\(\int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\) [2930]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}+\frac {4}{11} \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2}{11} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]

[Out]

4/121*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/121*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2
),1/33*1155^(1/2))*33^(1/2)+2/11*(2+3*x)^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)-20/121*(1-2*x)^(1/2)*(2+3*x)^(1/2)/
(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {101, 157, 164, 114, 120} \[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {2}{11} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {4}{11} \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {20 \sqrt {1-2 x} \sqrt {3 x+2}}{121 \sqrt {5 x+3}}+\frac {2 \sqrt {3 x+2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}} \]

[In]

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

[Out]

(2*Sqrt[2 + 3*x])/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (20*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(121*Sqrt[3 + 5*x]) + (4
*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/11 - (2*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*S
qrt[1 - 2*x]], 35/33])/11

Rule 101

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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 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 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 {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {2}{11} \int \frac {-\frac {11}{2}-\frac {15 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}+\frac {4}{121} \int \frac {-\frac {3}{4}-15 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = \frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}-\frac {12}{121} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx+\frac {3}{11} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = \frac {2 \sqrt {2+3 x}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {20 \sqrt {1-2 x} \sqrt {2+3 x}}{121 \sqrt {3+5 x}}+\frac {4}{11} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2}{11} \sqrt {\frac {3}{11}} 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 = 5.55 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {2+3 x} \sqrt {3+5 x} (1+20 x)-4 i \sqrt {33-66 x} (3+5 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+2 i \sqrt {33-66 x} (3+5 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{121 \sqrt {1-2 x} (3+5 x)} \]

[In]

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

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1 + 20*x) - (4*I)*Sqrt[33 - 66*x]*(3 + 5*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]
], -2/33] + (2*I)*Sqrt[33 - 66*x]*(3 + 5*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(121*Sqrt[1 - 2*x]*(3
+ 5*x))

Maple [A] (verified)

Time = 1.43 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05

method	result	size
default	\(\frac {2 \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (33 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-70 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2100 x^{2}-1505 x -70\right )}{4235 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(135\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (\frac {1}{1210}+\frac {2 x}{121}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}-\frac {2 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{4235 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {8 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{847 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(195\)

[In]

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

[Out]

2/4235*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(33*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))-70*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*Ellip
ticE((10+15*x)^(1/2),1/35*70^(1/2))-2100*x^2-1505*x-70)/(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 = 82, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {90 \, {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 37 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 180 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{5445 \, {\left (10 \, x^{2} + x - 3\right )}} \]

[In]

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

[Out]

-1/5445*(90*(20*x + 1)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 37*sqrt(-30)*(10*x^2 + x - 3)*weierstrassP
Inverse(1159/675, 38998/91125, x + 23/90) + 180*sqrt(-30)*(10*x^2 + x - 3)*weierstrassZeta(1159/675, 38998/911
25, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(10*x^2 + x - 3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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