\(\int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\) [1236]

Optimal result

Integrand size = 26, antiderivative size = 161 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {117649}{1056 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1838265689 \sqrt {1-2 x}}{79860000 (3+5 x)^{3/2}}-\frac {134456}{1331 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {1649184923 \sqrt {1-2 x}}{43923000 \sqrt {3+5 x}}-\frac {108621 \sqrt {1-2 x} \sqrt {3+5 x}}{40000}-\frac {729 \sqrt {1-2 x} x \sqrt {3+5 x}}{2000}+\frac {753543 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8000 \sqrt {10}} \] Output:

117649/1056/(1-2*x)^(3/2)/(3+5*x)^(3/2)-1838265689/79860000*(1-2*x)^(1/2)/ 
(3+5*x)^(3/2)-134456/1331/(1-2*x)^(1/2)/(3+5*x)^(1/2)[phone]/43923000* 
(1-2*x)^(1/2)/(3+5*x)^(1/2)-108621/40000*(1-2*x)^(1/2)*(3+5*x)^(1/2)-729/2 
000*(1-2*x)^(1/2)*x*(3+5*x)^(1/2)+753543/80000*arcsin(1/11*22^(1/2)*(3+5*x 
)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.49 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {44437106459+19932058554 x-274128335769 x^2-252342435560 x^3+97980793020 x^4+12807946800 x^5}{351384000 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {753543 \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{8000 \sqrt {10}} \] Input:

Integrate[(2 + 3*x)^6/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]
 

Output:

-1/351384000*(44437106459 + 19932058554*x - 274128335769*x^2 - 25234243556 
0*x^3 + 97980793020*x^4 + 12807946800*x^5)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2 
)) - (753543*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(8000*Sqrt[10])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.19, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {109, 27, 167, 27, 167, 27, 167, 27, 164, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{33} \int \frac {3 (3 x+2)^4 (239 x+136)}{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{22} \int \frac {(3 x+2)^4 (239 x+136)}{(1-2 x)^{3/2} (5 x+3)^{5/2}}dx\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{22} \left (-\frac {1}{11} \int -\frac {(3 x+2)^3 (38547 x+21610)}{2 \sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{22} \left (\frac {1}{22} \int \frac {(3 x+2)^3 (38547 x+21610)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {2}{165} \int \frac {7 (3 x+2)^2 (571191 x+335248)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \int \frac {(3 x+2)^2 (571191 x+335248)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \left (\frac {2}{55} \int \frac {3 (3 x+2) (6581765 x+4039402)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {74666 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \left (\frac {3}{55} \int \frac {(3 x+2) (6581765 x+4039402)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {74666 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \left (\frac {3}{55} \left (\frac {1576089009}{160} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (78981180 x+190406711)\right )+\frac {74666 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \left (\frac {3}{55} \left (\frac {1576089009}{400} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (78981180 x+190406711)\right )+\frac {74666 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{22} \left (\frac {1}{22} \left (\frac {7}{165} \left (\frac {3}{55} \left (\frac {1576089009 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80 \sqrt {10}}-\frac {1}{80} \sqrt {1-2 x} \sqrt {5 x+3} (78981180 x+190406711)\right )+\frac {74666 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}\right )+\frac {15182 \sqrt {1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}\right )-\frac {511 (3 x+2)^4}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

Input:

Int[(2 + 3*x)^6/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]
 

Output:

(7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + ((-511*(2 + 3*x)^4) 
/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((15182*Sqrt[1 - 2*x]*(2 + 3*x)^3)/( 
165*(3 + 5*x)^(3/2)) + (7*((74666*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(55*Sqrt[3 + 
5*x]) + (3*(-1/80*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(190406711 + 78981180*x)) + 
 (1576089009*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(80*Sqrt[10])))/55))/165)/2 
2)/22
 

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 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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 167

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

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 = 0.26 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {3309786918900 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}-256158936000 x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-1959615860400 x^{4} \sqrt {-10 x^{2}-x +3}-1952774282151 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}-198587215134 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-398641171080 x \sqrt {-10 x^{2}-x +3}-888742129180 \sqrt {-10 x^{2}-x +3}}{7027680000 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(192\)

