3.2623 \(\int \frac{1}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{400 \sqrt{1-2 x}}{3993 \sqrt{5 x+3}}+\frac{40}{363 \sqrt{5 x+3} \sqrt{1-2 x}}+\frac{2}{33 \sqrt{5 x+3} (1-2 x)^{3/2}} \]

[Out]

2/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 40/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (400*Sqrt[1 - 2*x])/(3993*Sqrt[3
+ 5*x])

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Rubi [A]  time = 0.0099669, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{400 \sqrt{1-2 x}}{3993 \sqrt{5 x+3}}+\frac{40}{363 \sqrt{5 x+3} \sqrt{1-2 x}}+\frac{2}{33 \sqrt{5 x+3} (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + 40/(363*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (400*Sqrt[1 - 2*x])/(3993*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)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac{2}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{20}{33} \int \frac{1}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{2}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{40}{363 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{200}{363} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{2}{33 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{40}{363 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{400 \sqrt{1-2 x}}{3993 \sqrt{3+5 x}}\\ \end{align*}

Mathematica [A]  time = 0.0085787, size = 32, normalized size = 0.48 \[ \frac{-1600 x^2+720 x+282}{3993 (1-2 x)^{3/2} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(282 + 720*x - 1600*x^2)/(3993*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])

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Maple [A]  time = 0.003, size = 27, normalized size = 0.4 \begin{align*} -{\frac{1600\,{x}^{2}-720\,x-282}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3993*(800*x^2-360*x-141)/(3+5*x)^(1/2)/(1-2*x)^(3/2)

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Maxima [A]  time = 2.00066, size = 86, normalized size = 1.28 \begin{align*} \frac{800 \, x}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{40}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{33 \,{\left (2 \, \sqrt{-10 \, x^{2} - x + 3} x - \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

800/3993*x/sqrt(-10*x^2 - x + 3) + 40/3993/sqrt(-10*x^2 - x + 3) - 2/33/(2*sqrt(-10*x^2 - x + 3)*x - sqrt(-10*
x^2 - x + 3))

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Fricas [A]  time = 1.79217, size = 120, normalized size = 1.79 \begin{align*} -\frac{2 \,{\left (800 \, x^{2} - 360 \, x - 141\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3993 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3993*(800*x^2 - 360*x - 141)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]  time = 22.9073, size = 230, normalized size = 3.43 \begin{align*} \begin{cases} - \frac{8000 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{2}}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} + \frac{13200 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} - \frac{3630 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{8000 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{2}}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} + \frac{13200 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} - \frac{3630 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{- 878460 x + 399300 \left (x + \frac{3}{5}\right )^{2} - 43923} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-8000*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**2/(-878460*x + 399300*(x + 3/5)**2 - 43923)
+ 13200*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)/(-878460*x + 399300*(x + 3/5)**2 - 43923) - 3630*sqrt(
10)*sqrt(-1 + 11/(10*(x + 3/5)))/(-878460*x + 399300*(x + 3/5)**2 - 43923), 11/(10*Abs(x + 3/5)) > 1), (-8000*
sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**2/(-878460*x + 399300*(x + 3/5)**2 - 43923) + 13200*sqrt(10)
*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)/(-878460*x + 399300*(x + 3/5)**2 - 43923) - 3630*sqrt(10)*I*sqrt(1 -
11/(10*(x + 3/5)))/(-878460*x + 399300*(x + 3/5)**2 - 43923), True))

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Giac [B]  time = 1.23539, size = 135, normalized size = 2.01 \begin{align*} -\frac{5 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2662 \, \sqrt{5 \, x + 3}} - \frac{8 \,{\left (5 \, \sqrt{5}{\left (5 \, x + 3\right )} - 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{19965 \,{\left (2 \, x - 1\right )}^{2}} + \frac{10 \, \sqrt{10} \sqrt{5 \, x + 3}}{1331 \,{\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)^(5/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-5/2662*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 8/19965*(5*sqrt(5)*(5*x + 3) - 33*sqrt(5
))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 10/1331*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(
22))