3.2276 \(\int \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=143 \[ -\frac{153}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}-\frac{9007 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9600}-\frac{99077 (1-2 x)^{3/2} \sqrt{5 x+3}}{25600}+\frac{1089847 \sqrt{1-2 x} \sqrt{5 x+3}}{256000}+\frac{11988317 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000 \sqrt{10}} \]

[Out]

(1089847*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256000 - (99077*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/25600 - (9007*(1 - 2*x)^(
3/2)*(3 + 5*x)^(3/2))/9600 - (153*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/800 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x
)^(5/2))/50 + (11988317*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0413263, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \[ -\frac{153}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}-\frac{9007 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9600}-\frac{99077 (1-2 x)^{3/2} \sqrt{5 x+3}}{25600}+\frac{1089847 \sqrt{1-2 x} \sqrt{5 x+3}}{256000}+\frac{11988317 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1089847*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256000 - (99077*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/25600 - (9007*(1 - 2*x)^(
3/2)*(3 + 5*x)^(3/2))/9600 - (153*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/800 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x
)^(5/2))/50 + (11988317*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000*Sqrt[10])

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

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)^2 (3+5 x)^{3/2} \, dx &=-\frac{3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}-\frac{1}{50} \int \left (-248-\frac{765 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac{153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac{9007 \int \sqrt{1-2 x} (3+5 x)^{3/2} \, dx}{1600}\\ &=-\frac{9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac{153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac{99077 \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx}{6400}\\ &=-\frac{99077 (1-2 x)^{3/2} \sqrt{3+5 x}}{25600}-\frac{9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac{153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac{1089847 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{51200}\\ &=\frac{1089847 \sqrt{1-2 x} \sqrt{3+5 x}}{256000}-\frac{99077 (1-2 x)^{3/2} \sqrt{3+5 x}}{25600}-\frac{9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac{153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac{11988317 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{512000}\\ &=\frac{1089847 \sqrt{1-2 x} \sqrt{3+5 x}}{256000}-\frac{99077 (1-2 x)^{3/2} \sqrt{3+5 x}}{25600}-\frac{9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac{153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac{11988317 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{256000 \sqrt{5}}\\ &=\frac{1089847 \sqrt{1-2 x} \sqrt{3+5 x}}{256000}-\frac{99077 (1-2 x)^{3/2} \sqrt{3+5 x}}{25600}-\frac{9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac{153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac{11988317 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{256000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0437319, size = 70, normalized size = 0.49 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6912000 x^4+16790400 x^3+13913120 x^2+2552540 x-4015809\right )-35964951 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{7680000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-4015809 + 2552540*x + 13913120*x^2 + 16790400*x^3 + 6912000*x^4) - 35964951*
Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/7680000

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 121, normalized size = 0.9 \begin{align*}{\frac{1}{15360000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-80316180\,\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)^2*(3+5*x)^(3/2)*(1-2*x)^(1/2),x)

[Out]

1/15360000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(138240000*x^4*(-10*x^2-x+3)^(1/2)[phone]*x^3*(-10*x^2-x+3)^(1/2)+27
8262400*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-80316180
*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.84255, size = 117, normalized size = 0.82 \begin{align*} -\frac{9}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{1677}{800} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{17971}{9600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{99077}{12800} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{11988317}{5120000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{99077}{256000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/10*(-10*x^2 - x + 3)^(3/2)*x^2 - 1677/800*(-10*x^2 - x + 3)^(3/2)*x - 17971/9600*(-10*x^2 - x + 3)^(3/2) +
99077/12800*sqrt(-10*x^2 - x + 3)*x - 11988317/5120000*sqrt(10)*arcsin(-20/11*x - 1/11) + 99077/256000*sqrt(-1
0*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.79392, size = 286, normalized size = 2. \begin{align*} \frac{1}{768000} \,{\left (6912000 \, x^{4} + 16790400 \, x^{3} + 13913120 \, x^{2} + 2552540 \, x - 4015809\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{11988317}{5120000} \, \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)^2*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/768000*(6912000*x^4 + 16790400*x^3 + 13913120*x^2 + 2552540*x - 4015809)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1198
8317/5120000*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 = 52.88, size = 488, normalized size = 3.41 \begin{align*} - \frac{539 \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 )}{16} + \frac{707 \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 )}{16} - \frac{309 \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 )}{16} + \frac{45 \sqrt{2} \left (\begin{cases} \frac{161051 \sqrt{5} \left (\frac{5 \sqrt{5} \left (1 - 2 x\right )^{\frac{5}{2}} \left (10 x + 6\right )^{\frac{5}{2}}}{322102} - \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 )}{7744} - \frac{3 \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 )}{3748096} + \frac{7 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{256}\right )}{3125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-539*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)))/16 + 707*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)))/16 - 309*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)))/16 + 45*sqrt(2)*Piecewise((161051*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(5/2)*(1
0*x + 6)**(5/2)/322102 - 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)/7744 - 3*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2
- 4719)/3748096 + 7*asin(sqrt(55)*sqrt(1 - 2*x)/11)/256)/3125, (x <= 1/2) & (x > -3/5)))/16

________________________________________________________________________________________

Giac [B]  time = 1.44205, size = 317, normalized size = 2.22 \begin{align*} \frac{3}{12800000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{29}{640000} \, \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{7}{3000} \, \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}{100} \, \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)^2*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

3/12800000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*(5*x + 3) + 60555)*sqrt(5*x
+ 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 29/640000*sqrt(5)*(2*(4*(8*(60*x
- 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*
sqrt(5*x + 3))) + 7/3000*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))) + 3/100*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)))