\(\int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\) [1209]

Optimal result

Integrand size = 26, antiderivative size = 140 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {123745}{256} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {123745 \sqrt {1-2 x} (3+5 x)^{3/2}}{2112}-\frac {1183 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}-\frac {3}{8} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}+\frac {272239}{256} \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \] Output:

-123745/256*(1-2*x)^(1/2)*(3+5*x)^(1/2)-123745/2112*(1-2*x)^(1/2)*(3+5*x)^ 
(3/2)-1183/66*(3+5*x)^(5/2)/(1-2*x)^(1/2)-3/8*(1-2*x)^(1/2)*(3+5*x)^(5/2)+ 
49/66*(3+5*x)^(7/2)/(1-2*x)^(3/2)+272239/512*arcsin(1/11*22^(1/2)*(3+5*x)^ 
(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.59 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {1}{768} \left (-\frac {\sqrt {3+5 x} \left (617319-1713440 x+497868 x^2+146160 x^3+28800 x^4\right )}{(1-2 x)^{3/2}}-816717 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )\right ) \] Input:

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

Output:

(-((Sqrt[3 + 5*x]*(617319 - 1713440*x + 497868*x^2 + 146160*x^3 + 28800*x^ 
4))/(1 - 2*x)^(3/2)) - 816717*Sqrt[10]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[11] - S 
qrt[5 - 10*x])])/768
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {100, 27, 87, 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.

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

\(\Big \downarrow \) 100

\(\displaystyle \frac {49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {1}{66} \int \frac {(5 x+3)^{5/2} (594 x+2069)}{2 (1-2 x)^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}-\frac {1}{132} \int \frac {(5 x+3)^{5/2} (594 x+2069)}{(1-2 x)^{3/2}}dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {1}{132} \left (\frac {74247}{11} \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x}}dx-\frac {4732 (5 x+3)^{7/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{132} \left (\frac {74247}{11} \left (\frac {55}{12} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x}}dx-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\right )-\frac {4732 (5 x+3)^{7/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{132} \left (\frac {74247}{11} \left (\frac {55}{12} \left (\frac {33}{8} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\right )-\frac {4732 (5 x+3)^{7/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{132} \left (\frac {74247}{11} \left (\frac {55}{12} \left (\frac {33}{8} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\right )-\frac {4732 (5 x+3)^{7/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{132} \left (\frac {74247}{11} \left (\frac {55}{12} \left (\frac {33}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\right )-\frac {4732 (5 x+3)^{7/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{132} \left (\frac {74247}{11} \left (\frac {55}{12} \left (\frac {33}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\right )-\frac {4732 (5 x+3)^{7/2}}{11 \sqrt {1-2 x}}\right )+\frac {49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}\)

Input:

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

Output:

(49*(3 + 5*x)^(7/2))/(66*(1 - 2*x)^(3/2)) + ((-4732*(3 + 5*x)^(7/2))/(11*S 
qrt[1 - 2*x]) + (74247*(-1/6*(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)) + (55*(-1/4*( 
Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (33*(-1/2*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + 
 (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2*Sqrt[10])))/8))/12))/11)/132
 

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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

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 = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.05

method	result	size
default	\(\frac {\left (-115200 x^{4} \sqrt {-10 x^{2}-x +3}+3266868 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-584640 x^{3} \sqrt {-10 x^{2}-x +3}-3266868 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -1991472 x^{2} \sqrt {-10 x^{2}-x +3}+816717 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+6853760 x \sqrt {-10 x^{2}-x +3}-2469276 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{3072 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {-10 x^{2}-x +3}}\)	\(147\)

Input:

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

Output:

1/3072*(-115200*x^4*(-10*x^2-x+3)^(1/2)+3266868*10^(1/2)*arcsin(20/11*x+1/ 
11)*x^2-584640*x^3*(-10*x^2-x+3)^(1/2)-3266868*10^(1/2)*arcsin(20/11*x+1/1 
1)*x-1991472*x^2*(-10*x^2-x+3)^(1/2)+816717*10^(1/2)*arcsin(20/11*x+1/11)+ 
6853760*x*(-10*x^2-x+3)^(1/2)-2469276*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)/( 
1-2*x)^(3/2)/(-10*x^2-x+3)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {816717 \, \sqrt {\frac {5}{2}} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {\frac {5}{2}} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{10 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 2 \, {\left (28800 \, x^{4} + 146160 \, x^{3} + 497868 \, x^{2} - 1713440 \, x + 617319\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1536 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:

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

Output:

-1/1536*(816717*sqrt(5/2)*(4*x^2 - 4*x + 1)*arctan(1/10*sqrt(5/2)*(20*x + 
1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 2*(28800*x^4 + 146160* 
x^3 + 497868*x^2 - 1713440*x + 617319)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^ 
2 - 4*x + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (99) = 198\).

Time = 0.15 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.76 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {272239}{1024} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {49 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{8 \, {\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac {21 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{8 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{8 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {5445}{256} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2695 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{96 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {1155 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {165 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{64 \, {\left (2 \, x - 1\right )}} + \frac {29645 \, \sqrt {-10 \, x^{2} - x + 3}}{192 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {104335 \, \sqrt {-10 \, x^{2} - x + 3}}{96 \, {\left (2 \, x - 1\right )}} \] Input:

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

Output:

272239/1024*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 49/8*(-10*x^2 - x + 3 
)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x + 1) - 21/8*(-10*x^2 - x + 3)^(5/2 
)/(8*x^3 - 12*x^2 + 6*x - 1) - 3/8*(-10*x^2 - x + 3)^(5/2)/(4*x^2 - 4*x + 
1) - 5445/256*sqrt(-10*x^2 - x + 3) - 2695/96*(-10*x^2 - x + 3)^(3/2)/(8*x 
^3 - 12*x^2 + 6*x - 1) + 1155/32*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) 
 + 165/64*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 29645/192*sqrt(-10*x^2 - x + 
 3)/(4*x^2 - 4*x + 1) + 104335/96*sqrt(-10*x^2 - x + 3)/(2*x - 1)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {272239}{512} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (3 \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 107 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 24749 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 2722390 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 44919435 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{96000 \, {\left (2 \, x - 1\right )}^{2}} \] Input:

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

Output:

272239/512*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/96000*(4*(3*(1 
2*(8*sqrt(5)*(5*x + 3) + 107*sqrt(5))*(5*x + 3) + 24749*sqrt(5))*(5*x + 3) 
 - 2722390*sqrt(5))*(5*x + 3) + 44919435*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x 
 + 5)/(2*x - 1)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.84 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {-1633434 \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right ) x +816717 \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )+57600 \sqrt {5 x +3}\, x^{4}+292320 \sqrt {5 x +3}\, x^{3}+995736 \sqrt {5 x +3}\, x^{2}-3426880 \sqrt {5 x +3}\, x +1234638 \sqrt {5 x +3}}{1536 \sqrt {-2 x +1}\, \left (2 x -1\right )} \] Input:

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

Output:

( - 1633434*sqrt( - 2*x + 1)*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt 
(11))*x + 816717*sqrt( - 2*x + 1)*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5)) 
/sqrt(11)) + 57600*sqrt(5*x + 3)*x**4 + 292320*sqrt(5*x + 3)*x**3 + 995736 
*sqrt(5*x + 3)*x**2 - 3426880*sqrt(5*x + 3)*x + 1234638*sqrt(5*x + 3))/(15 
36*sqrt( - 2*x + 1)*(2*x - 1))