\(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\) [2873]

Optimal result

Integrand size = 28, antiderivative size = 160 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {192}{49} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]

[Out]

6388/539*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+192/539*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)^(1/2)+288/49*(1-2*x)^(1/2)/(2+3*x)^(1
/2)/(3+5*x)^(1/2)-31940/539*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164, 114, 120} \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {192}{49} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {31940 \sqrt {1-2 x} \sqrt {3 x+2}}{539 \sqrt {5 x+3}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}} \]

[In]

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

[Out]

(2*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (288*Sqrt[1 - 2*x])/(49*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (
31940*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(539*Sqrt[3 + 5*x]) + (6388*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/49 + (192*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/49

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} \sqrt {3+5 x}}+\frac {2}{21} \int \frac {42-45 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {4}{147} \int \frac {\frac {3495}{2}-1080 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}-\frac {8 \int \frac {\frac {45495}{2}+\frac {71865 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1617} \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}-\frac {288}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {19164}{539} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {288 \sqrt {1-2 x}}{49 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {31940 \sqrt {1-2 x} \sqrt {2+3 x}}{539 \sqrt {3+5 x}}+\frac {6388}{49} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {192}{49} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {2}{539} \left (-\frac {\sqrt {1-2 x} \left (60635+186888 x+143730 x^2\right )}{(2+3 x)^{3/2} \sqrt {3+5 x}}-2 i \sqrt {33} \left (1597 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1645 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right ) \]

[In]

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

[Out]

(2*(-((Sqrt[1 - 2*x]*(60635 + 186888*x + 143730*x^2))/((2 + 3*x)^(3/2)*Sqrt[3 + 5*x])) - (2*I)*Sqrt[33]*(1597*
EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 1645*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/539

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.37

method	result	size
default	\(\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (9306 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9582 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+6204 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-6388 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-287460 x^{3}-230046 x^{2}+65618 x +60635\right )}{539 \left (2+3 x \right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )}\)	\(219\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21 \left (\frac {2}{3}+x \right )^{2}}-\frac {358 \left (-30 x^{2}-3 x +9\right )}{49 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {8088 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{3773 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {12776 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{3773 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {50 \left (-30 x^{2}-5 x +10\right )}{11 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(247\)

[In]

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

[Out]

2/539*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(9306*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)-9582*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)+6204*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))-6388*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))-287460*x^3-230046*x^2+65618*x+60635)/(2+3*x)^(3/2)/(10*x^2+x-3)

Fricas [C] (verification not implemented)

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

Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (45 \, {\left (143730 \, x^{2} + 186888 \, x + 60635\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 54259 \, \sqrt {-30} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 143730 \, \sqrt {-30} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{24255 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

[In]

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

[Out]

-2/24255*(45*(143730*x^2 + 186888*x + 60635)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 54259*sqrt(-30)*(45*
x^3 + 87*x^2 + 56*x + 12)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 143730*sqrt(-30)*(45*x^3 + 8
7*x^2 + 56*x + 12)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90
)))/(45*x^3 + 87*x^2 + 56*x + 12)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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