3.2162 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{125}{24} (1-2 x)^{3/2}-\frac{825}{8} \sqrt{1-2 x}-\frac{1815}{8 \sqrt{1-2 x}}+\frac{1331}{24 (1-2 x)^{3/2}} \]

[Out]

1331/(24*(1 - 2*x)^(3/2)) - 1815/(8*Sqrt[1 - 2*x]) - (825*Sqrt[1 - 2*x])/8 + (125*(1 - 2*x)^(3/2))/24

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Rubi [A]  time = 0.0096535, 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} \[ \frac{125}{24} (1-2 x)^{3/2}-\frac{825}{8} \sqrt{1-2 x}-\frac{1815}{8 \sqrt{1-2 x}}+\frac{1331}{24 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/(1 - 2*x)^(5/2),x]

[Out]

1331/(24*(1 - 2*x)^(3/2)) - 1815/(8*Sqrt[1 - 2*x]) - (825*Sqrt[1 - 2*x])/8 + (125*(1 - 2*x)^(3/2))/24

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)^{5/2}} \, dx &=\int \left (\frac{1331}{8 (1-2 x)^{5/2}}-\frac{1815}{8 (1-2 x)^{3/2}}+\frac{825}{8 \sqrt{1-2 x}}-\frac{125}{8} \sqrt{1-2 x}\right ) \, dx\\ &=\frac{1331}{24 (1-2 x)^{3/2}}-\frac{1815}{8 \sqrt{1-2 x}}-\frac{825}{8} \sqrt{1-2 x}+\frac{125}{24} (1-2 x)^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0102994, size = 28, normalized size = 0.53 \[ -\frac{125 x^3+1050 x^2-2505 x+808}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(808 - 2505*x + 1050*x^2 + 125*x^3)/(3*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.003, size = 25, normalized size = 0.5 \begin{align*} -{\frac{125\,{x}^{3}+1050\,{x}^{2}-2505\,x+808}{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)^3/(1-2*x)^(5/2),x)

[Out]

-1/3*(125*x^3+1050*x^2-2505*x+808)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.45519, size = 45, normalized size = 0.85 \begin{align*} \frac{125}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{825}{8} \, \sqrt{-2 \, x + 1} + \frac{121 \,{\left (45 \, x - 17\right )}}{12 \,{\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)^3/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

125/24*(-2*x + 1)^(3/2) - 825/8*sqrt(-2*x + 1) + 121/12*(45*x - 17)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.53885, size = 101, normalized size = 1.91 \begin{align*} -\frac{{\left (125 \, x^{3} + 1050 \, x^{2} - 2505 \, x + 808\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)^3/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(125*x^3 + 1050*x^2 - 2505*x + 808)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [B]  time = 0.673959, size = 102, normalized size = 1.92 \begin{align*} \frac{125 x^{3}}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} + \frac{1050 x^{2}}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} - \frac{2505 x}{6 x \sqrt{1 - 2 x} - 3 \sqrt{1 - 2 x}} + \frac{808}{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)**3/(1-2*x)**(5/2),x)

[Out]

125*x**3/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 1050*x**2/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) - 2505*x/(6*x
*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 808/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x))

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Giac [A]  time = 2.26125, size = 54, normalized size = 1.02 \begin{align*} \frac{125}{24} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{825}{8} \, \sqrt{-2 \, x + 1} - \frac{121 \,{\left (45 \, x - 17\right )}}{12 \,{\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)^3/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

125/24*(-2*x + 1)^(3/2) - 825/8*sqrt(-2*x + 1) - 121/12*(45*x - 17)/((2*x - 1)*sqrt(-2*x + 1))