\(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\) [2882]

Optimal result

Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}-\frac {1255552 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {37768 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{539 \sqrt {33}} \]

[Out]

-1255552/17787*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-37768/17787*EllipticF(1/7*21^(1/
2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/7*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2)+428/49*(1-2*x)^(1/2)/
(3+5*x)^(3/2)/(2+3*x)^(1/2)-94420/1617*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+6277760/17787*(1-2*x)^(1/2)*(
2+3*x)^(1/2)/(3+5*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} \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {37768 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {1255552 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}+\frac {6277760 \sqrt {1-2 x} \sqrt {3 x+2}}{17787 \sqrt {5 x+3}}-\frac {94420 \sqrt {1-2 x} \sqrt {3 x+2}}{1617 (5 x+3)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}} \]

[In]

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

[Out]

(2*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (428*Sqrt[1 - 2*x])/(49*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))
- (94420*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1617*(3 + 5*x)^(3/2)) + (6277760*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(17787*S
qrt[3 + 5*x]) - (1255552*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(539*Sqrt[33]) - (37768*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(539*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 {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2}{21} \int \frac {57-75 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {4}{147} \int \frac {\frac {8385}{2}-4815 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}-\frac {8 \int \frac {\frac {343365}{2}-\frac {212445 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{4851} \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {8942355}{4}+3531240 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{53361} \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}+\frac {18884}{539} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {1255552 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5929} \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}-\frac {1255552 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {37768 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.81 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {\sqrt {1-2 x} \left (35747225+169778606 x+268408770 x^2+141249600 x^3\right )}{(2+3 x)^{3/2} (3+5 x)^{3/2}}+4 i \sqrt {33} \left (156944 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-161665 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{17787} \]

[In]

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

[Out]

(2*((Sqrt[1 - 2*x]*(35747225 + 169778606*x + 268408770*x^2 + 141249600*x^3))/((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))
+ (4*I)*Sqrt[33]*(156944*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 161665*EllipticF[I*ArcSinh[Sqrt[9 + 15
*x]], -2/33])))/17787

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.22

method	result	size
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {1294}{51975}-\frac {136 x}{3465}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right )^{2}}-\frac {2 \left (15-30 x \right ) \left (-\frac {5966174}{266805}-\frac {627776 x}{17787}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}+\frac {1589752 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{124509 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2511104 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{124509 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(228\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (9145620 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9416640 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-11927744 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+3658248 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-3766656 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-282499200 x^{4}-395567940 x^{3}-71148442 x^{2}[phone] x [phone]\right )}{17787 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\)	\(311\)

[In]

int(1/(2+3*x)^(5/2)/(3+5*x)^(5/2)/(1-2*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)*((-1294/51975-136/3465*x)*(-30*x^3
-23*x^2+7*x+6)^(1/2)/(x^2+19/15*x+2/5)^2-2*(15-30*x)*(-5966174/266805-627776/17787*x)/((x^2+19/15*x+2/5)*(15-3
0*x))^(1/2)+1589752/124509*(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)*Ellipt
icF((10+15*x)^(1/2),1/35*70^(1/2))+2511104/124509*(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 = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (141249600 \, x^{3} + 268408770 \, x^{2} + 169778606 \, x + 35747225\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 10665286 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 28249920 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{800415 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

[In]

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

[Out]

2/800415*(45*(141249600*x^3 + 268408770*x^2 + 169778606*x + 35747225)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x +
1) - 10665286*sqrt(-30)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*weierstrassPInverse(1159/675, 38998/91125,
x + 23/90) + 28249920*sqrt(-30)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*weierstrassZeta(1159/675, 38998/911
25, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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