3.2006 \(\int \frac{3+5 x}{\sqrt{1-2 x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0054607, antiderivative size = 27, normalized size of antiderivative = 1., 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 = {43} \[ \frac{5}{6} (1-2 x)^{3/2}-\frac{11}{2} \sqrt{1-2 x} \]

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 43

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*}

Mathematica [A]  time = 0.0051807, size = 18, normalized size = 0.67 \[ -\frac{1}{3} \sqrt{1-2 x} (5 x+14) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/Sqrt[1 - 2*x],x]

[Out]

-(Sqrt[1 - 2*x]*(14 + 5*x))/3

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Maple [A]  time = 0.002, size = 15, normalized size = 0.6 \begin{align*} -{\frac{14+5\,x}{3}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(14+5*x)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.5814, size = 26, normalized size = 0.96 \begin{align*} \frac{5}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{11}{2} \, \sqrt{-2 \, x + 1} \end{align*}

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 = 1.4103, size = 43, normalized size = 1.59 \begin{align*} -\frac{1}{3} \,{\left (5 \, x + 14\right )} \sqrt{-2 \, x + 1} \end{align*}

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 [A]  time = 0.860873, size = 88, normalized size = 3.26 \begin{align*} \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}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \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{align*}

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, 10*Abs(x + 3/5)/11 > 1), (-
sqrt(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 = 1.91708, size = 26, normalized size = 0.96 \begin{align*} \frac{5}{6} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{11}{2} \, \sqrt{-2 \, x + 1} \end{align*}

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)