3.19.19 \(\int \sqrt {1-2 x} (3+5 x)^3 \, dx\) [1819]

Optimal. Leaf size=53 \[ -\frac {1331}{24} (1-2 x)^{3/2}+\frac {363}{8} (1-2 x)^{5/2}-\frac {825}{56} (1-2 x)^{7/2}+\frac {125}{72} (1-2 x)^{9/2} \]

[Out]

-1331/24*(1-2*x)^(3/2)+363/8*(1-2*x)^(5/2)-825/56*(1-2*x)^(7/2)+125/72*(1-2*x)^(9/2)

________________________________________________________________________________________

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}{72} (1-2 x)^{9/2}-\frac {825}{56} (1-2 x)^{7/2}+\frac {363}{8} (1-2 x)^{5/2}-\frac {1331}{24} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(3 + 5*x)^3,x]

[Out]

(-1331*(1 - 2*x)^(3/2))/24 + (363*(1 - 2*x)^(5/2))/8 - (825*(1 - 2*x)^(7/2))/56 + (125*(1 - 2*x)^(9/2))/72

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 \sqrt {1-2 x} (3+5 x)^3 \, dx &=\int \left (\frac {1331}{8} \sqrt {1-2 x}-\frac {1815}{8} (1-2 x)^{3/2}+\frac {825}{8} (1-2 x)^{5/2}-\frac {125}{8} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {1331}{24} (1-2 x)^{3/2}+\frac {363}{8} (1-2 x)^{5/2}-\frac {825}{56} (1-2 x)^{7/2}+\frac {125}{72} (1-2 x)^{9/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {1}{63} (1-2 x)^{3/2} \left (1454+2661 x+2400 x^2+875 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/63*((1 - 2*x)^(3/2)*(1454 + 2661*x + 2400*x^2 + 875*x^3))

________________________________________________________________________________________

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

method	result	size
gosper	\(-\frac {\left (875 x^{3}+2400 x^{2}+2661 x +1454\right ) \left (1-2 x \right )^{\frac {3}{2}}}{63}\)	\(25\)
trager	\(\left (\frac {250}{9} x^{4}+\frac {3925}{63} x^{3}+\frac {974}{21} x^{2}+\frac {247}{63} x -\frac {1454}{63}\right ) \sqrt {1-2 x}\)	\(29\)
risch	\(-\frac {\left (1750 x^{4}+3925 x^{3}+2922 x^{2}+247 x -1454\right ) \left (-1+2 x \right )}{63 \sqrt {1-2 x}}\)	\(35\)
derivativedivides	\(-\frac {1331 \left (1-2 x \right )^{\frac {3}{2}}}{24}+\frac {363 \left (1-2 x \right )^{\frac {5}{2}}}{8}-\frac {825 \left (1-2 x \right )^{\frac {7}{2}}}{56}+\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{72}\)	\(38\)
default	\(-\frac {1331 \left (1-2 x \right )^{\frac {3}{2}}}{24}+\frac {363 \left (1-2 x \right )^{\frac {5}{2}}}{8}-\frac {825 \left (1-2 x \right )^{\frac {7}{2}}}{56}+\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{72}\)	\(38\)
meijerg	\(\frac {9 \sqrt {\pi }-\frac {9 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{2}}{\sqrt {\pi }}-\frac {135 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{8 \sqrt {\pi }}+\frac {\frac {30 \sqrt {\pi }}{7}-\frac {15 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{28}}{\sqrt {\pi }}-\frac {125 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{32 \sqrt {\pi }}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1331/24*(1-2*x)^(3/2)+363/8*(1-2*x)^(5/2)-825/56*(1-2*x)^(7/2)+125/72*(1-2*x)^(9/2)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 37, normalized size = 0.70 \begin {gather*} \frac {125}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {825}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {363}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1331}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/72*(-2*x + 1)^(9/2) - 825/56*(-2*x + 1)^(7/2) + 363/8*(-2*x + 1)^(5/2) - 1331/24*(-2*x + 1)^(3/2)

________________________________________________________________________________________

Fricas [A]
time = 0.64, size = 29, normalized size = 0.55 \begin {gather*} \frac {1}{63} \, {\left (1750 \, x^{4} + 3925 \, x^{3} + 2922 \, x^{2} + 247 \, x - 1454\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/63*(1750*x^4 + 3925*x^3 + 2922*x^2 + 247*x - 1454)*sqrt(-2*x + 1)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.99, size = 235, normalized size = 4.43 \begin {gather*} \begin {cases} \frac {50 \sqrt {5} i \left (x + \frac {3}{5}\right )^{4} \sqrt {10 x - 5}}{9} - \frac {55 \sqrt {5} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{63} - \frac {121 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{105} - \frac {2662 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{1575} - \frac {29282 \sqrt {5} i \sqrt {10 x - 5}}{7875} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {50 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{4}}{9} - \frac {55 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{63} - \frac {121 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{105} - \frac {2662 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{1575} - \frac {29282 \sqrt {5} \sqrt {5 - 10 x}}{7875} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((50*sqrt(5)*I*(x + 3/5)**4*sqrt(10*x - 5)/9 - 55*sqrt(5)*I*(x + 3/5)**3*sqrt(10*x - 5)/63 - 121*sqrt
(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/105 - 2662*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/1575 - 29282*sqrt(5)*I*sqrt(10
*x - 5)/7875, Abs(x + 3/5) > 11/10), (50*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**4/9 - 55*sqrt(5)*sqrt(5 - 10*x)*(x
+ 3/5)**3/63 - 121*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/105 - 2662*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)/1575 - 2928
2*sqrt(5)*sqrt(5 - 10*x)/7875, True))

________________________________________________________________________________________

Giac [A]
time = 1.72, size = 58, normalized size = 1.09 \begin {gather*} \frac {125}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {825}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {363}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {1331}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/72*(2*x - 1)^4*sqrt(-2*x + 1) + 825/56*(2*x - 1)^3*sqrt(-2*x + 1) + 363/8*(2*x - 1)^2*sqrt(-2*x + 1) - 133
1/24*(-2*x + 1)^(3/2)

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {363\,{\left (1-2\,x\right )}^{5/2}}{8}-\frac {1331\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {825\,{\left (1-2\,x\right )}^{7/2}}{56}+\frac {125\,{\left (1-2\,x\right )}^{9/2}}{72} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)*(5*x + 3)^3,x)

[Out]

(363*(1 - 2*x)^(5/2))/8 - (1331*(1 - 2*x)^(3/2))/24 - (825*(1 - 2*x)^(7/2))/56 + (125*(1 - 2*x)^(9/2))/72

________________________________________________________________________________________