\(\int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\) [2926]

Optimal result

Integrand size = 28, antiderivative size = 191 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375}+\frac {175111 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{68750} \]

[Out]

2911577/378125*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+175111/756250*EllipticF(1/7*21^(
1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+7/11*(2+3*x)^(7/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)-37/605*(2+3*x)^(5/
2)*(1-2*x)^(1/2)/(3+5*x)^(1/2)+10851/15125*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+502941/151250*(1-2*x)^(1/
2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 191, 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 = {100, 155, 159, 164, 114, 120} \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {175111 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{68750}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375}+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {37 \sqrt {1-2 x} (3 x+2)^{5/2}}{605 \sqrt {5 x+3}}+\frac {10851 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{15125}+\frac {502941 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{151250} \]

[In]

Int[(2 + 3*x)^(9/2)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^(7/2))/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
+ (502941*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/151250 + (10851*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5
*x])/15125 + (2911577*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/34375 + (175111*Sqrt[3/11]
*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/68750

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 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 {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x)^{5/2} \left (\frac {367}{2}+312 x\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{605} \int \frac {(2+3 x)^{3/2} \left (\frac {13173}{4}+\frac {10851 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2 \int \frac {\left (-\frac {929925}{4}-\frac {1508823 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{15125} \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}-\frac {2 \int \frac {\frac {66357261}{8}+\frac {26204193 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{226875} \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}-\frac {525333 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{137500}-\frac {8734731 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{378125} \\ & = -\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2911577 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{34375}+\frac {175111 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{68750} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.37 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {-5 \sqrt {2+3 x} \sqrt {3+5 x} \left (-2892883-3684629 x+2188890 x^2+490050 x^3\right )-5823154 i \sqrt {33-66 x} (3+5 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+5998265 i \sqrt {33-66 x} (3+5 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{756250 \sqrt {1-2 x} (3+5 x)} \]

[In]

Integrate[(2 + 3*x)^(9/2)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]

[Out]

(-5*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-2892883 - 3684629*x + 2188890*x^2 + 490050*x^3) - (5823154*I)*Sqrt[33 - 66*x
]*(3 + 5*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (5998265*I)*Sqrt[33 - 66*x]*(3 + 5*x)*EllipticF[I*Ar
cSinh[Sqrt[9 + 15*x]], -2/33])/(756250*Sqrt[1 - 2*x]*(3 + 5*x))

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.76

method	result	size
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (5823154 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-5655507 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-7350750 x^{4}-37733850 x^{3}[phone] x^{2}[phone] x [phone]\right )}{756250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(145\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (\frac {900367}{1210000}+\frac {1500641 x}{1210000}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}+\frac {81 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{250}+\frac {3537 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2500}-\frac {7373029 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{5293750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {5823154 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2646875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(234\)

[In]

int((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/756250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5823154*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))-5655507*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))-7350750*x^4-37733850*x^3+33380535*x^2+80239535*x+28928830)/(3
0*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 = 92, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {450 \, {\left (490050 \, x^{3} + 2188890 \, x^{2} - 3684629 \, x - 2892883\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 197853763 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 524083860 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{68062500 \, {\left (10 \, x^{2} + x - 3\right )}} \]

[In]

integrate((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/68062500*(450*(490050*x^3 + 2188890*x^2 - 3684629*x - 2892883)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) +
197853763*sqrt(-30)*(10*x^2 + x - 3)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 524083860*sqrt(-3
0)*(10*x^2 + x - 3)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/9
0)))/(10*x^2 + x - 3)

Sympy [F(-1)]

Timed out. \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((2+3*x)**(9/2)/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)), x)

Giac [F]

\[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{9/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]

[In]

int((3*x + 2)^(9/2)/((1 - 2*x)^(3/2)*(5*x + 3)^(3/2)),x)

[Out]

int((3*x + 2)^(9/2)/((1 - 2*x)^(3/2)*(5*x + 3)^(3/2)), x)