3.2633 \(\int \frac{1}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{1600 \sqrt{1-2 x}}{43923 \sqrt{5 x+3}}-\frac{400 \sqrt{1-2 x}}{3993 (5 x+3)^{3/2}}+\frac{20}{121 (5 x+3)^{3/2} \sqrt{1-2 x}}+\frac{2}{33 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

[Out]

2/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 20/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (400*Sqrt[1 - 2*x])/(3993*(3
+ 5*x)^(3/2)) - (1600*Sqrt[1 - 2*x])/(43923*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.0157057, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac{1600 \sqrt{1-2 x}}{43923 \sqrt{5 x+3}}-\frac{400 \sqrt{1-2 x}}{3993 (5 x+3)^{3/2}}+\frac{20}{121 (5 x+3)^{3/2} \sqrt{1-2 x}}+\frac{2}{33 (5 x+3)^{3/2} (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 20/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (400*Sqrt[1 - 2*x])/(3993*(3
+ 5*x)^(3/2)) - (1600*Sqrt[1 - 2*x])/(43923*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)^{5/2}} \, dx &=\frac{2}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{10}{11} \int \frac{1}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac{2}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{20}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{200}{121} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{2}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{20}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{400 \sqrt{1-2 x}}{3993 (3+5 x)^{3/2}}+\frac{800 \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{3993}\\ &=\frac{2}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{20}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{400 \sqrt{1-2 x}}{3993 (3+5 x)^{3/2}}-\frac{1600 \sqrt{1-2 x}}{43923 \sqrt{3+5 x}}\\ \end{align*}

Mathematica [A]  time = 0.0114287, size = 37, normalized size = 0.42 \[ \frac{-32000 x^3-4800 x^2+14280 x+722}{43923 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(722 + 14280*x - 4800*x^2 - 32000*x^3)/(43923*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.001, size = 32, normalized size = 0.4 \begin{align*} -{\frac{32000\,{x}^{3}+4800\,{x}^{2}-14280\,x-722}{43923} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/43923*(16000*x^3+2400*x^2-7140*x-361)/(3+5*x)^(3/2)/(1-2*x)^(3/2)

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Maxima [A]  time = 2.74581, size = 80, normalized size = 0.9 \begin{align*} \frac{3200 \, x}{43923 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{160}{43923 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{40 \, x}{363 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2}{363 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3200/43923*x/sqrt(-10*x^2 - x + 3) + 160/43923/sqrt(-10*x^2 - x + 3) + 40/363*x/(-10*x^2 - x + 3)^(3/2) + 2/36
3/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.74142, size = 155, normalized size = 1.74 \begin{align*} -\frac{2 \,{\left (16000 \, x^{3} + 2400 \, x^{2} - 7140 \, x - 361\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{43923 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/43923*(16000*x^3 + 2400*x^2 - 7140*x - 361)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x +
9)

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Sympy [B]  time = 48.752, size = 391, normalized size = 4.39 \begin{align*} \begin{cases} - \frac{32000 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{3}}{5314683 x + 4392300 \left (x + \frac{3}{5}\right )^{3} - 9663060 \left (x + \frac{3}{5}\right )^{2} + \frac{15944049}{5}} + \frac{52800 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{2}}{5314683 x + 4392300 \left (x + \frac{3}{5}\right )^{3} - 9663060 \left (x + \frac{3}{5}\right )^{2} + \frac{15944049}{5}} - \frac{14520 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{5314683 x + 4392300 \left (x + \frac{3}{5}\right )^{3} - 9663060 \left (x + \frac{3}{5}\right )^{2} + \frac{15944049}{5}} - \frac{2662 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{5314683 x + 4392300 \left (x + \frac{3}{5}\right )^{3} - 9663060 \left (x + \frac{3}{5}\right )^{2} + \frac{15944049}{5}} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{32000 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{3}}{5314683 x + 4392300 \left (x + \frac{3}{5}\right )^{3} - 9663060 \left (x + \frac{3}{5}\right )^{2} + \frac{15944049}{5}} + \frac{52800 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{2}}{5314683 x + 4392300 \left (x + \frac{3}{5}\right )^{3} - 9663060 \left (x + \frac{3}{5}\right )^{2} + \frac{15944049}{5}} - \frac{14520 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )}{5314683 x + 4392300 \left (x + \frac{3}{5}\right )^{3} - 9663060 \left (x + \frac{3}{5}\right )^{2} + \frac{15944049}{5}} - \frac{2662 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{5314683 x + 4392300 \left (x + \frac{3}{5}\right )^{3} - 9663060 \left (x + \frac{3}{5}\right )^{2} + \frac{15944049}{5}} & \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)**(5/2),x)

[Out]

Piecewise((-32000*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**3/(5314683*x + 4392300*(x + 3/5)**3 - 96630
60*(x + 3/5)**2 + 15944049/5) + 52800*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**2/(5314683*x + 4392300*
(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15944049/5) - 14520*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)/(531
4683*x + 4392300*(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15944049/5) - 2662*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5))
)/(5314683*x + 4392300*(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15944049/5), 11/(10*Abs(x + 3/5)) > 1), (-32000*s
qrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**3/(5314683*x + 4392300*(x + 3/5)**3 - 9663060*(x + 3/5)**2 +
15944049/5) + 52800*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**2/(5314683*x + 4392300*(x + 3/5)**3 - 96
63060*(x + 3/5)**2 + 15944049/5) - 14520*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)/(5314683*x + 4392300
*(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15944049/5) - 2662*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/(5314683*x +
4392300*(x + 3/5)**3 - 9663060*(x + 3/5)**2 + 15944049/5), True))

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Giac [B]  time = 2.55063, size = 223, normalized size = 2.51 \begin{align*} -\frac{5 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{702768 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{5 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{5324 \, \sqrt{5 \, x + 3}} - \frac{8 \,{\left (16 \, \sqrt{5}{\left (5 \, x + 3\right )} - 99 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{219615 \,{\left (2 \, x - 1\right )}^{2}} + \frac{5 \,{\left (\frac{33 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{43923 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/702768*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 5/5324*sqrt(10)*(sqrt(2)*sqrt(-10*
x + 5) - sqrt(22))/sqrt(5*x + 3) - 8/219615*(16*sqrt(5)*(5*x + 3) - 99*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/
(2*x - 1)^2 + 5/43923*(33*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3
/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3