\(\int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\) [2832]

Optimal result

Integrand size = 28, antiderivative size = 191 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {8256877 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{56700}-\frac {62092 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{14175} \]

[Out]

-8256877/170100*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-62092/42525*EllipticF(1/7*21^(1
/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1/9*(2+3*x)^(3/2)*(3+5*x)^(5/2)*(1-2*x)^(1/2)-1877/630*(3+5*x)^(3/
2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-3/7*(3+5*x)^(5/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-62092/2835*(1-2*x)^(1/2)*(2+3*x)^
(1/2)*(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.06 (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 = {103, 159, 164, 114, 120} \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {62092 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{14175}-\frac {8256877 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{56700}-\frac {1}{9} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {62092 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{2835} \]

[In]

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

[Out]

(-62092*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2835 - (1877*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/6
30 - (3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/7 - (Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/9 - (
8256877*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/56700 - (62092*Sqrt[11/3]*EllipticF[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/14175

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b
*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[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 {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{9} \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {173}{2}+135 x\right )}{\sqrt {1-2 x}} \, dx \\ & = -\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{315} \int \frac {\left (-\frac {18455}{2}-\frac {28155 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {\int \frac {\sqrt {3+5 x} \left (\frac {2421135}{4}+931380 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{4725} \\ & = -\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {\int \frac {-\frac {78409965}{4}-\frac {123853155 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{42525} \\ & = -\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {341506 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{14175}+\frac {8256877 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{56700} \\ & = -\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {8256877 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{56700}-\frac {62092 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{14175} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.56 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.54 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\frac {-30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (208073+212175 x+148950 x^2+47250 x^3\right )+8256877 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-8505245 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{170100} \]

[In]

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

[Out]

(-30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(208073 + 212175*x + 148950*x^2 + 47250*x^3) + (8256877*I)*Sqrt
[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (8505245*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -
2/33])/170100

Maple [A] (verified)

Time = 1.35 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.81

method	result	size
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (8019231 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-8256877 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-81886830 x -37453140\right )}{170100 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(155\)
risch	\(\frac {\left (47250 x^{3}+148950 x^{2}+212175 x +208073\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{5670 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {5227331 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{623700 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8256877 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{623700 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(256\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-\frac {4715 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{126}-\frac {208073 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5670}+\frac {5227331 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{595350 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8256877 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{595350 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1655 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{63}-\frac {25 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\)	\(262\)

[In]

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

[Out]

-1/170100*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(8019231*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))-8256877*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))[phone]*x^6+166657500*x^5+283810500*x^4+293881950*x^3+722026
20*x^2-81886830*x-37453140)/(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 = 64, normalized size of antiderivative = 0.34 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{5670} \, {\left (47250 \, x^{3} + 148950 \, x^{2} + 212175 \, x + 208073\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {280551619}{15309000} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {8256877}{170100} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]

[In]

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

[Out]

-1/5670*(47250*x^3 + 148950*x^2 + 212175*x + 208073)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 280551619/15
309000*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 8256877/170100*sqrt(-30)*weierstrassZ
eta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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