3.2338 \(\int (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=165 \[ -\frac{1}{20} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{259 (5 x+3)^{5/2} (1-2 x)^{5/2}}{2000}-\frac{3101 (5 x+3)^{3/2} (1-2 x)^{5/2}}{6400}-\frac{34111 \sqrt{5 x+3} (1-2 x)^{5/2}}{25600}+\frac{375221 \sqrt{5 x+3} (1-2 x)^{3/2}}{512000}+\frac{12382293 \sqrt{5 x+3} \sqrt{1-2 x}}{5120000}+\frac{136205223 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120000 \sqrt{10}} \]

[Out]

(12382293*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5120000 + (375221*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/512000 - (34111*(1 - 2
*x)^(5/2)*Sqrt[3 + 5*x])/25600 - (3101*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/6400 - (259*(1 - 2*x)^(5/2)*(3 + 5*x)^
(5/2))/2000 - ((1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(5/2))/20 + (136205223*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5
120000*Sqrt[10])

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Rubi [A]  time = 0.0510146, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{1}{20} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac{259 (5 x+3)^{5/2} (1-2 x)^{5/2}}{2000}-\frac{3101 (5 x+3)^{3/2} (1-2 x)^{5/2}}{6400}-\frac{34111 \sqrt{5 x+3} (1-2 x)^{5/2}}{25600}+\frac{375221 \sqrt{5 x+3} (1-2 x)^{3/2}}{512000}+\frac{12382293 \sqrt{5 x+3} \sqrt{1-2 x}}{5120000}+\frac{136205223 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12382293*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5120000 + (375221*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/512000 - (34111*(1 - 2
*x)^(5/2)*Sqrt[3 + 5*x])/25600 - (3101*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/6400 - (259*(1 - 2*x)^(5/2)*(3 + 5*x)^
(5/2))/2000 - ((1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(5/2))/20 + (136205223*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5
120000*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 (3+5 x)^{3/2} \, dx &=-\frac{1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}-\frac{1}{60} \int \left (-252-\frac{777 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac{3101}{800} \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac{259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac{102333 \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx}{12800}\\ &=-\frac{34111 (1-2 x)^{5/2} \sqrt{3+5 x}}{25600}-\frac{3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac{259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac{375221 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{51200}\\ &=\frac{375221 (1-2 x)^{3/2} \sqrt{3+5 x}}{512000}-\frac{34111 (1-2 x)^{5/2} \sqrt{3+5 x}}{25600}-\frac{3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac{259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac{12382293 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{1024000}\\ &=\frac{12382293 \sqrt{1-2 x} \sqrt{3+5 x}}{5120000}+\frac{375221 (1-2 x)^{3/2} \sqrt{3+5 x}}{512000}-\frac{34111 (1-2 x)^{5/2} \sqrt{3+5 x}}{25600}-\frac{3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac{259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac{136205223 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{10240000}\\ &=\frac{12382293 \sqrt{1-2 x} \sqrt{3+5 x}}{5120000}+\frac{375221 (1-2 x)^{3/2} \sqrt{3+5 x}}{512000}-\frac{34111 (1-2 x)^{5/2} \sqrt{3+5 x}}{25600}-\frac{3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac{259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac{136205223 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{5120000 \sqrt{5}}\\ &=\frac{12382293 \sqrt{1-2 x} \sqrt{3+5 x}}{5120000}+\frac{375221 (1-2 x)^{3/2} \sqrt{3+5 x}}{512000}-\frac{34111 (1-2 x)^{5/2} \sqrt{3+5 x}}{25600}-\frac{3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac{259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac{136205223 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{5120000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0515089, size = 75, normalized size = 0.45 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (76800000 x^5+132864000 x^4+27804800 x^3-66492960 x^2-37288220 x+8705457\right )-136205223 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{51200000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8705457 - 37288220*x - 66492960*x^2 + 27804800*x^3 + 132864000*x^4 + 7680000
0*x^5) - 136205223*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/51200000

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Maple [A]  time = 0.009, size = 138, normalized size = 0.8 \begin{align*}{\frac{1}{102400000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -1536000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-2657280000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-556096000\,{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}-174109140\,\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)^(3/2),x)

[Out]

1/102400000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-1536000000*x^5*(-10*x^2-x+3)^(1/2)-2657280000*x^4*(-10*x^2-x+3)^(1/2
)-556096000*x^3*(-10*x^2-x+3)^(1/2)[phone]*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)-174109140*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.64712, size = 134, normalized size = 0.81 \begin{align*} -\frac{3}{20} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{459}{2000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3101}{3200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{3101}{64000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{1125663}{256000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{136205223}{102400000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1125663}{5120000} \, \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)^(3/2),x, algorithm="maxima")

[Out]

-3/20*(-10*x^2 - x + 3)^(5/2)*x - 459/2000*(-10*x^2 - x + 3)^(5/2) + 3101/3200*(-10*x^2 - x + 3)^(3/2)*x + 310
1/64000*(-10*x^2 - x + 3)^(3/2) + 1125663/256000*sqrt(-10*x^2 - x + 3)*x - 136205223/102400000*sqrt(10)*arcsin
(-20/11*x - 1/11) + 1125663/5120000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.46585, size = 317, normalized size = 1.92 \begin{align*} -\frac{1}{5120000} \,{\left (76800000 \, x^{5} + 132864000 \, x^{4} + 27804800 \, x^{3} - 66492960 \, x^{2} - 37288220 \, x + 8705457\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{136205223}{102400000} \, \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)^(3/2),x, algorithm="fricas")

[Out]

-1/5120000*(76800000*x^5 + 132864000*x^4 + 27804800*x^3 - 66492960*x^2 - 37288220*x + 8705457)*sqrt(5*x + 3)*s
qrt(-2*x + 1) - 136205223/102400000*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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.22416, size = 427, normalized size = 2.59 \begin{align*} -\frac{3}{256000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{43}{64000000} \, \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}{76800} \, \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}{750} \, \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((1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-3/256000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27999)*(5*x + 3) - 318159)*(5*x + 3) + 3237255)
*(5*x + 3) - 2656665)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) -
43/64000000*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))) - 1/76800*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)*s
qrt(5*x + 3))) + 1/750*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*a
rcsin(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)))