\(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx\) [2950]

Optimal result

Integrand size = 28, antiderivative size = 125 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=\frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}+\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{1617 \sqrt {1-2 x}}+\frac {31 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}+\frac {4 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{49 \sqrt {33}} \]

[Out]

31/1617*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/1617*EllipticF(1/7*21^(1/2)*(1-2*x)^(
1/2),1/33*1155^(1/2))*33^(1/2)+2/21*(2+3*x)^(1/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2)+62/1617*(2+3*x)^(1/2)*(3+5*x)^(1
/2)/(1-2*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 = {101, 157, 164, 114, 120} \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=\frac {4 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{49 \sqrt {33}}+\frac {31 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}+\frac {62 \sqrt {3 x+2} \sqrt {5 x+3}}{1617 \sqrt {1-2 x}}+\frac {2 \sqrt {3 x+2} \sqrt {5 x+3}}{21 (1-2 x)^{3/2}} \]

[In]

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

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)) + (62*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(1617*Sqrt[1 - 2*x]) +
(31*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(49*Sqrt[33]) + (4*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/(49*Sqrt[33])

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} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}-\frac {2}{21} \int \frac {-4-\frac {15 x}{2}}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = \frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}+\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{1617 \sqrt {1-2 x}}+\frac {4 \int \frac {-\frac {345}{4}-\frac {465 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1617} \\ & = \frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}+\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{1617 \sqrt {1-2 x}}-\frac {2}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {31}{539} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}+\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{1617 \sqrt {1-2 x}}+\frac {31 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}+\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.74 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=-\frac {4 \sqrt {2+3 x} \sqrt {3+5 x} (-54+31 x)-31 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+35 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{1617 (1-2 x)^{3/2}} \]

[In]

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

[Out]

-1/1617*(4*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-54 + 31*x) - (31*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*EllipticE[I*ArcSinh[Sq
rt[9 + 15*x]], -2/33] + (35*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(1 - 2*
x)^(3/2)

Maple [B] (verified)

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

Time = 4.63 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.75

method	result	size
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{42 \left (x -\frac {1}{2}\right )^{2}}-\frac {31 \left (-30 x^{2}-38 x -12\right )}{1617 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {46 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{11319 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {62 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{11319 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(219\)
default	\(\frac {\left (66 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-62 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-33 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+31 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1860 x^{3}+884 x^{2}+3360 x +1296\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}}{1617 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\)	\(228\)

[In]

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

[Out]

(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)*(1/42*(-30*x^3-23*x^2+7*x+6)^(1/2)
/(x-1/2)^2-31/1617*(-30*x^2-38*x-12)/((x-1/2)*(-30*x^2-38*x-12))^(1/2)-46/11319*(10+15*x)^(1/2)*(21-42*x)^(1/2
)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))-62/11319*(10+15*x)^(1/
2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))
+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(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 = 88, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=-\frac {360 \, {\left (31 \, x - 54\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1357 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2790 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{145530 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

[In]

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

[Out]

-1/145530*(360*(31*x - 54)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1357*sqrt(-30)*(4*x^2 - 4*x + 1)*weier
strassPInverse(1159/675, 38998/91125, x + 23/90) + 2790*sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassZeta(1159/675,
38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(4*x^2 - 4*x + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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