\(\int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx\) [2671]

   Optimal result
   Rubi [A] (verified)
   Mathematica [C] (verified)
   Maple [A] (verified)
   Fricas [C] (verification not implemented)
   Sympy [F(-1)]
   Maxima [F]
   Giac [F]
   Mupad [F(-1)]

Optimal result

Integrand size = 28, antiderivative size = 249 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac {1305025844 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {37904696 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{47647845 \sqrt {33}} \]

[Out]

-1305025844/1572378885*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-37904696/1572378885*Elli
pticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-118/2079*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)-
2/33*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2)-13022/305613*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+627806/
10696455*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+19417096/74875185*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)
+1305025844/524126295*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=-\frac {37904696 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {1305025844 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {5 x+3}}{524126295 \sqrt {3 x+2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac {13022 \sqrt {1-2 x} \sqrt {5 x+3}}{305613 (3 x+2)^{7/2}} \]

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^(13/2),x]

[Out]

(-13022*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(305613*(2 + 3*x)^(7/2)) + (627806*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10696455
*(2 + 3*x)^(5/2)) + (19417096*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(74875185*(2 + 3*x)^(3/2)) + (1305025844*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x])/(524126295*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2079*(2 + 3*x)^(9/2)) - (2
*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) - (1305025844*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
 35/33])/(47647845*Sqrt[33]) - (37904696*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(47647845*Sqrt[33]
)

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 155

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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

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} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx \\ & = -\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {4 \int \frac {\left (-\frac {189}{4}-\frac {5025 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{6237} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {8 \int \frac {-\frac {676497}{8}-185700 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{916839} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {16 \int \frac {\frac {1286433}{2}-\frac {14125635 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{32089365} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {32 \int \frac {\frac {687512943}{16}-\frac {109221165 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{673876665} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {64 \int \frac {\frac {2319498765}{4}+\frac {14681540745 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4717136655} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {18952348 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{47647845}+\frac {1305025844 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{524126295} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac {1305025844 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {37904696 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.45 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (21813966691+162787885893 x+484598540169 x^2+719808574005 x^3+534040213536 x^4+158560640046 x^5\right )}{2 (2+3 x)^{11/2}}+i \sqrt {33} \left (326256461 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-335732635 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{1572378885} \]

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^(13/2),x]

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(21813966691 + 162787885893*x + 484598540169*x^2 + 719808574005*x^3 + 53404
0213536*x^4 + 158560640046*x^5))/(2*(2 + 3*x)^(11/2)) + I*Sqrt[33]*(326256461*EllipticE[I*ArcSinh[Sqrt[9 + 15*
x]], -2/33] - 335732635*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/1572378885

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.27

method result size
elliptic \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{6}}+\frac {538 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1515591 \left (\frac {2}{3}+x \right )^{5}}-\frac {93382 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{24754653 \left (\frac {2}{3}+x \right )^{4}}+\frac {627806 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{288804285 \left (\frac {2}{3}+x \right )^{3}}+\frac {19417096 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{673876665 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {2610051688}{104825259} x^{2}-\frac {1305025844}{524126295} x +\frac {1305025844}{174708765}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {1649421344 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{11006652195 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2610051688 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{11006652195 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) \(315\)
default \(-\frac {2 \left (153842285718 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-158560640046 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+512807619060 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-528535466820 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+683743492080 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-704713955760 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+455828994720 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-469809303840 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+151942998240 \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}-156603101280 \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}-4756819201380 x^{7}+20259066432 \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 )-20880413504 \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 )-16496888326218 x^{6}-21769332100344 x^{5}-11891020005261 x^{4}+140844968748 x^{3}+3218604203112 x^{2}+1399649072964 x +196325700219\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{1572378885 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) \(599\)

[In]

int((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x,method=_RETURNVERBOSE)

[Out]

(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)*(-2/216513*(-30*x^3-23*x^2+7*x+6)^
(1/2)/(2/3+x)^6+538/1515591*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^5-93382/24754653*(-30*x^3-23*x^2+7*x+6)^(1/2)
/(2/3+x)^4+627806/288804285*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+19417096/673876665*(-30*x^3-23*x^2+7*x+6)^(
1/2)/(2/3+x)^2+1305025844/1572378885*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+1649421344/11006652195*(1
0+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(
1/2))+2610051688/11006652195*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-7/
6*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))))

Fricas [C] (verification not implemented)

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

Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\frac {2 \, {\left (135 \, {\left (158560640046 \, x^{5} + 534040213536 \, x^{4} + 719808574005 \, x^{3} + 484598540169 \, x^{2} + 162787885893 \, x + 21813966691\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 11052091517 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 29363081490 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{70757049825 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="fricas")

[Out]

2/70757049825*(135*(158560640046*x^5 + 534040213536*x^4 + 719808574005*x^3 + 484598540169*x^2 + 162787885893*x
 + 21813966691)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 11052091517*sqrt(-30)*(729*x^6 + 2916*x^5 + 4860*
x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 29363081490*sq
rt(-30)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*weierstrassZeta(1159/675, 38998/911
25, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x
^2 + 576*x + 64)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**(13/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(13/2), x)

Giac [F]

\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(13/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \]

[In]

int(((1 - 2*x)^(1/2)*(5*x + 3)^(5/2))/(3*x + 2)^(13/2),x)

[Out]

int(((1 - 2*x)^(1/2)*(5*x + 3)^(5/2))/(3*x + 2)^(13/2), x)