Input:

int((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/7027680000*(3309786918900*10^(1/2)*arcsin(20/11*x+1/11)*x^4-256158936000 
*x^5*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^3-19 
59615860400*x^4*(-10*x^2-x+3)^(1/2)-1952774282151*10^(1/2)*arcsin(20/11*x+ 
1/11)*x^2+5046848711200*x^3*(-10*x^2-x+3)^(1/2)-198587215134*10^(1/2)*arcs 
in(20/11*x+1/11)*x[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^( 
1/2)*arcsin(20/11*x+1/11)-398641171080*x*(-10*x^2-x+3)^(1/2)-888742129180* 
(-10*x^2-x+3)^(1/2))/(1-2*x)^(3/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.78 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {33097869189 \, \sqrt {10} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (12807946800 \, x^{5} + 97980793020 \, x^{4} - 252342435560 \, x^{3} - 274128335769 \, x^{2} + 19932058554 \, x + 44437106459\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7027680000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \] Input:

integrate((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

-1/7027680000*(33097869189*sqrt(10)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)* 
arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 
 3)) + 20*(12807946800*x^5 + 97980793020*x^4 - 252342435560*x^3 - 27412833 
5769*x^2 + 19932058554*x + 44437106459)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(100 
*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
 

Sympy [F]

\[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {\left (3 x + 2\right )^{6}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((2+3*x)**6/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)
 

Output:

Integral((3*x + 2)**6/((1 - 2*x)**(5/2)*(5*x + 3)**(5/2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.33 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {729 \, x^{5}}{20 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {111537 \, x^{4}}{400 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {251181}{234256000} \, x {\left (\frac {7220 \, x}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {361}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} - \frac {753543}{160000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {90676341}{117128000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {170985889 \, x}{7027680 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {766611 \, x^{2}}{1000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1005653687}{878460000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {416356591 \, x}{3630000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {496819753}{3630000 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:

integrate((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")
 

Output:

-729/20*x^5/(-10*x^2 - x + 3)^(3/2) - 111537/400*x^4/(-10*x^2 - x + 3)^(3/ 
2) + 251181/234256000*x*(7220*x/sqrt(-10*x^2 - x + 3) + 439230*x^2/(-10*x^ 
2 - x + 3)^(3/2) + 361/sqrt(-10*x^2 - x + 3) + 21901*x/(-10*x^2 - x + 3)^( 
3/2) - 87483/(-10*x^2 - x + 3)^(3/2)) - 753543/160000*sqrt(10)*arcsin(-20/ 
11*x - 1/11) + 90676341/117128000*sqrt(-10*x^2 - x + 3) - 170985889/702768 
0*x/sqrt(-10*x^2 - x + 3) + 766611/1000*x^2/(-10*x^2 - x + 3)^(3/2) + 1005 
653687/878460000/sqrt(-10*x^2 - x + 3) + 416356591/3630000*x/(-10*x^2 - x 
+ 3)^(3/2) - 496819753/3630000/(-10*x^2 - x + 3)^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.27 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{2196150000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {753543}{80000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {37 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{16637500 \, \sqrt {5 \, x + 3}} - \frac {{\left (4 \, {\left (32019867 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 93 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 110347010662 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1820310410259 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{219615000000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1221 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{137259375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \] Input:

integrate((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")
 

Output:

-1/2196150000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3 
/2) + 753543/80000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 37/16637 
500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/219615 
000000*(4*(32019867*(4*sqrt(5)*(5*x + 3) + 93*sqrt(5))*(5*x + 3) - 1103470 
10662*sqrt(5))*(5*x + 3) + 1820310410259*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x 
 + 5)/(2*x - 1)^2 + 1/137259375*sqrt(10)*(5*x + 3)^(3/2)*(1221*(sqrt(2)*sq 
rt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt 
(22))^3
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^6}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:

int((3*x + 2)^6/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)),x)
 

Output:

int((3*x + 2)^6/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {-330978691890 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right ) x^{2}-33097869189 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right ) x [phone] \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )[phone] x^{5}[phone] x^{4}-2523424355600 x^{3}-2741283357690 x^{2}[phone] x [phone]}{3513840000 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (10 x^{2}+x -3\right )} \] Input:

int((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)
 

Output:

( - 330978691890*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(10)*asin((sqrt( - 2*x 
 + 1)*sqrt(5))/sqrt(11))*x**2 - 33097869189*sqrt(5*x + 3)*sqrt( - 2*x + 1) 
*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))*x + 99293607567*sqrt(5 
*x + 3)*sqrt( - 2*x + 1)*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11) 
) + 128079468000*x**5 + 979807930200*x**4 - 2523424355600*x**3 - 274128335 
7690*x**2 + 199320585540*x + 444371064590)/(3513840000*sqrt(5*x + 3)*sqrt( 
 - 2*x + 1)*(10*x**2 + x - 3))