\(\int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\) [2870]

Optimal result

Integrand size = 28, antiderivative size = 81 \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {2 \sqrt {\frac {7}{5}} \sqrt {-3-5 x} E\left (\arcsin \left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right )}{11 \sqrt {3+5 x}} \]

[Out]

2/55*EllipticE(5^(1/2)*(2+3*x)^(1/2),1/35*70^(1/2))*35^(1/2)*(-3-5*x)^(1/2)/(3+5*x)^(1/2)-2/11*(1-2*x)^(1/2)*(
2+3*x)^(1/2)/(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {101, 21, 115, 114} \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {\frac {7}{5}} \sqrt {-5 x-3} E\left (\arcsin \left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right )}{11 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}} \]

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + (2*Sqrt[7/5]*Sqrt[-3 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqr
t[2 + 3*x]], 2/35])/(11*Sqrt[3 + 5*x])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x
]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d*x]*Sqrt[b*((e + f*x)/(b*e - a*f))])), Int[Sqrt[b*(e/(b*e - a*f)
) + b*f*(x/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]), x], 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {2}{11} \int \frac {\frac {3}{2}-3 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {3}{11} \int \frac {\sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {\left (3 \sqrt {7} \sqrt {-3-5 x}\right ) \int \frac {\sqrt {\frac {3}{7}-\frac {6 x}{7}}}{\sqrt {-9-15 x} \sqrt {2+3 x}} \, dx}{11 \sqrt {3+5 x}} \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {2 \sqrt {\frac {7}{5}} \sqrt {-3-5 x} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right )}{11 \sqrt {3+5 x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.62 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\frac {2}{55} \left (-\frac {5 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}-i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )\right ) \]

[In]

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

[Out]

(2*((-5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x] - I*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/
55

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14

method	result	size
default	\(-\frac {2 \sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (\sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+30 x^{2}+5 x -10\right )}{55 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(92\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-30 x^{2}-5 x +10\right )}{55 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {2 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{385 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{385 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(195\)

[In]

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

[Out]

-2/55*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*El
lipticE((10+15*x)^(1/2),1/35*70^(1/2))+30*x^2+5*x-10)/(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 = 68, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (34 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 45 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) + 225 \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}\right )}}{2475 \, {\left (5 \, x + 3\right )}} \]

[In]

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

[Out]

-2/2475*(34*sqrt(-30)*(5*x + 3)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 45*sqrt(-30)*(5*x + 3)
*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)) + 225*sqrt(5*x
+ 3)*sqrt(3*x + 2)*sqrt(-2*x + 1))/(5*x + 3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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