3.2328 \(\int (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=143 \[ -\frac{3}{50} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{567 (5 x+3)^{3/2} (1-2 x)^{5/2}}{4000}-\frac{4123 \sqrt{5 x+3} (1-2 x)^{5/2}}{9600}+\frac{45353 \sqrt{5 x+3} (1-2 x)^{3/2}}{192000}+\frac{498883 \sqrt{5 x+3} \sqrt{1-2 x}}{640000}+\frac{5487713 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640000 \sqrt{10}} \]

[Out]

(498883*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/640000 + (45353*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/192000 - (4123*(1 - 2*x)^(
5/2)*Sqrt[3 + 5*x])/9600 - (567*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/4000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)
^(3/2))/50 + (5487713*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640000*Sqrt[10])

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Rubi [A]  time = 0.0397672, 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} \[ -\frac{3}{50} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{567 (5 x+3)^{3/2} (1-2 x)^{5/2}}{4000}-\frac{4123 \sqrt{5 x+3} (1-2 x)^{5/2}}{9600}+\frac{45353 \sqrt{5 x+3} (1-2 x)^{3/2}}{192000}+\frac{498883 \sqrt{5 x+3} \sqrt{1-2 x}}{640000}+\frac{5487713 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(498883*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/640000 + (45353*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/192000 - (4123*(1 - 2*x)^(
5/2)*Sqrt[3 + 5*x])/9600 - (567*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/4000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)
^(3/2))/50 + (5487713*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640000*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 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x} \, dx &=-\frac{3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}-\frac{1}{50} \int \left (-182-\frac{567 x}{2}\right ) (1-2 x)^{3/2} \sqrt{3+5 x} \, dx\\ &=-\frac{567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac{3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac{4123 \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx}{1600}\\ &=-\frac{4123 (1-2 x)^{5/2} \sqrt{3+5 x}}{9600}-\frac{567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac{3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac{45353 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{19200}\\ &=\frac{45353 (1-2 x)^{3/2} \sqrt{3+5 x}}{192000}-\frac{4123 (1-2 x)^{5/2} \sqrt{3+5 x}}{9600}-\frac{567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac{3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac{498883 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{128000}\\ &=\frac{498883 \sqrt{1-2 x} \sqrt{3+5 x}}{640000}+\frac{45353 (1-2 x)^{3/2} \sqrt{3+5 x}}{192000}-\frac{4123 (1-2 x)^{5/2} \sqrt{3+5 x}}{9600}-\frac{567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac{3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac{5487713 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1280000}\\ &=\frac{498883 \sqrt{1-2 x} \sqrt{3+5 x}}{640000}+\frac{45353 (1-2 x)^{3/2} \sqrt{3+5 x}}{192000}-\frac{4123 (1-2 x)^{5/2} \sqrt{3+5 x}}{9600}-\frac{567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac{3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac{5487713 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{640000 \sqrt{5}}\\ &=\frac{498883 \sqrt{1-2 x} \sqrt{3+5 x}}{640000}+\frac{45353 (1-2 x)^{3/2} \sqrt{3+5 x}}{192000}-\frac{4123 (1-2 x)^{5/2} \sqrt{3+5 x}}{9600}-\frac{567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac{3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac{5487713 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{640000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0463192, size = 70, normalized size = 0.49 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6912000 x^4+7286400 x^3-3141280 x^2-4872460 x+382101\right )-16463139 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{19200000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(382101 - 4872460*x - 3141280*x^2 + 7286400*x^3 + 6912000*x^4) - 16463139*Sqr
t[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/19200000

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Maple [A]  time = 0.01, size = 121, normalized size = 0.9 \begin{align*}{\frac{1}{38400000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-145728000\,{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}-7642020\,\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((1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(1/2),x)

[Out]

1/38400000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-138240000*x^4*(-10*x^2-x+3)^(1/2)-145728000*x^3*(-10*x^2-x+3)^(1/2)+6
2825600*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)-7642020*
(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.20805, size = 117, normalized size = 0.82 \begin{align*} \frac{9}{25} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{687}{2000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{2159}{24000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{45353}{32000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{5487713}{12800000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{45353}{640000} \, \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)*(2+3*x)^2*(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

9/25*(-10*x^2 - x + 3)^(3/2)*x^2 + 687/2000*(-10*x^2 - x + 3)^(3/2)*x - 2159/24000*(-10*x^2 - x + 3)^(3/2) + 4
5353/32000*sqrt(-10*x^2 - x + 3)*x - 5487713/12800000*sqrt(10)*arcsin(-20/11*x - 1/11) + 45353/640000*sqrt(-10
*x^2 - x + 3)

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Fricas [A]  time = 1.53896, size = 285, normalized size = 1.99 \begin{align*} -\frac{1}{1920000} \,{\left (6912000 \, x^{4} + 7286400 \, x^{3} - 3141280 \, x^{2} - 4872460 \, x + 382101\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{5487713}{12800000} \, \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)*(2+3*x)^2*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/1920000*(6912000*x^4 + 7286400*x^3 - 3141280*x^2 - 4872460*x + 382101)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 54877
13/12800000*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 = 50.9703, size = 490, normalized size = 3.43 \begin{align*} \frac{22 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{121} + \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{3125} + \frac{128 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (- \frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{1936} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{3125} + \frac{174 \sqrt{5} \left (\begin{cases} \frac{14641 \sqrt{2} \left (- \frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{3872} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{3125} - \frac{36 \sqrt{5} \left (\begin{cases} \frac{161051 \sqrt{2} \left (\frac{2 \sqrt{2} \left (5 - 10 x\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{805255} - \frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{7744} - \frac{3 \sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{3748096} + \frac{7 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{256}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{3125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

22*sqrt(5)*Piecewise((121*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/121 + asin(sqrt(22)*sqrt(
5*x + 3)/11))/32, (x >= -3/5) & (x < 1/2)))/3125 + 128*sqrt(5)*Piecewise((1331*sqrt(2)*(-sqrt(2)*(5 - 10*x)**(
3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 + asin(sqrt(22)*sqrt(5*x +
3)/11)/16)/8, (x >= -3/5) & (x < 1/2)))/3125 + 174*sqrt(5)*Piecewise((14641*sqrt(2)*(-sqrt(2)*(5 - 10*x)**(3/2
)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/3872 - sqrt(2)*sqrt(5 - 10*x)*sqrt(
5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 + 5*asin(sqrt(22)*sqrt(5*x + 3)/11)/
128)/16, (x >= -3/5) & (x < 1/2)))/3125 - 36*sqrt(5)*Piecewise((161051*sqrt(2)*(2*sqrt(2)*(5 - 10*x)**(5/2)*(5
*x + 3)**(5/2)/805255 - sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*s
qrt(5*x + 3)/7744 - 3*sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 -
5929)/3748096 + 7*asin(sqrt(22)*sqrt(5*x + 3)/11)/256)/32, (x >= -3/5) & (x < 1/2)))/3125

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Giac [B]  time = 2.32101, size = 317, normalized size = 2.22 \begin{align*} -\frac{3}{32000000} \, \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{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{1}{6000} \, \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}{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((1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

-3/32000000*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))) - 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) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*
sqrt(5*x + 3))) + 1/6000*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))) + 1/100*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)))