3.2566 \(\int \frac{1}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{2}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{20 \sqrt{1-2 x}}{121 \sqrt{5 x+3}} \]

[Out]

2/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (20*Sqrt[1 - 2*x])/(121*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.0058214, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{2}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{20 \sqrt{1-2 x}}{121 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (20*Sqrt[1 - 2*x])/(121*Sqrt[3 + 5*x])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac{2}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{10}{11} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{2}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{20 \sqrt{1-2 x}}{121 \sqrt{3+5 x}}\\ \end{align*}

Mathematica [A]  time = 0.005777, size = 27, normalized size = 0.6 \[ \frac{2 (20 x+1)}{121 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 + 20*x))/(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]  time = 0.003, size = 22, normalized size = 0.5 \begin{align*}{\frac{2+40\,x}{121}{\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/121*(1+20*x)/(3+5*x)^(1/2)/(1-2*x)^(1/2)

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Maxima [A]  time = 2.09259, size = 41, normalized size = 0.91 \begin{align*} \frac{40 \, x}{121 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2}{121 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

40/121*x/sqrt(-10*x^2 - x + 3) + 2/121/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.50946, size = 88, normalized size = 1.96 \begin{align*} -\frac{2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{121 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/121*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)

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Sympy [A]  time = 2.91594, size = 116, normalized size = 2.58 \begin{align*} \begin{cases} - \frac{40 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{1210 x - 605} + \frac{22 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1210 x - 605} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{40 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{1210 x - 605} + \frac{22 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1210 x - 605} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-40*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)/(1210*x - 605) + 22*sqrt(10)*sqrt(-1 + 11/(10*(
x + 3/5)))/(1210*x - 605), 11/(10*Abs(x + 3/5)) > 1), (-40*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)/(1
210*x - 605) + 22*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/(1210*x - 605), True))

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Giac [B]  time = 1.5405, size = 117, normalized size = 2.6 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{242 \, \sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{605 \,{\left (2 \, x - 1\right )}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{121 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/242*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4/605*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x +
5)/(2*x - 1) + 2/121*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))