\(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\) [2978]

Optimal result

Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}-\frac {184636 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}}-\frac {9124 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{26411 \sqrt {33}} \]

[Out]

-184636/871563*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-9124/871563*EllipticF(1/7*21^(1/
2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/231*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2)+1072/17787*(3+5*x)^
(1/2)/(2+3*x)^(3/2)/(1-2*x)^(1/2)+974/41503*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+184636/290521*(1-2*x)^(1
/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164, 114, 120} \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {9124 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{26411 \sqrt {33}}-\frac {184636 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}}+\frac {184636 \sqrt {1-2 x} \sqrt {5 x+3}}{290521 \sqrt {3 x+2}}+\frac {974 \sqrt {1-2 x} \sqrt {5 x+3}}{41503 (3 x+2)^{3/2}}+\frac {1072 \sqrt {5 x+3}}{17787 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {4 \sqrt {5 x+3}}{231 (1-2 x)^{3/2} (3 x+2)^{3/2}} \]

[In]

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

[Out]

(4*Sqrt[3 + 5*x])/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (1072*Sqrt[3 + 5*x])/(17787*Sqrt[1 - 2*x]*(2 + 3*x)^
(3/2)) + (974*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41503*(2 + 3*x)^(3/2)) + (184636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(290
521*Sqrt[2 + 3*x]) - (184636*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(26411*Sqrt[33]) - (9124*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(26411*Sqrt[33])

Rule 106

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)/((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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
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 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 {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac {2}{231} \int \frac {-\frac {193}{2}-75 x}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx \\ & = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {4 \int \frac {\frac {17541}{4}+6030 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{17787} \\ & = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {8 \int \frac {\frac {123867}{4}-\frac {21915 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{373527} \\ & = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}+\frac {16 \int \frac {\frac {2718405}{8}+\frac {2077155 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2614689} \\ & = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}+\frac {4562 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{26411}+\frac {184636 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{290521} \\ & = \frac {4 \sqrt {3+5 x}}{231 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {1072 \sqrt {3+5 x}}{17787 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {974 \sqrt {1-2 x} \sqrt {3+5 x}}{41503 (2+3 x)^{3/2}}+\frac {184636 \sqrt {1-2 x} \sqrt {3+5 x}}{290521 \sqrt {2+3 x}}-\frac {184636 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}}-\frac {9124 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{26411 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2 \left (\frac {\sqrt {3+5 x} \left (597945-1478206 x-1066908 x^2+3323448 x^3\right )}{(1-2 x)^{3/2} (2+3 x)^{3/2}}+2 i \sqrt {33} \left (46159 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-48440 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{871563} \]

[In]

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

[Out]

(2*((Sqrt[3 + 5*x]*(597945 - 1478206*x - 1066908*x^2 + 3323448*x^3))/((1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (2*I)
*Sqrt[33]*(46159*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 48440*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/3
3])))/871563

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.22

method	result	size
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {107}{29106}-\frac {31 x}{4851}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right )^{2}}-\frac {2 \left (-18-30 x \right ) \left (\frac {135068}{2614689}-\frac {92318 x}{871563}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}+\frac {241636 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6100941 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {369272 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{6100941 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(228\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (548064 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-553908 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+91344 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-92318 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-182688 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+184636 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-16617240 x^{4}-4635804 x^{3}[phone] x^{2}+1444893 x -1793835\right )}{871563 \left (2+3 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\)	\(311\)

[In]

int(1/(1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*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)*((107/29106-31/4851*x)*(-30*x^3-23
*x^2+7*x+6)^(1/2)/(x^2+1/6*x-1/3)^2-2*(-18-30*x)*(135068/2614689-92318/871563*x)/((x^2+1/6*x-1/3)*(-18-30*x))^
(1/2)+241636/6100941*(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((1
0+15*x)^(1/2),1/35*70^(1/2))+369272/6100941*(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 = 128, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (45 \, {\left (3323448 \, x^{3} - 1066908 \, x^{2} - 1478206 \, x + 597945\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1656748 \, \sqrt {-30} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 4154310 \, \sqrt {-30} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{39220335 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]

[In]

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

[Out]

2/39220335*(45*(3323448*x^3 - 1066908*x^2 - 1478206*x + 597945)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1
656748*sqrt(-30)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) +
4154310*sqrt(-30)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInve
rse(1159/675, 38998/91125, x + 23/90)))/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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