3.2330 \(\int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=94 \[ -\frac{1}{6} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{11}{120} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{121}{400} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1331 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

[Out]

(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 + (11*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/120 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x
])/6 + (1331*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*Sqrt[10])

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Rubi [A]  time = 0.021234, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac{1}{6} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{11}{120} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{121}{400} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1331 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)*Sqrt[3 + 5*x],x]

[Out]

(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 + (11*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/120 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x
])/6 + (1331*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400*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 (1-2 x)^{3/2} \sqrt{3+5 x} \, dx &=-\frac{1}{6} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{11}{12} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{11}{120} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{1}{6} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{121}{80} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=\frac{121}{400} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11}{120} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{1}{6} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{1331}{800} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{121}{400} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11}{120} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{1}{6} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{1331 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{400 \sqrt{5}}\\ &=\frac{121}{400} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11}{120} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{1}{6} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{1331 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{400 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0538249, size = 69, normalized size = 0.73 \[ \frac{10 \sqrt{5 x+3} \left (1600 x^3-1960 x^2+34 x+273\right )-3993 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{12000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(273 + 34*x - 1960*x^2 + 1600*x^3) - 3993*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/
(12000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.004, size = 88, normalized size = 0.9 \begin{align*}{\frac{1}{15} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}}+{\frac{11}{100} \left ( 3+5\,x \right ) ^{{\frac{3}{2}}}\sqrt{1-2\,x}}-{\frac{121}{400}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{1331\,\sqrt{10}}{8000}\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]

1/15*(1-2*x)^(3/2)*(3+5*x)^(3/2)+11/100*(3+5*x)^(3/2)*(1-2*x)^(1/2)-121/400*(1-2*x)^(1/2)*(3+5*x)^(1/2)+1331/8
000*((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 = 3.71772, size = 74, normalized size = 0.79 \begin{align*} \frac{1}{15} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{11}{20} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1331}{8000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{11}{400} \, \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]

1/15*(-10*x^2 - x + 3)^(3/2) + 11/20*sqrt(-10*x^2 - x + 3)*x - 1331/8000*sqrt(10)*arcsin(-20/11*x - 1/11) + 11
/400*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.45619, size = 219, normalized size = 2.33 \begin{align*} -\frac{1}{1200} \,{\left (800 \, x^{2} - 580 \, x - 273\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{1331}{8000} \, \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]

-1/1200*(800*x^2 - 580*x - 273)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1331/8000*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 = 4.60366, size = 230, normalized size = 2.45 \begin{align*} \begin{cases} - \frac{20 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{3 \sqrt{10 x - 5}} + \frac{121 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{6 \sqrt{10 x - 5}} - \frac{2057 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{120 \sqrt{10 x - 5}} + \frac{1331 i \sqrt{x + \frac{3}{5}}}{400 \sqrt{10 x - 5}} - \frac{1331 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{4000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{1331 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{4000} + \frac{20 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{3 \sqrt{5 - 10 x}} - \frac{121 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{6 \sqrt{5 - 10 x}} + \frac{2057 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{120 \sqrt{5 - 10 x}} - \frac{1331 \sqrt{x + \frac{3}{5}}}{400 \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((-20*I*(x + 3/5)**(7/2)/(3*sqrt(10*x - 5)) + 121*I*(x + 3/5)**(5/2)/(6*sqrt(10*x - 5)) - 2057*I*(x +
3/5)**(3/2)/(120*sqrt(10*x - 5)) + 1331*I*sqrt(x + 3/5)/(400*sqrt(10*x - 5)) - 1331*sqrt(10)*I*acosh(sqrt(110
)*sqrt(x + 3/5)/11)/4000, 10*Abs(x + 3/5)/11 > 1), (1331*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/4000 + 20*(
x + 3/5)**(7/2)/(3*sqrt(5 - 10*x)) - 121*(x + 3/5)**(5/2)/(6*sqrt(5 - 10*x)) + 2057*(x + 3/5)**(3/2)/(120*sqrt
(5 - 10*x)) - 1331*sqrt(x + 3/5)/(400*sqrt(5 - 10*x)), True))

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Giac [A]  time = 2.45683, size = 135, normalized size = 1.44 \begin{align*} -\frac{1}{12000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{400} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, 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="giac")

[Out]

-1/12000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqr
t(22)*sqrt(5*x + 3))) + 1/400*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sq
rt(22)*sqrt(5*x + 3)))