\(\int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx\) [2960]

Optimal result

Integrand size = 28, antiderivative size = 191 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}+\frac {496 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}-\frac {582 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2401} \]

[Out]

-582/26411*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+496/7203*EllipticE(1/7*21^(1/2)*(1-2
*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+11/21*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2)+58/147*(3+5*x)^(1/2)/(2+3*
x)^(3/2)/(1-2*x)^(1/2)-89/343*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)-496/2401*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(
2+3*x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 157, 164, 114, 120} \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=-\frac {582 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2401}+\frac {496 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}-\frac {496 \sqrt {1-2 x} \sqrt {5 x+3}}{2401 \sqrt {3 x+2}}-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{343 (3 x+2)^{3/2}}+\frac {58 \sqrt {5 x+3}}{147 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}} \]

[In]

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

[Out]

(11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (58*Sqrt[3 + 5*x])/(147*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2
)) - (89*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(343*(2 + 3*x)^(3/2)) - (496*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2401*Sqrt[2 +
3*x]) + (496*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2401 - (582*Sqrt[3/11]*EllipticF[A
rcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2401

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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 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 157

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac {1}{21} \int \frac {-\frac {169}{2}-150 x}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx \\ & = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {2 \int \frac {\frac {16203}{4}+\frac {14355 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1617} \\ & = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}+\frac {4 \int \frac {\frac {10593}{2}+\frac {44055 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{33957} \\ & = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}+\frac {8 \int \frac {-\frac {60885}{8}-30690 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{237699} \\ & = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}-\frac {496 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2401}+\frac {873 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2401} \\ & = \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {58 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{343 (2+3 x)^{3/2}}-\frac {496 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 \sqrt {2+3 x}}+\frac {496 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401}-\frac {582 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.52 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=\frac {2 \left (-\frac {11 \sqrt {3+5 x} \left (-885-4616 x+762 x^2+8928 x^3\right )}{(1-2 x)^{3/2} (2+3 x)^{3/2}}-i \sqrt {33} \left (2728 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1855 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{79233} \]

[In]

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

[Out]

(2*((-11*Sqrt[3 + 5*x]*(-885 - 4616*x + 762*x^2 + 8928*x^3))/((1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) - I*Sqrt[33]*(2
728*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 1855*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/79233

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.19

method	result	size
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {23}{2646}+\frac {31 x}{2646}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right )^{2}}-\frac {2 \left (-18-30 x \right ) \left (-\frac {121}{43218}+\frac {248 x}{7203}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}-\frac {82 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{16807 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {992 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{50421 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(228\)
default	\(\frac {2 \sqrt {1-2 x}\, \left (954 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1488 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+159 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-248 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-318 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+496 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-44640 x^{4}-30594 x^{3}+20794 x^{2}+18273 x +2655\right )}{7203 \left (2+3 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\)	\(311\)

[In]

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

[Out]

(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)*((23/2646+31/2646*x)*(-30*x^3-23*x
^2+7*x+6)^(1/2)/(x^2+1/6*x-1/3)^2-2*(-18-30*x)*(-121/43218+248/7203*x)/((x^2+1/6*x-1/3)*(-18-30*x))^(1/2)-82/1
6807*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/
35*70^(1/2))-992/50421*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-7/6*Elli
pticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))))

Fricas [C] (verification not implemented)

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

Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx=-\frac {90 \, {\left (8928 \, x^{3} + 762 \, x^{2} - 4616 \, x - 885\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 169 \, \sqrt {-30} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 22320 \, \sqrt {-30} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{324135 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]

[In]

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

[Out]

-1/324135*(90*(8928*x^3 + 762*x^2 - 4616*x - 885)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 169*sqrt(-30)*(
36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 22320*sqrt(-30)*(3
6*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/
91125, x + 23/90)))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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