3.20.59 \(\int (1-2 x)^{5/2} (3+5 x)^3 \, dx\) [1959]

Optimal. Leaf size=53 \[ -\frac {1331}{56} (1-2 x)^{7/2}+\frac {605}{24} (1-2 x)^{9/2}-\frac {75}{8} (1-2 x)^{11/2}+\frac {125}{104} (1-2 x)^{13/2} \]

[Out]

-1331/56*(1-2*x)^(7/2)+605/24*(1-2*x)^(9/2)-75/8*(1-2*x)^(11/2)+125/104*(1-2*x)^(13/2)

________________________________________________________________________________________

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 = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \begin {gather*} \frac {125}{104} (1-2 x)^{13/2}-\frac {75}{8} (1-2 x)^{11/2}+\frac {605}{24} (1-2 x)^{9/2}-\frac {1331}{56} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)*(3 + 5*x)^3,x]

[Out]

(-1331*(1 - 2*x)^(7/2))/56 + (605*(1 - 2*x)^(9/2))/24 - (75*(1 - 2*x)^(11/2))/8 + (125*(1 - 2*x)^(13/2))/104

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (1-2 x)^{5/2} (3+5 x)^3 \, dx &=\int \left (\frac {1331}{8} (1-2 x)^{5/2}-\frac {1815}{8} (1-2 x)^{7/2}+\frac {825}{8} (1-2 x)^{9/2}-\frac {125}{8} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {1331}{56} (1-2 x)^{7/2}+\frac {605}{24} (1-2 x)^{9/2}-\frac {75}{8} (1-2 x)^{11/2}+\frac {125}{104} (1-2 x)^{13/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {1}{273} (1-2 x)^{7/2} \left (1838+5495 x+6300 x^2+2625 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/273*((1 - 2*x)^(7/2)*(1838 + 5495*x + 6300*x^2 + 2625*x^3))

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 38, normalized size = 0.72

method	result	size
gosper	\(-\frac {\left (2625 x^{3}+6300 x^{2}+5495 x +1838\right ) \left (1-2 x \right )^{\frac {7}{2}}}{273}\)	\(25\)
derivativedivides	\(-\frac {1331 \left (1-2 x \right )^{\frac {7}{2}}}{56}+\frac {605 \left (1-2 x \right )^{\frac {9}{2}}}{24}-\frac {75 \left (1-2 x \right )^{\frac {11}{2}}}{8}+\frac {125 \left (1-2 x \right )^{\frac {13}{2}}}{104}\)	\(38\)
default	\(-\frac {1331 \left (1-2 x \right )^{\frac {7}{2}}}{56}+\frac {605 \left (1-2 x \right )^{\frac {9}{2}}}{24}-\frac {75 \left (1-2 x \right )^{\frac {11}{2}}}{8}+\frac {125 \left (1-2 x \right )^{\frac {13}{2}}}{104}\)	\(38\)
trager	\(\left (\frac {1000}{13} x^{6}+\frac {900}{13} x^{5}-\frac {2270}{39} x^{4}-\frac {16061}{273} x^{3}+\frac {1538}{91} x^{2}+\frac {5533}{273} x -\frac {1838}{273}\right ) \sqrt {1-2 x}\)	\(39\)
risch	\(-\frac {\left (21000 x^{6}+18900 x^{5}-15890 x^{4}-16061 x^{3}+4614 x^{2}+5533 x -1838\right ) \left (-1+2 x \right )}{273 \sqrt {1-2 x}}\)	\(45\)
meijerg	\(\frac {\frac {27 \sqrt {\pi }}{7}-\frac {27 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{14}}{\sqrt {\pi }}-\frac {2025 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{32 \sqrt {\pi }}+\frac {\frac {50 \sqrt {\pi }}{77}-\frac {25 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{616}}{\sqrt {\pi }}-\frac {1875 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}\)	\(184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1331/56*(1-2*x)^(7/2)+605/24*(1-2*x)^(9/2)-75/8*(1-2*x)^(11/2)+125/104*(1-2*x)^(13/2)

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 37, normalized size = 0.70 \begin {gather*} \frac {125}{104} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {75}{8} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {605}{24} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {1331}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/104*(-2*x + 1)^(13/2) - 75/8*(-2*x + 1)^(11/2) + 605/24*(-2*x + 1)^(9/2) - 1331/56*(-2*x + 1)^(7/2)

________________________________________________________________________________________

Fricas [A]
time = 0.74, size = 39, normalized size = 0.74 \begin {gather*} \frac {1}{273} \, {\left (21000 \, x^{6} + 18900 \, x^{5} - 15890 \, x^{4} - 16061 \, x^{3} + 4614 \, x^{2} + 5533 \, x - 1838\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/273*(21000*x^6 + 18900*x^5 - 15890*x^4 - 16061*x^3 + 4614*x^2 + 5533*x - 1838)*sqrt(-2*x + 1)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (46) = 92\).
time = 0.28, size = 100, normalized size = 1.89 \begin {gather*} \frac {1000 x^{6} \sqrt {1 - 2 x}}{13} + \frac {900 x^{5} \sqrt {1 - 2 x}}{13} - \frac {2270 x^{4} \sqrt {1 - 2 x}}{39} - \frac {16061 x^{3} \sqrt {1 - 2 x}}{273} + \frac {1538 x^{2} \sqrt {1 - 2 x}}{91} + \frac {5533 x \sqrt {1 - 2 x}}{273} - \frac {1838 \sqrt {1 - 2 x}}{273} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1000*x**6*sqrt(1 - 2*x)/13 + 900*x**5*sqrt(1 - 2*x)/13 - 2270*x**4*sqrt(1 - 2*x)/39 - 16061*x**3*sqrt(1 - 2*x)
/273 + 1538*x**2*sqrt(1 - 2*x)/91 + 5533*x*sqrt(1 - 2*x)/273 - 1838*sqrt(1 - 2*x)/273

________________________________________________________________________________________

Giac [A]
time = 0.65, size = 65, normalized size = 1.23 \begin {gather*} \frac {125}{104} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {75}{8} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {605}{24} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {1331}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/104*(2*x - 1)^6*sqrt(-2*x + 1) + 75/8*(2*x - 1)^5*sqrt(-2*x + 1) + 605/24*(2*x - 1)^4*sqrt(-2*x + 1) + 133
1/56*(2*x - 1)^3*sqrt(-2*x + 1)

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 37, normalized size = 0.70 \begin {gather*} \frac {605\,{\left (1-2\,x\right )}^{9/2}}{24}-\frac {1331\,{\left (1-2\,x\right )}^{7/2}}{56}-\frac {75\,{\left (1-2\,x\right )}^{11/2}}{8}+\frac {125\,{\left (1-2\,x\right )}^{13/2}}{104} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)*(5*x + 3)^3,x)

[Out]

(605*(1 - 2*x)^(9/2))/24 - (1331*(1 - 2*x)^(7/2))/56 - (75*(1 - 2*x)^(11/2))/8 + (125*(1 - 2*x)^(13/2))/104

________________________________________________________________________________________