3.2140 \(\int \frac{3+5 x}{(1-2 x)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.005132, 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} \[ \frac{11}{6 (1-2 x)^{3/2}}-\frac{5}{2 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/(1 - 2*x)^(5/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0068459, size = 18, normalized size = 0.67 \[ \frac{15 x-2}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2 + 15*x)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.002, size = 15, normalized size = 0.6 \begin{align*}{\frac{-2+15\,x}{3} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-2+15*x)/(1-2*x)^(3/2)

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Maxima [A]  time = 2.19352, size = 19, normalized size = 0.7 \begin{align*} \frac{15 \, x - 2}{3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(15*x - 2)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.5908, size = 66, normalized size = 2.44 \begin{align*} \frac{{\left (15 \, x - 2\right )} \sqrt{-2 \, x + 1}}{3 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(15*x - 2)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [B]  time = 0.568682, size = 48, normalized size = 1.78 \begin{align*} - \frac{15 x}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} + \frac{2}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15*x/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 2/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x))

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Giac [A]  time = 2.2528, size = 28, normalized size = 1.04 \begin{align*} -\frac{15 \, x - 2}{3 \,{\left (2 \, x - 1\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)^(5/2),x, algorithm="giac")

[Out]

-1/3*(15*x - 2)/((2*x - 1)*sqrt(-2*x + 1))