3.2374 \(\int \frac{(1-2 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{2 (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{6}{25} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{33}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-2*(1 - 2*x)^(3/2))/(5*Sqrt[3 + 5*x]) - (6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25 - (33*Sqrt[2/5]*ArcSin[Sqrt[2/11]*
Sqrt[3 + 5*x]])/25

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Rubi [A]  time = 0.0152444, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {47, 50, 54, 216} \[ -\frac{2 (1-2 x)^{3/2}}{5 \sqrt{5 x+3}}-\frac{6}{25} \sqrt{5 x+3} \sqrt{1-2 x}-\frac{33}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2))/(5*Sqrt[3 + 5*x]) - (6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25 - (33*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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{3/2}}{5 \sqrt{3+5 x}}-\frac{6}{5} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{5 \sqrt{3+5 x}}-\frac{6}{25} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{33}{25} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2}}{5 \sqrt{3+5 x}}-\frac{6}{25} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{66 \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}}{5 \sqrt{3+5 x}}-\frac{6}{25} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{33}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0093829, size = 39, normalized size = 0.53 \[ -\frac{2}{55} \sqrt{\frac{2}{11}} (1-2 x)^{5/2} \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{5}{11} (1-2 x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2/11]*(1 - 2*x)^(5/2)*Hypergeometric2F1[3/2, 5/2, 7/2, (5*(1 - 2*x))/11])/55

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.67958, size = 84, normalized size = 1.14 \begin{align*} -\frac{33}{250} \, \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}}}{5 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{33 \, \sqrt{-10 \, x^{2} - x + 3}}{25 \,{\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)^(3/2),x, algorithm="maxima")

[Out]

-33/250*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 1/5*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) - 33/25*sqrt(
-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 1.44403, size = 242, normalized size = 3.27 \begin{align*} \frac{33 \, \sqrt{5} \sqrt{2}{\left (5 \, x + 3\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 ) - 20 \,{\left (5 \, x + 14\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{250 \,{\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)^(3/2),x, algorithm="fricas")

[Out]

1/250*(33*sqrt(5)*sqrt(2)*(5*x + 3)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^
2 + x - 3)) - 20*(5*x + 14)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [A]  time = 2.73386, size = 187, normalized size = 2.53 \begin{align*} \begin{cases} - \frac{4 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{5 \sqrt{10 x - 5}} - \frac{22 i \sqrt{x + \frac{3}{5}}}{25 \sqrt{10 x - 5}} + \frac{33 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{125} + \frac{242 i}{125 \sqrt{x + \frac{3}{5}} \sqrt{10 x - 5}} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{33 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{125} + \frac{4 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{5 \sqrt{5 - 10 x}} + \frac{22 \sqrt{x + \frac{3}{5}}}{25 \sqrt{5 - 10 x}} - \frac{242}{125 \sqrt{5 - 10 x} \sqrt{x + \frac{3}{5}}} & \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)**(3/2),x)

[Out]

Piecewise((-4*I*(x + 3/5)**(3/2)/(5*sqrt(10*x - 5)) - 22*I*sqrt(x + 3/5)/(25*sqrt(10*x - 5)) + 33*sqrt(10)*I*a
cosh(sqrt(110)*sqrt(x + 3/5)/11)/125 + 242*I/(125*sqrt(x + 3/5)*sqrt(10*x - 5)), 10*Abs(x + 3/5)/11 > 1), (-33
*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/125 + 4*(x + 3/5)**(3/2)/(5*sqrt(5 - 10*x)) + 22*sqrt(x + 3/5)/(25*
sqrt(5 - 10*x)) - 242/(125*sqrt(5 - 10*x)*sqrt(x + 3/5)), True))

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Giac [A]  time = 2.56867, size = 132, normalized size = 1.78 \begin{align*} -\frac{2}{125} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{33}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{250 \, \sqrt{5 \, x + 3}} + \frac{22 \, \sqrt{10} \sqrt{5 \, x + 3}}{125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/125*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 33/125*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 11/250*sq
rt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 22/125*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))