3.18.96 \(\int \sqrt {1-2 x} (3+5 x) \, dx\) [1796]

Optimal. Leaf size=27 \[ -\frac {11}{6} (1-2 x)^{3/2}+\frac {1}{2} (1-2 x)^{5/2} \]

[Out]

-11/6*(1-2*x)^(3/2)+1/2*(1-2*x)^(5/2)

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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} \frac {1}{2} (1-2 x)^{5/2}-\frac {11}{6} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(3 + 5*x),x]

[Out]

(-11*(1 - 2*x)^(3/2))/6 + (1 - 2*x)^(5/2)/2

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) \, dx &=\int \left (\frac {11}{2} \sqrt {1-2 x}-\frac {5}{2} (1-2 x)^{3/2}\right ) \, dx\\ &=-\frac {11}{6} (1-2 x)^{3/2}+\frac {1}{2} (1-2 x)^{5/2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} -\frac {1}{3} (1-2 x)^{3/2} (4+3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]*(3 + 5*x),x]

[Out]

-1/3*((1 - 2*x)^(3/2)*(4 + 3*x))

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Maple [A]
time = 0.08, size = 20, normalized size = 0.74

method	result	size
gosper	\(-\frac {\left (3 x +4\right ) \left (1-2 x \right )^{\frac {3}{2}}}{3}\)	\(15\)
trager	\(\left (2 x^{2}+\frac {5}{3} x -\frac {4}{3}\right ) \sqrt {1-2 x}\)	\(19\)
derivativedivides	\(-\frac {11 \left (1-2 x \right )^{\frac {3}{2}}}{6}+\frac {\left (1-2 x \right )^{\frac {5}{2}}}{2}\)	\(20\)
default	\(-\frac {11 \left (1-2 x \right )^{\frac {3}{2}}}{6}+\frac {\left (1-2 x \right )^{\frac {5}{2}}}{2}\)	\(20\)
risch	\(-\frac {\left (6 x^{2}+5 x -4\right ) \left (-1+2 x \right )}{3 \sqrt {1-2 x}}\)	\(25\)
meijerg	\(\frac {\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{2}}{\sqrt {\pi }}-\frac {5 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{8 \sqrt {\pi }}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/6*(1-2*x)^(3/2)+1/2*(1-2*x)^(5/2)

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Maxima [A]
time = 0.28, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{2} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {11}{6} \, {\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)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

1/2*(-2*x + 1)^(5/2) - 11/6*(-2*x + 1)^(3/2)

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Fricas [A]
time = 1.04, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{3} \, {\left (6 \, x^{2} + 5 \, x - 4\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(6*x^2 + 5*x - 4)*sqrt(-2*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 136, normalized size = 5.04 \begin {gather*} \begin {cases} \frac {2 \sqrt {5} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{5} - \frac {11 \sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{75} - \frac {121 \sqrt {5} i \sqrt {10 x - 5}}{375} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {2 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{5} - \frac {11 \sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{75} - \frac {121 \sqrt {5} \sqrt {5 - 10 x}}{375} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*sqrt(5)*I*(x + 3/5)**2*sqrt(10*x - 5)/5 - 11*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/75 - 121*sqrt(5)*
I*sqrt(10*x - 5)/375, Abs(x + 3/5) > 11/10), (2*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)**2/5 - 11*sqrt(5)*sqrt(5 - 10
*x)*(x + 3/5)/75 - 121*sqrt(5)*sqrt(5 - 10*x)/375, True))

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Giac [A]
time = 1.03, size = 26, normalized size = 0.96 \begin {gather*} \frac {1}{2} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {11}{6} \, {\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)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

1/2*(2*x - 1)^2*sqrt(-2*x + 1) - 11/6*(-2*x + 1)^(3/2)

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Mupad [B]
time = 0.02, size = 12, normalized size = 0.44 \begin {gather*} -{\left (1-2\,x\right )}^{3/2}\,\left (x+\frac {4}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)*(5*x + 3),x)

[Out]

-(1 - 2*x)^(3/2)*(x + 4/3)

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