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

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

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Rubi [A]  time = 0.01, 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 = {43} \begin {gather*} \frac {11}{6 (1-2 x)^{3/2}}-\frac {5}{2 \sqrt {1-2 x}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} \frac {15 x-2}{3 (1-2 x)^{3/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {11-15 (1-2 x)}{6 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11 - 15*(1 - 2*x))/(6*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.32, size = 26, normalized size = 0.96 \begin {gather*} \frac {{\left (15 \, x - 2\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

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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giac [A]  time = 1.25, size = 21, normalized size = 0.78 \begin {gather*} -\frac {15 \, x - 2}{3 \, {\left (2 \, x - 1\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)^(5/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 15, normalized size = 0.56 \begin {gather*} \frac {15 x -2}{3 \left (-2 x +1\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.51, size = 14, normalized size = 0.52 \begin {gather*} \frac {15 \, x - 2}{3 \, {\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)^(5/2),x, algorithm="maxima")

[Out]

1/3*(15*x - 2)/(-2*x + 1)^(3/2)

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mupad [B]  time = 0.02, size = 13, normalized size = 0.48 \begin {gather*} \frac {5\,x-\frac {2}{3}}{{\left (1-2\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)/(1 - 2*x)^(5/2),x)

[Out]

(5*x - 2/3)/(1 - 2*x)^(3/2)

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sympy [B]  time = 0.59, size = 48, normalized size = 1.78 \begin {gather*} - \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 {gather*}

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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