\(\int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\) [2941]

Optimal result

Integrand size = 28, antiderivative size = 156 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=\frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}-\frac {3896 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}-\frac {164 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{847 \sqrt {33}} \]

[Out]

-3896/27951*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-164/27951*EllipticF(1/7*21^(1/2)*(1
-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/77*(2+3*x)^(1/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)-410/2541*(1-2*x)^(1/2)*(2
+3*x)^(1/2)/(3+5*x)^(3/2)+19480/27951*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 156, 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} \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=-\frac {164 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{847 \sqrt {33}}-\frac {3896 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}+\frac {19480 \sqrt {1-2 x} \sqrt {3 x+2}}{27951 \sqrt {5 x+3}}-\frac {410 \sqrt {1-2 x} \sqrt {3 x+2}}{2541 (5 x+3)^{3/2}}+\frac {4 \sqrt {3 x+2}}{77 \sqrt {1-2 x} (5 x+3)^{3/2}} \]

[In]

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

[Out]

(4*Sqrt[2 + 3*x])/(77*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (410*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(2541*(3 + 5*x)^(3/2)
) + (19480*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27951*Sqrt[3 + 5*x]) - (3896*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/(847*Sqrt[33]) - (164*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(847*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 {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {95}{2}-45 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx \\ & = \frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {4 \int \frac {-\frac {605}{2}+\frac {615 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2541} \\ & = \frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {18885}{4}-7305 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{27951} \\ & = \frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}+\frac {82}{847} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {3896 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{9317} \\ & = \frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {410 \sqrt {1-2 x} \sqrt {2+3 x}}{2541 (3+5 x)^{3/2}}+\frac {19480 \sqrt {1-2 x} \sqrt {2+3 x}}{27951 \sqrt {3+5 x}}-\frac {3896 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}}-\frac {164 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{847 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.44 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {\sqrt {2+3 x} \left (27691-5230 x-97400 x^2\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}}+2 i \sqrt {33} \left (974 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1015 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{27951} \]

[In]

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

[Out]

(2*((Sqrt[2 + 3*x]*(27691 - 5230*x - 97400*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (2*I)*Sqrt[33]*(974*Ellipti
cE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 1015*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/27951

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.40

method	result	size
default	\(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (9570 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9740 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+5742 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-5844 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-292200 x^{3}-210490 x^{2}+72613 x +55382\right )}{27951 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\)	\(219\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {5036 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{195657 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {7792 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{195657 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{363 \left (x +\frac {3}{5}\right )^{2}}+\frac {-\frac {5600}{1331} x^{2}-\frac {2800}{3993} x +\frac {5600}{3993}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {8 \left (-30 x^{2}-38 x -12\right )}{9317 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(247\)

[In]

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

[Out]

-2/27951*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(9570*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)-9740*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)+5742*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))-5844*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))-292200*x^3-210490*x^2+72613*x+55382)/(3+5*x)^(3/2)/(6*x^2+x-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 = 108, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (97400 \, x^{2} + 5230 \, x - 27691\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 34253 \, \sqrt {-30} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 87660 \, \sqrt {-30} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{1257795 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

[In]

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

[Out]

2/1257795*(45*(97400*x^2 + 5230*x - 27691)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 34253*sqrt(-30)*(50*x^
3 + 35*x^2 - 12*x - 9)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 87660*sqrt(-30)*(50*x^3 + 35*x^
2 - 12*x - 9)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(
50*x^3 + 35*x^2 - 12*x - 9)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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