3.2265 \(\int \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=94 \[ -\frac{1}{10} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{37}{80} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{407}{800} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{4477 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]

[Out]

(407*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/800 - (37*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/80 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/
2))/10 + (4477*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(800*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0229561, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{1}{10} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{37}{80} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{407}{800} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{4477 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(407*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/800 - (37*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/80 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/
2))/10 + (4477*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(800*Sqrt[10])

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x} \, dx &=-\frac{1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac{37}{20} \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx\\ &=-\frac{37}{80} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac{407}{160} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=\frac{407}{800} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{37}{80} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac{4477 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1600}\\ &=\frac{407}{800} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{37}{80} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac{4477 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{800 \sqrt{5}}\\ &=\frac{407}{800} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{37}{80} (1-2 x)^{3/2} \sqrt{3+5 x}-\frac{1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac{4477 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{800 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0422527, size = 60, normalized size = 0.64 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (800 x^2+820 x-203\right )-4477 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{8000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-203 + 820*x + 800*x^2) - 4477*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/800
0

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 87, normalized size = 0.9 \begin{align*}{\frac{1}{16000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 16000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4477\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +16400\,x\sqrt{-10\,{x}^{2}-x+3}-4060\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(16000*x^2*(-10*x^2-x+3)^(1/2)+4477*10^(1/2)*arcsin(20/11*x+1/11)+16400*x*
(-10*x^2-x+3)^(1/2)-4060*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 3.32366, size = 74, normalized size = 0.79 \begin{align*} -\frac{1}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{37}{40} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{4477}{16000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{37}{800} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(-10*x^2 - x + 3)^(3/2) + 37/40*sqrt(-10*x^2 - x + 3)*x - 4477/16000*sqrt(10)*arcsin(-20/11*x - 1/11) +
37/800*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.72897, size = 217, normalized size = 2.31 \begin{align*} \frac{1}{800} \,{\left (800 \, x^{2} + 820 \, x - 203\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{4477}{16000} \, \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((2+3*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/800*(800*x^2 + 820*x - 203)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 4477/16000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x +
1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

________________________________________________________________________________________

Sympy [A]  time = 92.618, size = 168, normalized size = 1.79 \begin{align*} - \frac{7 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (- \frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6} \left (20 x + 1\right )}{121} + \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}\right )}{200} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{4} + \frac{3 \sqrt{2} \left (\begin{cases} \frac{1331 \sqrt{5} \left (- \frac{5 \sqrt{5} \left (1 - 2 x\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} - \frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6} \left (20 x + 1\right )}{1936} + \frac{\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*sqrt(2)*Piecewise((121*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/121 + asin(sqrt(55)*sqrt(1
- 2*x)/11))/200, (x <= 1/2) & (x > -3/5)))/4 + 3*sqrt(2)*Piecewise((1331*sqrt(5)*(-5*sqrt(5)*(1 - 2*x)**(3/2)
*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/1936 + asin(sqrt(55)*sqrt(1 - 2*x)/1
1)/16)/125, (x <= 1/2) & (x > -3/5)))/4

________________________________________________________________________________________

Giac [A]  time = 2.26629, size = 135, normalized size = 1.44 \begin{align*} \frac{1}{8000} \, \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}{200} \, \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((2+3*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

1/8000*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*sqrt(
22)*sqrt(5*x + 3))) + 1/200*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt
(22)*sqrt(5*x + 3)))