3.21.6 \(\int \frac {3+5 x}{\sqrt {1-2 x}} \, dx\) [2006]

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

[Out]

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

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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 {5}{6} (1-2 x)^{3/2}-\frac {11}{2} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/Sqrt[1 - 2*x],x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} -\frac {1}{3} \sqrt {1-2 x} (14+5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/Sqrt[1 - 2*x],x]

[Out]

-1/3*(Sqrt[1 - 2*x]*(14 + 5*x))

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Maple [A]
time = 0.09, size = 20, normalized size = 0.74

method	result	size
trager	\(\left (-\frac {14}{3}-\frac {5 x}{3}\right ) \sqrt {1-2 x}\)	\(14\)
gosper	\(-\frac {\left (14+5 x \right ) \sqrt {1-2 x}}{3}\)	\(15\)
derivativedivides	\(\frac {5 \left (1-2 x \right )^{\frac {3}{2}}}{6}-\frac {11 \sqrt {1-2 x}}{2}\)	\(20\)
default	\(\frac {5 \left (1-2 x \right )^{\frac {3}{2}}}{6}-\frac {11 \sqrt {1-2 x}}{2}\)	\(20\)
risch	\(\frac {\left (-1+2 x \right ) \left (14+5 x \right )}{3 \sqrt {1-2 x}}\)	\(20\)
meijerg	\(-\frac {3 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{2 \sqrt {\pi }}+\frac {\frac {5 \sqrt {\pi }}{3}-\frac {5 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{24}}{\sqrt {\pi }}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 19, normalized size = 0.70 \begin {gather*} \frac {5}{6} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {11}{2} \, \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="maxima")

[Out]

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

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Fricas [A]
time = 0.96, size = 14, normalized size = 0.52 \begin {gather*} -\frac {1}{3} \, {\left (5 \, x + 14\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*(5*x + 14)*sqrt(-2*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 87, normalized size = 3.22 \begin {gather*} \begin {cases} - \frac {\sqrt {5} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{3} - \frac {11 \sqrt {5} i \sqrt {10 x - 5}}{15} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\- \frac {\sqrt {5} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{3} - \frac {11 \sqrt {5} \sqrt {5 - 10 x}}{15} & \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((-sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/3 - 11*sqrt(5)*I*sqrt(10*x - 5)/15, Abs(x + 3/5) > 11/10), (-sq
rt(5)*sqrt(5 - 10*x)*(x + 3/5)/3 - 11*sqrt(5)*sqrt(5 - 10*x)/15, True))

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Giac [A]
time = 0.62, size = 19, normalized size = 0.70 \begin {gather*} \frac {5}{6} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {11}{2} \, \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="giac")

[Out]

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

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Mupad [B]
time = 0.02, size = 14, normalized size = 0.52 \begin {gather*} -\frac {\sqrt {1-2\,x}\,\left (10\,x+28\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)/(1 - 2*x)^(1/2),x)

[Out]

-((1 - 2*x)^(1/2)*(10*x + 28))/6

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