\(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx\) [2777]

Optimal result

Integrand size = 28, antiderivative size = 218 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=-\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {146222113 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}}-\frac {1654421 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1913625 \sqrt {33}} \]

[Out]

-146222113/126299250*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1654421/63149625*EllipticF
(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+74/891*(1-2*x)^(3/2)*(3+5*x)^(5/2)*(2+3*x)^(1/2)+2/33*(1
-2*x)^(5/2)*(3+5*x)^(5/2)*(2+3*x)^(1/2)-146963/467775*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)+9698/93555*(3+
5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-1654421/4209975*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {103, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=-\frac {1654421 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1913625 \sqrt {33}}-\frac {146222113 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {9698 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{93555}-\frac {146963 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{467775}-\frac {1654421 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{4209975} \]

[In]

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

[Out]

(-1654421*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/4209975 - (146963*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(
3/2))/467775 + (9698*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/93555 + (74*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3
+ 5*x)^(5/2))/891 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/33 - (146222113*EllipticE[ArcSin[Sqrt[3
/7]*Sqrt[1 - 2*x]], 35/33])/(3827250*Sqrt[33]) - (1654421*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(
1913625*Sqrt[33])

Rule 103

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

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*Rt[-(b
*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /;
FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-(b*c - a*d)/d, 0] &&
!(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a
+ b*x] && GtQ[((-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[((-d)*e + c*f)/f,
0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f/b]))

Rule 159

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

Rule 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{33} \int \frac {\left (-50-\frac {185 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx \\ & = \frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-\frac {9245}{4}-\frac {24245 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx}{4455} \\ & = \frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (\frac {84395}{4}-\frac {2204445 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{467775} \\ & = -\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {44328915}{16}+\frac {24816315 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{7016625} \\ & = -\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {-\frac {685297455}{8}-\frac {2193331695 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{63149625} \\ & = -\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {1654421 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3827250}+\frac {146222113 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{42099750} \\ & = -\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {146222113 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}}-\frac {1654421 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1913625 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.50 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (3748468+9143865 x-16381350 x^2-12379500 x^3+25515000 x^4\right )[phone] i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-149530955 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{126299250} \]

[In]

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

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(3748468 + 9143865*x - 16381350*x^2 - 12379500*x^3 + 25515000*x^
4) + (146222113*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (149530955*I)*Sqrt[33]*EllipticF[I*A
rcSinh[Sqrt[9 + 15*x]], -2/33])/126299250

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.73

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-22963500000 x^{7}[phone] \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-146222113 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-6463800000 x^{6}[phone] x^{5}[phone] x^{4}-15351281550 x^{3}-3614874270 x^{2}[phone] x [phone]\right )}{126299250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(160\)
risch	\(-\frac {\left (25515000 x^{4}-12379500 x^{3}-16381350 x^{2}+9143865 x +3748468\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{4209975 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {45686497 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{231548625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {146222113 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{463097250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(262\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {203197 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{93555}+\frac {3748468 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4209975}+\frac {91372994 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{442047375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {146222113 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{442047375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {72806 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18711}+\frac {200 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{33}-\frac {2620 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{891}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(272\)

[In]

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

[Out]

-1/126299250*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(-22963500000*x^7+140986329*5^(1/2)*(2+3*x)^(1/2)*7^(1/
2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))-146222113*5^(1/2)*(2+3*x)^(1/2)*7^(1/
2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))-6463800000*x^6+28643220000*x^5+506665
8000*x^4-15351281550*x^3-3614874270*x^2+2433073980*x+674724240)/(30*x^3+23*x^2-7*x-6)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.32 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=\frac {1}{4209975} \, {\left (25515000 \, x^{4} - 12379500 \, x^{3} - 16381350 \, x^{2} + 9143865 \, x + 3748468\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {4860460861}{11366932500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {146222113}{126299250} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]

[In]

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

[Out]

1/4209975*(25515000*x^4 - 12379500*x^3 - 16381350*x^2 + 9143865*x + 3748468)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(
-2*x + 1) - 4860460861/11366932500*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 146222113
/126299250*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/
90))

Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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