\(\int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx\) [2694]

Optimal result

Integrand size = 28, antiderivative size = 125 \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}+\frac {62 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}+\frac {8 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{25 \sqrt {33}} \]

[Out]

62/825*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+8/825*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/
2),1/33*1155^(1/2))*33^(1/2)-2/15*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)-62/165*(1-2*x)^(1/2)*(2+3*x)^(1/2)
/(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 125, 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 = {99, 157, 164, 114, 120} \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\frac {8 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{25 \sqrt {33}}+\frac {62 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}-\frac {62 \sqrt {1-2 x} \sqrt {3 x+2}}{165 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}} \]

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(15*(3 + 5*x)^(3/2)) - (62*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(165*Sqrt[3 + 5*x]) +
(62*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(25*Sqrt[33]) + (8*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/(25*Sqrt[33])

Rule 99

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/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 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 {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {-\frac {1}{2}-6 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}-\frac {4}{165} \int \frac {\frac {69}{2}+\frac {93 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}-\frac {4}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {62}{275} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}+\frac {62 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}+\frac {8 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=\frac {2}{825} \left (-\frac {5 \sqrt {1-2 x} \sqrt {2+3 x} (104+155 x)}{(3+5 x)^{3/2}}-31 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+35 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \]

[In]

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

[Out]

(2*((-5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(104 + 155*x))/(3 + 5*x)^(3/2) - (31*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[
9 + 15*x]], -2/33] + (35*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/825

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(218\) vs. \(2(93)=186\).

Time = 1.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.75

method	result	size
default	\(\frac {2 \left (165 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-155 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+99 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-93 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4650 x^{3}-3895 x^{2}+1030 x +1040\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}}{825 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\)	\(219\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{375 \left (x +\frac {3}{5}\right )^{2}}-\frac {62 \left (-30 x^{2}-5 x +10\right )}{825 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {92 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{5775 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {124 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{5775 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(219\)

[In]

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

[Out]

2/825*(165*5^(1/2)*7^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))*x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/
2)-155*5^(1/2)*7^(1/2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))*x*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)+9
9*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))-93*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))-4650*x^3-3895*x
^2+1030*x+1040)*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

Fricas [C] (verification not implemented)

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

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx=-\frac {450 \, {\left (155 \, x + 104\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1357 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2790 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{37125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

[In]

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

[Out]

-1/37125*(450*(155*x + 104)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1357*sqrt(-30)*(25*x^2 + 30*x + 9)*we
ierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 2790*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassZeta(1159/
675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(25*x^2 + 30*x + 9)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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