\(\int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\) [2964]

Optimal result

Integrand size = 28, antiderivative size = 189 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {91}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]

[Out]

1/3*(2+3*x)^(3/2)*(3+5*x)^(5/2)/(1-2*x)^(3/2)-12101/60*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*3
3^(1/2)-91/15*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-137/33*(3+5*x)^(5/2)*(2+3*x)^(1/2
)/(1-2*x)^(1/2)-817/66*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-91*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 159, 164, 114, 120} \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {91}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {(3 x+2)^{3/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {137 \sqrt {3 x+2} (5 x+3)^{5/2}}{33 \sqrt {1-2 x}}-\frac {817}{66} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-91 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \]

[In]

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

[Out]

-91*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - (817*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/66 - (137*Sq
rt[2 + 3*x]*(3 + 5*x)^(5/2))/(33*Sqrt[1 - 2*x]) + ((2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (121
01*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/20 - (91*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/5

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))), x] - Dist[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 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 155

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

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+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {77}{2}+60 x\right )}{(1-2 x)^{3/2}} \, dx \\ & = -\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {8033}{2}-\frac {12255 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1}{495} \int \frac {\sqrt {3+5 x} \left (\frac {1053855}{4}+405405 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\int \frac {-\frac {34129755}{4}-\frac {53909955 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4455} \\ & = -91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1001}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {12101}{20} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {91}{5} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.60 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (957-2579 x+438 x^2+90 x^3\right )+12101 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-12465 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{60 (1-2 x)^{3/2}} \]

[In]

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

[Out]

-1/60*(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(957 - 2579*x + 438*x^2 + 90*x^3) + (12101*I)*Sqrt[33 - 66*x]*(-1 + 2*x)
*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (12465*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*EllipticF[I*ArcSinh[Sqrt[9
+ 15*x]], -2/33])/(1 - 2*x)^(3/2)

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.26

method	result	size
default	\(-\frac {\left (164538 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-169414 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-82269 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+84707 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+94500 x^{5}+579600 x^{4}-2087610 x^{3}-2241260 x^{2}+189630 x +401940\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}}{420 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\)	\(238\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-\frac {15 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4}-22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}+\frac {7661 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{210 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {12101 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{210 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {847 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{96 \left (x -\frac {1}{2}\right )^{2}}+\frac {-\frac {20735}{8} x^{2}-\frac {78793}{24} x -\frac {4147}{4}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\)	\(270\)

[In]

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

[Out]

-1/420*(164538*5^(1/2)*7^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))*x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)
^(1/2)-169414*5^(1/2)*7^(1/2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))*x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^
(1/2)-82269*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)
)+84707*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))+94
500*x^5+579600*x^4-2087610*x^3-2241260*x^2+189630*x+401940)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)/(-1+2*x)
^2/(15*x^2+19*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 = 98, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {900 \, {\left (90 \, x^{3} + 438 \, x^{2} - 2579 \, x + 957\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 411167 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 1089090 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{5400 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

[In]

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

[Out]

-1/5400*(900*(90*x^3 + 438*x^2 - 2579*x + 957)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 411167*sqrt(-30)*(
4*x^2 - 4*x + 1)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 1089090*sqrt(-30)*(4*x^2 - 4*x + 1)*w
eierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(4*x^2 - 4*x + 1
)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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