3.2383 \(\int \frac{(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{2 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{4 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}+\frac{4}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-2*(1 - 2*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + (4*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*x]) + (4*Sqrt[2/5]*ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]])/25

________________________________________________________________________________________

Rubi [A]  time = 0.0142955, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {47, 54, 216} \[ -\frac{2 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{4 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}+\frac{4}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + (4*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*x]) + (4*Sqrt[2/5]*ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]])/25

Rule 47

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

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{3/2}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}-\frac{2}{5} \int \frac{\sqrt{1-2 x}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac{4 \sqrt{1-2 x}}{25 \sqrt{3+5 x}}+\frac{4}{25} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac{4 \sqrt{1-2 x}}{25 \sqrt{3+5 x}}+\frac{8 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{25 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{3/2}}{15 (3+5 x)^{3/2}}+\frac{4 \sqrt{1-2 x}}{25 \sqrt{3+5 x}}+\frac{4}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0874287, size = 70, normalized size = 0.95 \[ -\frac{2 \left (400 x^2-70 x+6 \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-65\right )}{375 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-65 - 70*x + 400*x^2 + 6*Sqrt[10 - 20*x]*(3 + 5*x)^(3/2)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]]))/(375*Sqrt[1 -
2*x]*(3 + 5*x)^(3/2))

________________________________________________________________________________________

Maple [F]  time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.6465, size = 126, normalized size = 1.7 \begin{align*} \frac{2}{125} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{15 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} - \frac{11 \, \sqrt{-10 \, x^{2} - x + 3}}{75 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{14 \, \sqrt{-10 \, x^{2} - x + 3}}{75 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/125*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/15*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 135*x + 27) -
11/75*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) + 14/75*sqrt(-10*x^2 - x + 3)/(5*x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.51329, size = 269, normalized size = 3.64 \begin{align*} -\frac{2 \,{\left (3 \, \sqrt{5} \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 5 \,{\left (40 \, x + 13\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{375 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/375*(3*sqrt(5)*sqrt(2)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) - 5*(40*x + 13)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

________________________________________________________________________________________

Sympy [C]  time = 4.88326, size = 204, normalized size = 2.76 \begin{align*} \begin{cases} \frac{16 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{375} - \frac{22 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1875 \left (x + \frac{3}{5}\right )} + \frac{2 \sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{125} + \frac{2 \sqrt{10} i \log{\left (x + \frac{3}{5} \right )}}{125} + \frac{4 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{125} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\\frac{16 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{375} - \frac{22 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1875 \left (x + \frac{3}{5}\right )} + \frac{2 \sqrt{10} i \log{\left (\frac{1}{x + \frac{3}{5}} \right )}}{125} - \frac{4 \sqrt{10} i \log{\left (\sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} + 1 \right )}}{125} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/375 - 22*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/(1875*(x +
3/5)) + 2*sqrt(10)*I*log(1/(x + 3/5))/125 + 2*sqrt(10)*I*log(x + 3/5)/125 + 4*sqrt(10)*asin(sqrt(110)*sqrt(x +
3/5)/11)/125, 11/(10*Abs(x + 3/5)) > 1), (16*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/375 - 22*sqrt(10)*I*sqrt(
1 - 11/(10*(x + 3/5)))/(1875*(x + 3/5)) + 2*sqrt(10)*I*log(1/(x + 3/5))/125 - 4*sqrt(10)*I*log(sqrt(1 - 11/(10
*(x + 3/5))) + 1)/125, True))

________________________________________________________________________________________

Giac [B]  time = 2.36109, size = 194, normalized size = 2.62 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{6000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{4}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{100 \, \sqrt{5 \, x + 3}} - \frac{{\left (\frac{15 \, \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}}}{375 \,{\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((1-2*x)^(3/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/6000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 4/125*sqrt(10)*arcsin(1/11*sqrt(22)*
sqrt(5*x + 3)) + 1/100*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/375*(15*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