}}\)	\(37\)
derivativedivides	\(\frac {3773}{48 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {207 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {3283}{8 \sqrt {1-2 x}}-\frac {1071 \sqrt {1-2 x}}{4}\)	\(47\)
default	\(\frac {3773}{48 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {207 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {27 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {3283}{8 \sqrt {1-2 x}}-\frac {1071 \sqrt {1-2 x}}{4}\)	\(47\)
meijerg	\(-\frac {16 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {\frac {148 \sqrt {\pi }}{3}-\frac {37 \sqrt {\pi }\, \left (-24 x +8\right )}{6 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {57 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {234 \sqrt {\pi }-\frac {117 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{64 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {45 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{8 \sqrt {\pi }}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3773/48/(1-2*x)^(3/2)+207/8*(1-2*x)^(3/2)-27/16*(1-2*x)^(5/2)-3283/8/(1-2*x)^(1/2)-1071/4*(1-2*x)^(1/2)

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Maxima [A]
time = 0.27, size = 42, normalized size = 0.64 \begin {gather*} -\frac {27}{16} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {207}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1071}{4} \, \sqrt {-2 \, x + 1} + \frac {49 \, {\left (804 \, x - 325\right )}}{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)^(5/2),x, algorithm="maxima")

[Out]

-27/16*(-2*x + 1)^(5/2) + 207/8*(-2*x + 1)^(3/2) - 1071/4*sqrt(-2*x + 1) + 49/48*(804*x - 325)/(-2*x + 1)^(3/2
)

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Fricas [A]
time = 1.11, size = 41, normalized size = 0.62 \begin {gather*} -\frac {{\left (81 \, x^{4} + 459 \, x^{3} + 2403 \, x^{2} - 5250 \, x + 1726\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (4 \, x^{2} - 4 \, 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)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(81*x^4 + 459*x^3 + 2403*x^2 - 5250*x + 1726)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]
time = 10.65, size = 58, normalized size = 0.88 \begin {gather*} - \frac {27 \left (1 - 2 x\right )^{\frac {5}{2}}}{16} + \frac {207 \left (1 - 2 x\right )^{\frac {3}{2}}}{8} - \frac {1071 \sqrt {1 - 2 x}}{4} - \frac {3283}{8 \sqrt {1 - 2 x}} + \frac {3773}{48 \left (1 - 2 x\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)**(5/2),x)

[Out]

-27*(1 - 2*x)**(5/2)/16 + 207*(1 - 2*x)**(3/2)/8 - 1071*sqrt(1 - 2*x)/4 - 3283/(8*sqrt(1 - 2*x)) + 3773/(48*(1
- 2*x)**(3/2))

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Giac [A]
time = 2.90, size = 56, normalized size = 0.85 \begin {gather*} -\frac {27}{16} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {207}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {1071}{4} \, \sqrt {-2 \, x + 1} - \frac {49 \, {\left (804 \, x - 325\right )}}{48 \, {\left (2 \, x - 1\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)^(5/2),x, algorithm="giac")

[Out]

-27/16*(2*x - 1)^2*sqrt(-2*x + 1) + 207/8*(-2*x + 1)^(3/2) - 1071/4*sqrt(-2*x + 1) - 49/48*(804*x - 325)/((2*x
- 1)*sqrt(-2*x + 1))

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Mupad [B]
time = 0.04, size = 41, normalized size = 0.62 \begin {gather*} \frac {\frac {3283\,x}{4}-\frac {15925}{48}}{{\left (1-2\,x\right )}^{3/2}}-\frac {1071\,\sqrt {1-2\,x}}{4}+\frac {207\,{\left (1-2\,x\right )}^{3/2}}{8}-\frac {27\,{\left (1-2\,x\right )}^{5/2}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3283*x)/4 - 15925/48)/(1 - 2*x)^(3/2) - (1071*(1 - 2*x)^(1/2))/4 + (207*(1 - 2*x)^(3/2))/8 - (27*(1 - 2*x)^(
5/2))/16

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