3.2364 \(\int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{10} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{33}{100} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{100 \sqrt{10}} \]

[Out]

(33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/100 + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (363*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(100*Sqrt[10])

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Rubi [A]  time = 0.0147747, antiderivative size = 72, 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 = {50, 54, 216} \[ \frac{1}{10} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{33}{100} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{100 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)/Sqrt[3 + 5*x],x]

[Out]

(33*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/100 + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (363*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]])/(100*Sqrt[10])

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}}{\sqrt{3+5 x}} \, dx &=\frac{1}{10} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{33}{20} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=\frac{33}{100} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{1}{10} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{363}{200} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{33}{100} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{1}{10} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{363 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{100 \sqrt{5}}\\ &=\frac{33}{100} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{1}{10} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{363 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{100 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0240494, size = 64, normalized size = 0.89 \[ \frac{10 \sqrt{5 x+3} \left (40 x^2-106 x+43\right )-363 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(43 - 106*x + 40*x^2) - 363*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1000*Sqrt[1 -
2*x])

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Maple [A]  time = 0.003, size = 72, normalized size = 1. \begin{align*}{\frac{1}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}\sqrt{3+5\,x}}+{\frac{33}{100}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{363\,\sqrt{10}}{2000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\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((1-2*x)^(3/2)/(3+5*x)^(1/2),x)

[Out]

1/10*(1-2*x)^(3/2)*(3+5*x)^(1/2)+33/100*(1-2*x)^(1/2)*(3+5*x)^(1/2)+363/2000*((1-2*x)*(3+5*x))^(1/2)/(3+5*x)^(
1/2)/(1-2*x)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

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Maxima [A]  time = 1.52133, size = 55, normalized size = 0.76 \begin{align*} -\frac{1}{5} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{363}{2000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{43}{100} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*sqrt(-10*x^2 - x + 3)*x - 363/2000*sqrt(10)*arcsin(-20/11*x - 1/11) + 43/100*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.48667, size = 200, normalized size = 2.78 \begin{align*} -\frac{1}{100} \,{\left (20 \, x - 43\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{363}{2000} \, \sqrt{10} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/100*(20*x - 43)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 363/2000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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Sympy [A]  time = 2.78965, size = 184, normalized size = 2.56 \begin{align*} \begin{cases} - \frac{2 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{\sqrt{10 x - 5}} + \frac{77 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{10 \sqrt{10 x - 5}} - \frac{121 i \sqrt{x + \frac{3}{5}}}{20 \sqrt{10 x - 5}} - \frac{363 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{1000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{363 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{1000} + \frac{2 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{\sqrt{5 - 10 x}} - \frac{77 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{10 \sqrt{5 - 10 x}} + \frac{121 \sqrt{x + \frac{3}{5}}}{20 \sqrt{5 - 10 x}} & \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)**(1/2),x)

[Out]

Piecewise((-2*I*(x + 3/5)**(5/2)/sqrt(10*x - 5) + 77*I*(x + 3/5)**(3/2)/(10*sqrt(10*x - 5)) - 121*I*sqrt(x + 3
/5)/(20*sqrt(10*x - 5)) - 363*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/1000, 10*Abs(x + 3/5)/11 > 1), (363
*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/1000 + 2*(x + 3/5)**(5/2)/sqrt(5 - 10*x) - 77*(x + 3/5)**(3/2)/(10*
sqrt(5 - 10*x)) + 121*sqrt(x + 3/5)/(20*sqrt(5 - 10*x)), True))

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Giac [A]  time = 2.08407, size = 116, normalized size = 1.61 \begin{align*} -\frac{1}{1000} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{50} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1000*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))
) + 1/50*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))