\(\int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\) [2818]

Optimal result

Integrand size = 28, antiderivative size = 31 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

-1/3*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {114} \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[In]

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

[Out]

-(Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])

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])

Rubi steps \begin{align*} \text {integral}& = -\sqrt {\frac {11}{3}} E\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 = 1.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=i \sqrt {\frac {11}{3}} \left (E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \]

[In]

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

[Out]

I*Sqrt[11/3]*(EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(23)=46\).

Time = 1.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74

method	result	size
default	\(\frac {\left (33 F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-35 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )\right ) \sqrt {-3-5 x}\, \sqrt {7}\, \sqrt {5}}{105 \sqrt {3+5 x}}\)	\(54\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{35 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(167\)

[In]

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

[Out]

1/105*(33*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))-35*EllipticE((10+15*x)^(1/2),1/35*70^(1/2)))*(-3-5*x)^(1/2)
*7^(1/2)*5^(1/2)/(3+5*x)^(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 = 26, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\frac {31}{270} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {1}{3} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]

[In]

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

[Out]

-31/270*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 1/3*sqrt(-30)*weierstrassZeta(1159/6
75, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90))

Sympy [F]

\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int \frac {\sqrt {5 x + 3}}{\sqrt {1 - 2 x} \sqrt {3 x + 2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{\sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{\sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int \frac {\sqrt {5\,x+3}}{\sqrt {1-2\,x}\,\sqrt {3\,x+2}} \,d x \]

[In]

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

[Out]

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