\(\int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\) [2789]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx=-\frac {1088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3375}-\frac {4}{75} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {53194 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16875}-\frac {34154 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{16875} \]

[Out]

53194/50625*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-34154/50625*EllipticF(1/7*21^(1/2)*
(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-4/75*(1-2*x)^(3/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-1088/3375*(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 = {104, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx=-\frac {34154 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{16875}+\frac {53194 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16875}-\frac {4}{75} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {1088 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{3375} \]

[In]

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

[Out]

(-1088*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3375 - (4*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/75 +
(53194*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/16875 - (34154*Sqrt[11/3]*EllipticF[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/16875

Rule 104

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]

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 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && 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 {4}{75} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2}{75} \int \frac {\left (\frac {41}{2}-272 x\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {1088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3375}-\frac {4}{75} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4 \int \frac {\frac {5653}{4}-\frac {26597 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3375} \\ & = -\frac {1088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3375}-\frac {4}{75} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {53194 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{16875}+\frac {187847 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{16875} \\ & = -\frac {1088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3375}-\frac {4}{75} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {53194 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16875}-\frac {34154 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16875} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.59 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx=\frac {60 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} (-317+90 x)-53194 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+19040 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{50625} \]

[In]

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

[Out]

(60*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-317 + 90*x) - (53194*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 +
15*x]], -2/33] + (19040*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/50625

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (8976 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-26597 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+81000 x^{4}-223200 x^{3}-237630 x^{2}+50370 x +57060\right )}{50625 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(145\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {8 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{75}-\frac {1268 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3375}+\frac {11306 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{354375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {106388 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{354375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(206\)
risch	\(-\frac {4 \left (-317+90 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{3375 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {5653 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{185625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {53194 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{185625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(247\)

[In]

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

[Out]

2/50625*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(8976*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))-26597*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))+81000*x^4-223200*x^3-237630*x^2+50370*x+57060)/(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 = 54, normalized size of antiderivative = 0.42 \[ \int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx=\frac {4}{3375} \, {\left (90 \, x - 317\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {866116}{2278125} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - \frac {53194}{50625} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]

[In]

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

[Out]

4/3375*(90*x - 317)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 866116/2278125*sqrt(-30)*weierstrassPInverse(
1159/675, 38998/91125, x + 23/90) - 53194/50625*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPI
nverse(1159/675, 38998/91125, x + 23/90))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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