3.2098 \(\int \frac{(3+5 x)^3}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{25}{8} (1-2 x)^{5/2}-\frac{275}{8} (1-2 x)^{3/2}+\frac{1815}{8} \sqrt{1-2 x}+\frac{1331}{8 \sqrt{1-2 x}} \]

[Out]

1331/(8*Sqrt[1 - 2*x]) + (1815*Sqrt[1 - 2*x])/8 - (275*(1 - 2*x)^(3/2))/8 + (25*(1 - 2*x)^(5/2))/8

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Rubi [A]  time = 0.0097423, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ \frac{25}{8} (1-2 x)^{5/2}-\frac{275}{8} (1-2 x)^{3/2}+\frac{1815}{8} \sqrt{1-2 x}+\frac{1331}{8 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/(1 - 2*x)^(3/2),x]

[Out]

1331/(8*Sqrt[1 - 2*x]) + (1815*Sqrt[1 - 2*x])/8 - (275*(1 - 2*x)^(3/2))/8 + (25*(1 - 2*x)^(5/2))/8

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)^3}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac{1331}{8 (1-2 x)^{3/2}}-\frac{1815}{8 \sqrt{1-2 x}}+\frac{825}{8} \sqrt{1-2 x}-\frac{125}{8} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac{1331}{8 \sqrt{1-2 x}}+\frac{1815}{8} \sqrt{1-2 x}-\frac{275}{8} (1-2 x)^{3/2}+\frac{25}{8} (1-2 x)^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0106631, size = 25, normalized size = 0.47 \[ \frac{-25 x^3-100 x^2-335 x+362}{\sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(362 - 335*x - 100*x^2 - 25*x^3)/Sqrt[1 - 2*x]

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Maple [A]  time = 0.004, size = 25, normalized size = 0.5 \begin{align*} -{(25\,{x}^{3}+100\,{x}^{2}+335\,x-362){\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(25*x^3+100*x^2+335*x-362)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.08185, size = 50, normalized size = 0.94 \begin{align*} \frac{25}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{275}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1815}{8} \, \sqrt{-2 \, x + 1} + \frac{1331}{8 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/8*(-2*x + 1)^(5/2) - 275/8*(-2*x + 1)^(3/2) + 1815/8*sqrt(-2*x + 1) + 1331/8/sqrt(-2*x + 1)

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Fricas [A]  time = 1.57746, size = 80, normalized size = 1.51 \begin{align*} \frac{{\left (25 \, x^{3} + 100 \, x^{2} + 335 \, x - 362\right )} \sqrt{-2 \, x + 1}}{2 \, x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(25*x^3 + 100*x^2 + 335*x - 362)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [B]  time = 1.89912, size = 435, normalized size = 8.21 \begin{align*} \begin{cases} \frac{125 \sqrt{55} i \left (x + \frac{3}{5}\right )^{3} \sqrt{10 x - 5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{275 \sqrt{55} i \left (x + \frac{3}{5}\right )^{2} \sqrt{10 x - 5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{1210 \sqrt{55} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} - \frac{26620 \sqrt{5} \left (x + \frac{3}{5}\right )}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} - \frac{2662 \sqrt{55} i \sqrt{10 x - 5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{29282 \sqrt{5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{125 \sqrt{55} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{3}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{275 \sqrt{55} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )^{2}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{1210 \sqrt{55} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} - \frac{2662 \sqrt{55} \sqrt{5 - 10 x}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} - \frac{26620 \sqrt{5} \left (x + \frac{3}{5}\right )}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} + \frac{29282 \sqrt{5}}{50 \sqrt{11} \left (x + \frac{3}{5}\right ) - 55 \sqrt{11}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((125*sqrt(55)*I*(x + 3/5)**3*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 275*sqrt(55)*I*(
x + 3/5)**2*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 1210*sqrt(55)*I*(x + 3/5)*sqrt(10*x - 5)/(5
0*sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 26620*sqrt(5)*(x + 3/5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 2662*sqr
t(55)*I*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 29282*sqrt(5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(
11)), 10*Abs(x + 3/5)/11 > 1), (125*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)**3/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11))
+ 275*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)**2/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 1210*sqrt(55)*sqrt(5 - 10*
x)*(x + 3/5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 2662*sqrt(55)*sqrt(5 - 10*x)/(50*sqrt(11)*(x + 3/5) - 55*
sqrt(11)) - 26620*sqrt(5)*(x + 3/5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 29282*sqrt(5)/(50*sqrt(11)*(x + 3/
5) - 55*sqrt(11)), True))

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Giac [A]  time = 1.6329, size = 59, normalized size = 1.11 \begin{align*} \frac{25}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{275}{8} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1815}{8} \, \sqrt{-2 \, x + 1} + \frac{1331}{8 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/8*(2*x - 1)^2*sqrt(-2*x + 1) - 275/8*(-2*x + 1)^(3/2) + 1815/8*sqrt(-2*x + 1) + 1331/8/sqrt(-2*x + 1)