\(\int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx\) [974]

Optimal result

Integrand size = 26, antiderivative size = 138 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{5/2}}{1375 \sqrt {3+5 x}}+\frac {35511 \sqrt {1-2 x} \sqrt {3+5 x}}{160000}+\frac {11837 (1-2 x)^{3/2} \sqrt {3+5 x}}{176000}-\frac {207}{800} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {27}{400} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {390621 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{160000 \sqrt {10}} \] Output:

-2/1375*(1-2*x)^(5/2)/(3+5*x)^(1/2)+35511/160000*(1-2*x)^(1/2)*(3+5*x)^(1/ 
2)+11837/176000*(1-2*x)^(3/2)*(3+5*x)^(1/2)-207/800*(1-2*x)^(5/2)*(3+5*x)^ 
(1/2)+27/400*(1-2*x)^(7/2)*(3+5*x)^(1/2)+390621/1600000*arcsin(1/11*22^(1/ 
2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.62 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {-5 \sqrt {1-2 x} \left (-46783-317125 x-287460 x^2+439200 x^3+432000 x^4\right )-390621 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{800000 \sqrt {3+5 x}} \] Input:

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

Output:

(-5*Sqrt[1 - 2*x]*(-46783 - 317125*x - 287460*x^2 + 439200*x^3 + 432000*x^ 
4) - 390621*Sqrt[30 + 50*x]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10* 
x])])/(800000*Sqrt[3 + 5*x])
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {108, 27, 170, 27, 164, 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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {2}{5} \int \frac {3 (1-9 x) \sqrt {1-2 x} (3 x+2)^2}{\sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{5} \int \frac {(1-9 x) \sqrt {1-2 x} (3 x+2)^2}{\sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {6}{5} \left (\frac {9}{40} (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}-\frac {1}{40} \int -\frac {7 (10-3 x) \sqrt {1-2 x} (3 x+2)}{2 \sqrt {5 x+3}}dx\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{5} \left (\frac {7}{80} \int \frac {(10-3 x) \sqrt {1-2 x} (3 x+2)}{\sqrt {5 x+3}}dx+\frac {9}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {6}{5} \left (\frac {7}{80} \left (\frac {1691}{160} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx-\frac {3}{80} (35-8 x) (1-2 x)^{3/2} \sqrt {5 x+3}\right )+\frac {9}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {6}{5} \left (\frac {7}{80} \left (\frac {1691}{160} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{80} (35-8 x) (1-2 x)^{3/2} \sqrt {5 x+3}\right )+\frac {9}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {6}{5} \left (\frac {7}{80} \left (\frac {1691}{160} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{80} (35-8 x) (1-2 x)^{3/2} \sqrt {5 x+3}\right )+\frac {9}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {6}{5} \left (\frac {7}{80} \left (\frac {1691}{160} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {3}{80} (35-8 x) (1-2 x)^{3/2} \sqrt {5 x+3}\right )+\frac {9}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}\)

Input:

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

Output:

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) + (6*((9*(1 - 2*x)^(3/2 
)*(2 + 3*x)^2*Sqrt[3 + 5*x])/40 + (7*((-3*(35 - 8*x)*(1 - 2*x)^(3/2)*Sqrt[ 
3 + 5*x])/80 + (1691*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/ 
11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/160))/80))/5
 

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 108

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 
*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

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 = 0.23 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {\left (-8640000 x^{4} \sqrt {-10 x^{2}-x +3}-8784000 x^{3} \sqrt {-10 x^{2}-x +3}+1953105 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +5749200 x^{2} \sqrt {-10 x^{2}-x +3}+1171863 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+6342500 x \sqrt {-10 x^{2}-x +3}+935660 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{3200000 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\)	\(133\)

Input:

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

Output:

1/3200000*(-8640000*x^4*(-10*x^2-x+3)^(1/2)-8784000*x^3*(-10*x^2-x+3)^(1/2 
)+1953105*10^(1/2)*arcsin(20/11*x+1/11)*x+5749200*x^2*(-10*x^2-x+3)^(1/2)+ 
1171863*10^(1/2)*arcsin(20/11*x+1/11)+6342500*x*(-10*x^2-x+3)^(1/2)+935660 
*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=-\frac {390621 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (432000 \, x^{4} + 439200 \, x^{3} - 287460 \, x^{2} - 317125 \, x - 46783\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3200000 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

-1/3200000*(390621*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt 
(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(432000*x^4 + 439200*x^3 - 
 287460*x^2 - 317125*x - 46783)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.33 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {27}{500} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {35937}{1000000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {23}{11}\right ) + \frac {1378113}{16000000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {171}{10000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {297}{2500} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} x + \frac {9801}{40000} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {6831}{50000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} + \frac {28809}{800000} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1250 \, {\left (5 \, x + 3\right )}} - \frac {33 \, \sqrt {-10 \, x^{2} - x + 3}}{3125 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

27/500*(-10*x^2 - x + 3)^(3/2)*x - 35937/1000000*I*sqrt(5)*sqrt(2)*arcsin( 
20/11*x + 23/11) + 1378113/16000000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) 
 + 171/10000*(-10*x^2 - x + 3)^(3/2) + 297/2500*sqrt(10*x^2 + 23*x + 51/5) 
*x + 9801/40000*sqrt(-10*x^2 - x + 3)*x + 6831/50000*sqrt(10*x^2 + 23*x + 
51/5) + 28809/800000*sqrt(-10*x^2 - x + 3) + 1/625*(-10*x^2 - x + 3)^(3/2) 
/(25*x^2 + 30*x + 9) + 9/1250*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 33/3125* 
sqrt(-10*x^2 - x + 3)/(5*x + 3)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=-\frac {1}{4000000} \, {\left (36 \, {\left (8 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 83 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 805 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 128915 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {390621}{1600000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{31250 \, \sqrt {5 \, x + 3}} + \frac {22 \, \sqrt {10} \sqrt {5 \, x + 3}}{15625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \] Input:

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

Output:

-1/4000000*(36*(8*(12*sqrt(5)*(5*x + 3) - 83*sqrt(5))*(5*x + 3) - 805*sqrt 
(5))*(5*x + 3) + 128915*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 390621/16 
00000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 11/31250*sqrt(10)*(sq 
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 22/15625*sqrt(10)*sqrt(5 
*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.62 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx=\frac {-390621 \sqrt {5 x +3}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )-4320000 \sqrt {-2 x +1}\, x^{4}-4392000 \sqrt {-2 x +1}\, x^{3}+2874600 \sqrt {-2 x +1}\, x^{2}+3171250 \sqrt {-2 x +1}\, x +467830 \sqrt {-2 x +1}}{1600000 \sqrt {5 x +3}} \] Input:

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

Output:

( - 390621*sqrt(5*x + 3)*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11) 
) - 4320000*sqrt( - 2*x + 1)*x**4 - 4392000*sqrt( - 2*x + 1)*x**3 + 287460 
0*sqrt( - 2*x + 1)*x**2 + 3171250*sqrt( - 2*x + 1)*x + 467830*sqrt( - 2*x 
+ 1))/(1600000*sqrt(5*x + 3))