\(\int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx\) [2799]

Optimal result

Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {81164 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125}-\frac {28174 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{28125} \]

[Out]

81164/84375*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-28174/84375*EllipticF(1/7*21^(1/2)*
(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/5*(1-2*x)^(5/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)-24/125*(1-2*x)^(3/2)*(2+
3*x)^(1/2)*(3+5*x)^(1/2)-3028/5625*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.04 (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 = {99, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=-\frac {28174 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{28125}+\frac {81164 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125}-\frac {2 \sqrt {3 x+2} (1-2 x)^{5/2}}{5 \sqrt {5 x+3}}-\frac {24}{125} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {3028 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{5625} \]

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) - (3028*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5625 -
(24*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/125 + (81164*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/28125 - (28174*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/28125

Rule 99

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

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 {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {\left (-\frac {17}{2}-18 x\right ) (1-2 x)^{3/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{375} \int \frac {\left (-\frac {1887}{4}-\frac {2271 x}{2}\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {8 \int \frac {-\frac {53121}{8}-\frac {60873 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{16875} \\ & = -\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {81164 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{28125}+\frac {154957 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{28125} \\ & = -\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {81164 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125}-\frac {28174 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.39 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\frac {2 \left (\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \left (-7287-2530 x+900 x^2\right )}{\sqrt {3+5 x}}-40582 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+26495 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{84375} \]

[In]

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

[Out]

(2*((15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-7287 - 2530*x + 900*x^2))/Sqrt[3 + 5*x] - (40582*I)*Sqrt[33]*EllipticE[I
*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (26495*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/84375

Maple [A] (verified)

Time = 1.39 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (24981 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-40582 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+81000 x^{4}-214200 x^{3}-720780 x^{2}-33405 x +218610\right )}{84375 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(145\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {8 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}-\frac {1228 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5625}-\frac {35414 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{590625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {162328 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{590625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \left (-30 x^{2}-5 x +10\right )}{625 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(234\)

[In]

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

[Out]

2/84375*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(24981*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))-40582*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^4-214200*x^3-720780*x^2-33405*x+218610)/(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 = 78, normalized size of antiderivative = 0.49 \[ \int \frac {(1-2 x)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{3/2}} \, dx=\frac {1350 \, {\left (900 \, x^{2} - 2530 \, x - 7287\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 136571 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 3652380 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{3796875 \, {\left (5 \, x + 3\right )}} \]

[In]

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

[Out]

1/3796875*(1350*(900*x^2 - 2530*x - 7287)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 136571*sqrt(-30)*(5*x +
3)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 3652380*sqrt(-30)*(5*x + 3)*weierstrassZeta(1159/6
75, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(5*x + 3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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