\(\int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx\) [2722]

Optimal result

Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=-\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2894 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835}-\frac {1061 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2835} \]

[Out]

-2894/8505*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1061/8505*EllipticF(1/7*21^(1/2)*(1-
2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/3*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(1/2)+202/63*(3+5*x)^(3/2)*(1-2*x
)^(1/2)*(2+3*x)^(1/2)-32/63*(3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-1061/567*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+
5*x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 191, 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 = {99, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=-\frac {1061 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2835}-\frac {2894 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835}-\frac {32}{63} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}+\frac {202}{63} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {1061}{567} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \]

[In]

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

[Out]

(-1061*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/567 + (202*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/63 -
(2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(3*Sqrt[2 + 3*x]) - (32*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/63 -
(2894*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2835 - (1061*Sqrt[11/3]*EllipticF[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2835

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)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {4}{315} \int \frac {\left (\frac {4495}{4}-\frac {7575 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (\frac {1125}{2}-\frac {79575 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{4725} \\ & = -\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {4 \int \frac {\frac {435525}{8}+\frac {108525 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{42525} \\ & = -\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2894 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2835}+\frac {11671 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5670} \\ & = -\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2894 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835}-\frac {1061 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.80 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\frac {2894 i \sqrt {33} (2+3 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5 \left (3 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-200-1767 x-180 x^2+2700 x^3\right )+791 i \sqrt {33} (2+3 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{8505 (2+3 x)} \]

[In]

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

[Out]

((2894*I)*Sqrt[33]*(2 + 3*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 5*(3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sq
rt[3 + 5*x]*(-200 - 1767*x - 180*x^2 + 2700*x^3) + (791*I)*Sqrt[33]*(2 + 3*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*
x]], -2/33]))/(8505*(2 + 3*x))

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (3729 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2894 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+405000 x^{5}+13500 x^{4}-389250 x^{3}-48405 x^{2}+76515 x +9000\right )}{8505 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(150\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {220 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{189}+\frac {149 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{567}+\frac {5807 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{59535 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5788 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{59535 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {100 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{63}-\frac {14 \left (-30 x^{2}-3 x +9\right )}{243 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(256\)

[In]

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

[Out]

-1/8505*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(3729*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))-2894*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))+405000*x^5+13500*x^4-389250*x^3-48405*x^2+76515*x+9000)/(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 = 83, normalized size of antiderivative = 0.43 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=-\frac {1350 \, {\left (2700 \, x^{3} - 180 \, x^{2} - 1767 \, x - 200\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 194753 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 260460 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{765450 \, {\left (3 \, x + 2\right )}} \]

[In]

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

[Out]

-1/765450*(1350*(2700*x^3 - 180*x^2 - 1767*x - 200)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 194753*sqrt(-
30)*(3*x + 2)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 260460*sqrt(-30)*(3*x + 2)*weierstrassZe
ta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(3*x + 2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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