3.2277 \(\int \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=116 \[ -\frac{3}{40} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{181}{480} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac{1991 (1-2 x)^{3/2} \sqrt{5 x+3}}{1280}+\frac{21901 \sqrt{1-2 x} \sqrt{5 x+3}}{12800}+\frac{240911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800 \sqrt{10}} \]

[Out]

(21901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800 - (1991*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1280 - (181*(1 - 2*x)^(3/2)*(
3 + 5*x)^(3/2))/480 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/40 + (240911*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1280
0*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0287285, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{3}{40} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{181}{480} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac{1991 (1-2 x)^{3/2} \sqrt{5 x+3}}{1280}+\frac{21901 \sqrt{1-2 x} \sqrt{5 x+3}}{12800}+\frac{240911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800 - (1991*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1280 - (181*(1 - 2*x)^(3/2)*(
3 + 5*x)^(3/2))/480 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/40 + (240911*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1280
0*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) (3+5 x)^{3/2} \, dx &=-\frac{3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{181}{80} \int \sqrt{1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac{181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{1991}{320} \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx\\ &=-\frac{1991 (1-2 x)^{3/2} \sqrt{3+5 x}}{1280}-\frac{181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{21901 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{2560}\\ &=\frac{21901 \sqrt{1-2 x} \sqrt{3+5 x}}{12800}-\frac{1991 (1-2 x)^{3/2} \sqrt{3+5 x}}{1280}-\frac{181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{240911 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{25600}\\ &=\frac{21901 \sqrt{1-2 x} \sqrt{3+5 x}}{12800}-\frac{1991 (1-2 x)^{3/2} \sqrt{3+5 x}}{1280}-\frac{181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{240911 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{12800 \sqrt{5}}\\ &=\frac{21901 \sqrt{1-2 x} \sqrt{3+5 x}}{12800}-\frac{1991 (1-2 x)^{3/2} \sqrt{3+5 x}}{1280}-\frac{181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac{240911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{12800 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0364117, size = 65, normalized size = 0.56 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (144000 x^3+245600 x^2+99380 x-63387\right )-722733 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{384000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-63387 + 99380*x + 245600*x^2 + 144000*x^3) - 722733*Sqrt[10]*ArcSin[Sqrt[5/1
1]*Sqrt[1 - 2*x]])/384000

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{768000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2880000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+4912000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+722733\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1987600\,x\sqrt{-10\,{x}^{2}-x+3}-1267740\,\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)*(3+5*x)^(3/2)*(1-2*x)^(1/2),x)

[Out]

1/768000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2880000*x^3*(-10*x^2-x+3)^(1/2)+4912000*x^2*(-10*x^2-x+3)^(1/2)+722733*1
0^(1/2)*arcsin(20/11*x+1/11)+1987600*x*(-10*x^2-x+3)^(1/2)-1267740*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.8064, size = 95, normalized size = 0.82 \begin{align*} -\frac{3}{8} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{289}{480} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{1991}{640} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{240911}{256000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1991}{12800} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*(-10*x^2 - x + 3)^(3/2)*x - 289/480*(-10*x^2 - x + 3)^(3/2) + 1991/640*sqrt(-10*x^2 - x + 3)*x - 240911/2
56000*sqrt(10)*arcsin(-20/11*x - 1/11) + 1991/12800*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.80256, size = 251, normalized size = 2.16 \begin{align*} \frac{1}{38400} \,{\left (144000 \, x^{3} + 245600 \, x^{2} + 99380 \, x - 63387\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{240911}{256000} \, \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)*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/38400*(144000*x^3 + 245600*x^2 + 99380*x - 63387)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 240911/256000*sqrt(10)*arct
an(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

________________________________________________________________________________________

Sympy [A]  time = 29.4917, size = 314, normalized size = 2.71 \begin{align*} - \frac{77 \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 )}{8} + \frac{17 \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 )}{2} - \frac{15 \sqrt{2} \left (\begin{cases} \frac{14641 \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 )}{3872} - \frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{128}\right )}{625} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-77*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)))/8 + 17*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)
/11)/16)/125, (x <= 1/2) & (x > -3/5)))/2 - 15*sqrt(2)*Piecewise((14641*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)/3872 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x
+ 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/1874048 + 5*asin(sqrt(55)*sqrt(1 - 2*x)/11)/128
)/625, (x <= 1/2) & (x > -3/5)))/8

________________________________________________________________________________________

Giac [A]  time = 2.04736, size = 220, normalized size = 1.9 \begin{align*} \frac{1}{128000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{19}{24000} \, \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{3}{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)*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

1/128000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4537
5*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 19/24000*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))) + 3/200*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)))