3.2400 \(\int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=187 \[ -\frac{3}{70} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{263 (5 x+3)^{5/2} (1-2 x)^{7/2}}{2800}-\frac{2287 (5 x+3)^{3/2} (1-2 x)^{7/2}}{8000}-\frac{75471 \sqrt{5 x+3} (1-2 x)^{7/2}}{128000}+\frac{276727 \sqrt{5 x+3} (1-2 x)^{5/2}}{1280000}+\frac{3043997 \sqrt{5 x+3} (1-2 x)^{3/2}}{5120000}+\frac{100451901 \sqrt{5 x+3} \sqrt{1-2 x}}{51200000}+\frac{1104970911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200000 \sqrt{10}} \]

[Out]

(100451901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200000 + (3043997*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/5120000 + (276727*(
1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/1280000 - (75471*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/128000 - (2287*(1 - 2*x)^(7/2)*(
3 + 5*x)^(3/2))/8000 - (263*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/2800 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(5/
2))/70 + (1104970911*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200000*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0621344, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{3}{70} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{263 (5 x+3)^{5/2} (1-2 x)^{7/2}}{2800}-\frac{2287 (5 x+3)^{3/2} (1-2 x)^{7/2}}{8000}-\frac{75471 \sqrt{5 x+3} (1-2 x)^{7/2}}{128000}+\frac{276727 \sqrt{5 x+3} (1-2 x)^{5/2}}{1280000}+\frac{3043997 \sqrt{5 x+3} (1-2 x)^{3/2}}{5120000}+\frac{100451901 \sqrt{5 x+3} \sqrt{1-2 x}}{51200000}+\frac{1104970911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(100451901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200000 + (3043997*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/5120000 + (276727*(
1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/1280000 - (75471*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/128000 - (2287*(1 - 2*x)^(7/2)*(
3 + 5*x)^(3/2))/8000 - (263*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/2800 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(5/
2))/70 + (1104970911*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200000*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)^{5/2} (2+3 x)^2 (3+5 x)^{3/2} \, dx &=-\frac{3}{70} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{5/2}-\frac{1}{70} \int \left (-256-\frac{789 x}{2}\right ) (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{263 (1-2 x)^{7/2} (3+5 x)^{5/2}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{5/2}+\frac{2287}{800} \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{2287 (1-2 x)^{7/2} (3+5 x)^{3/2}}{8000}-\frac{263 (1-2 x)^{7/2} (3+5 x)^{5/2}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{5/2}+\frac{75471 \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx}{16000}\\ &=-\frac{75471 (1-2 x)^{7/2} \sqrt{3+5 x}}{128000}-\frac{2287 (1-2 x)^{7/2} (3+5 x)^{3/2}}{8000}-\frac{263 (1-2 x)^{7/2} (3+5 x)^{5/2}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{5/2}+\frac{830181 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{256000}\\ &=\frac{276727 (1-2 x)^{5/2} \sqrt{3+5 x}}{1280000}-\frac{75471 (1-2 x)^{7/2} \sqrt{3+5 x}}{128000}-\frac{2287 (1-2 x)^{7/2} (3+5 x)^{3/2}}{8000}-\frac{263 (1-2 x)^{7/2} (3+5 x)^{5/2}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{5/2}+\frac{3043997 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{512000}\\ &=\frac{3043997 (1-2 x)^{3/2} \sqrt{3+5 x}}{5120000}+\frac{276727 (1-2 x)^{5/2} \sqrt{3+5 x}}{1280000}-\frac{75471 (1-2 x)^{7/2} \sqrt{3+5 x}}{128000}-\frac{2287 (1-2 x)^{7/2} (3+5 x)^{3/2}}{8000}-\frac{263 (1-2 x)^{7/2} (3+5 x)^{5/2}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{5/2}+\frac{100451901 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{10240000}\\ &=\frac{100451901 \sqrt{1-2 x} \sqrt{3+5 x}}{51200000}+\frac{3043997 (1-2 x)^{3/2} \sqrt{3+5 x}}{5120000}+\frac{276727 (1-2 x)^{5/2} \sqrt{3+5 x}}{1280000}-\frac{75471 (1-2 x)^{7/2} \sqrt{3+5 x}}{128000}-\frac{2287 (1-2 x)^{7/2} (3+5 x)^{3/2}}{8000}-\frac{263 (1-2 x)^{7/2} (3+5 x)^{5/2}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{5/2}+\frac{1104970911 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{102400000}\\ &=\frac{100451901 \sqrt{1-2 x} \sqrt{3+5 x}}{51200000}+\frac{3043997 (1-2 x)^{3/2} \sqrt{3+5 x}}{5120000}+\frac{276727 (1-2 x)^{5/2} \sqrt{3+5 x}}{1280000}-\frac{75471 (1-2 x)^{7/2} \sqrt{3+5 x}}{128000}-\frac{2287 (1-2 x)^{7/2} (3+5 x)^{3/2}}{8000}-\frac{263 (1-2 x)^{7/2} (3+5 x)^{5/2}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{5/2}+\frac{1104970911 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{51200000 \sqrt{5}}\\ &=\frac{100451901 \sqrt{1-2 x} \sqrt{3+5 x}}{51200000}+\frac{3043997 (1-2 x)^{3/2} \sqrt{3+5 x}}{5120000}+\frac{276727 (1-2 x)^{5/2} \sqrt{3+5 x}}{1280000}-\frac{75471 (1-2 x)^{7/2} \sqrt{3+5 x}}{128000}-\frac{2287 (1-2 x)^{7/2} (3+5 x)^{3/2}}{8000}-\frac{263 (1-2 x)^{7/2} (3+5 x)^{5/2}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{5/2}+\frac{1104970911 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{51200000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0579676, size = 80, normalized size = 0.43 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (9216000000 x^6+10112000000 x^5-6123776000 x^4-8717155200 x^3+1291331040 x^2+2994263780 x-104420943\right )-7734796377 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3584000000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-104420943 + 2994263780*x + 1291331040*x^2 - 8717155200*x^3 - 6123776000*x^4
+ 10112000000*x^5 + 9216000000*x^6) - 7734796377*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/3584000000

