\(\int (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx\) [931]

Optimal result

Integrand size = 26, antiderivative size = 138 \[ \int (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {498883 \sqrt {1-2 x} \sqrt {3+5 x}}{640000}+\frac {45353 (1-2 x)^{3/2} \sqrt {3+5 x}}{192000}-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {1407 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}+\frac {9}{100} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {5487713 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{640000 \sqrt {10}} \] Output:

498883/640000*(1-2*x)^(1/2)*(3+5*x)^(1/2)+45353/192000*(1-2*x)^(3/2)*(3+5* 
x)^(1/2)-4123/9600*(1-2*x)^(5/2)*(3+5*x)^(1/2)-1407/4000*(1-2*x)^(5/2)*(3+ 
5*x)^(3/2)+9/100*(1-2*x)^(7/2)*(3+5*x)^(3/2)+5487713/6400000*arcsin(1/11*2 
2^(1/2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.60 \[ \int (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {-10 \sqrt {1-2 x} \left (1146303-12706875 x-33786140 x^2+6152800 x^3+57168000 x^4+34560000 x^5\right )-16463139 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{19200000 \sqrt {3+5 x}} \] Input:

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

Output:

(-10*Sqrt[1 - 2*x]*(1146303 - 12706875*x - 33786140*x^2 + 6152800*x^3 + 57 
168000*x^4 + 34560000*x^5) - 16463139*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5* 
x]/Sqrt[3 + 5*x]])/(19200000*Sqrt[3 + 5*x])
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {101, 27, 90, 60, 60, 60, 64, 223}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 101

\(\displaystyle -\frac {1}{50} \int -\frac {7}{2} (1-2 x)^{3/2} \sqrt {5 x+3} (81 x+52)dx-\frac {3}{50} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7}{100} \int (1-2 x)^{3/2} \sqrt {5 x+3} (81 x+52)dx-\frac {3}{50} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{3/2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {7}{100} \left (\frac {589}{16} \int (1-2 x)^{3/2} \sqrt {5 x+3}dx-\frac {81}{40} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {7}{100} \left (\frac {589}{16} \left (\frac {11}{12} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {81}{40} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {7}{100} \left (\frac {589}{16} \left (\frac {11}{12} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {81}{40} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {7}{100} \left (\frac {589}{16} \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {81}{40} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{3/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {7}{100} \left (\frac {589}{16} \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {81}{40} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{3/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {7}{100} \left (\frac {589}{16} \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {81}{40} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {3}{50} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{3/2}\)

Input:

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

Output:

(-3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/50 + (7*((-81*(1 - 2*x)^(5/ 
2)*(3 + 5*x)^(3/2))/40 + (589*(-1/6*((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]) + (11* 
(((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (33*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 
 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/12))/16))/100
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 64

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp 
[2/b   Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] 
 /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] 
 || PosQ[b])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(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 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[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 223

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]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {\left (6912000 x^{4}+7286400 x^{3}-3141280 x^{2}-4872460 x +382101\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1920000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {5487713 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{12800000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(108\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-138240000 x^{4} \sqrt {-10 x^{2}-x +3}-145728000 x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-7642020 \sqrt {-10 x^{2}-x +3}\right )}{38400000 \sqrt {-10 x^{2}-x +3}}\)	\(121\)

Input:

int((1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/1920000*(6912000*x^4+7286400*x^3-3141280*x^2-4872460*x+382101)*(-1+2*x)* 
(3+5*x)^(1/2)/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1 
/2)+5487713/12800000*10^(1/2)*arcsin(20/11*x+1/11)*((1-2*x)*(3+5*x))^(1/2) 
/(1-2*x)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.56 \[ \int (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=-\frac {1}{1920000} \, {\left (6912000 \, x^{4} + 7286400 \, x^{3} - 3141280 \, x^{2} - 4872460 \, x + 382101\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {5487713}{12800000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \] Input:

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

Output:

-1/1920000*(6912000*x^4 + 7286400*x^3 - 3141280*x^2 - 4872460*x + 382101)* 
sqrt(5*x + 3)*sqrt(-2*x + 1) - 5487713/12800000*sqrt(10)*arctan(1/20*sqrt( 
10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

Sympy [F]

\[ \int (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\int \left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2} \sqrt {5 x + 3}\, dx \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.63 \[ \int (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {9}{25} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {687}{2000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {2159}{24000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {45353}{32000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {5487713}{12800000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {45353}{640000} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

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

Output:

9/25*(-10*x^2 - x + 3)^(3/2)*x^2 + 687/2000*(-10*x^2 - x + 3)^(3/2)*x - 21 
59/24000*(-10*x^2 - x + 3)^(3/2) + 45353/32000*sqrt(-10*x^2 - x + 3)*x - 5 
487713/12800000*sqrt(10)*arcsin(-20/11*x - 1/11) + 45353/640000*sqrt(-10*x 
^2 - x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (99) = 198\).

Time = 0.17 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=-\frac {3}{32000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {43}{3200000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{4800} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {2}{125} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {6}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \] Input:

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

Output:

-3/32000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) 
 - 506185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*s 
qrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 43/3200000*sqrt(5)*(2*(4*(8* 
(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x 
 + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/4800*sqrt( 
5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 478 
5*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 2/125*sqrt(5)*(2*(20*x - 
23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt( 
5*x + 3))) + 6/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) 
+ 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
 

Mupad [F(-1)]

Timed out. \[ \int (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2\,\sqrt {5\,x+3} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.72 \[ \int (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=-\frac {5487713 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{6400000}-\frac {18 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{5}-\frac {759 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{200}+\frac {19633 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{12000}+\frac {243623 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{96000}-\frac {127367 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{640000} \] Input:

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

Output:

( - 16463139*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) - 69120000 
*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 72864000*sqrt(5*x + 3)*sqrt( - 2*x 
+ 1)*x**3 + 31412800*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 48724600*sqrt(5 
*x + 3)*sqrt( - 2*x + 1)*x - 3821010*sqrt(5*x + 3)*sqrt( - 2*x + 1))/19200 
000