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 155, normalized size = 0.8 \begin{align*}{\frac{1}{7168000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 184320000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-122475520000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-174343104000\,{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}-2088418860\,\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)^(5/2)*(2+3*x)^2*(3+5*x)^(3/2),x)

[Out]

1/7168000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(184320000000*(-10*x^2-x+3)^(1/2)*x^6+202240000000*x^5*(-10*x^2-x+3)^
(1/2)-122475520000*x^4*(-10*x^2-x+3)^(1/2)-174343104000*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)-2088418860*(-10*x^2-x+3)^(1/2
))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 3.92602, size = 157, normalized size = 0.84 \begin{align*} \frac{9}{35} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{323}{1400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{9141}{140000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{25157}{32000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{25157}{640000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{9131991}{2560000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1104970911}{1024000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{9131991}{51200000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/35*(-10*x^2 - x + 3)^(5/2)*x^2 + 323/1400*(-10*x^2 - x + 3)^(5/2)*x - 9141/140000*(-10*x^2 - x + 3)^(5/2) +
25157/32000*(-10*x^2 - x + 3)^(3/2)*x + 25157/640000*(-10*x^2 - x + 3)^(3/2) + 9131991/2560000*sqrt(-10*x^2 -
x + 3)*x - 1104970911/1024000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 9131991/51200000*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.51095, size = 360, normalized size = 1.93 \begin{align*} \frac{1}{358400000} \,{\left (9216000000 \, x^{6} + 10112000000 \, x^{5} - 6123776000 \, x^{4} - 8717155200 \, x^{3} + 1291331040 \, x^{2} + 2994263780 \, x - 104420943\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{1104970911}{1024000000} \, \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)^(5/2)*(2+3*x)^2*(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/358400000*(9216000000*x^6 + 10112000000*x^5 - 6123776000*x^4 - 8717155200*x^3 + 1291331040*x^2 + 2994263780*
x - 104420943)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1104970911/1024000000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*s
qrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

________________________________________________________________________________________

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)**(5/2)*(2+3*x)**2*(3+5*x)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.64674, size = 548, normalized size = 2.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/17920000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 359)*(5*x + 3) + 63769)*(5*x + 3) - 3968469)*(5*x + 3) + 33
617829)*(5*x + 3) - 276044685)*(5*x + 3) + 87356115)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 960917265*sqrt(2)*arcsin(
1/11*sqrt(22)*sqrt(5*x + 3))) + 7/320000000*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))) - 79/192000000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 13
6405)*(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) -
89/1920000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4
5375*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/3000*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)))