\(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx\) [2917]

Optimal result

Integrand size = 28, antiderivative size = 253 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}-\frac {1199452 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{22235661} \]

[Out]

-6036028/66706983*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1199452/66706983*EllipticF(1/
7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+11/7*(3+5*x)^(3/2)/(2+3*x)^(9/2)/(1-2*x)^(1/2)+295/1323*(1-
2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)-67345/64827*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-167228/453789*(1-
2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)-392998/3176523*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+6036028/222356
61*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 253, 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 = {100, 155, 157, 164, 114, 120} \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=-\frac {1199452 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{22235661}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {5 x+3}}{22235661 \sqrt {3 x+2}}-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{3176523 (3 x+2)^{3/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{453789 (3 x+2)^{5/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {5 x+3}}{64827 (3 x+2)^{7/2}}+\frac {295 \sqrt {1-2 x} \sqrt {5 x+3}}{1323 (3 x+2)^{9/2}} \]

[In]

Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(11/2)),x]

[Out]

(295*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1323*(2 + 3*x)^(9/2)) - (67345*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(64827*(2 + 3*x
)^(7/2)) - (167228*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(453789*(2 + 3*x)^(5/2)) - (392998*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(3176523*(2 + 3*x)^(3/2)) + (6036028*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(22235661*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(
3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)) - (6036028*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/22235661 - (1199452*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/22235661

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {1}{7} \int \frac {\left (-\frac {555}{2}-490 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {2 \int \frac {-\frac {169585}{4}-\frac {144025 x}{2}}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx}{1323} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {4 \int \frac {-245765-\frac {1683625 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{64827} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {8 \int \frac {-\frac {7378905}{8}-\frac {3135525 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{2268945} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {16 \int \frac {-\frac {17369985}{8}-\frac {14737425 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{47647845} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {32 \int \frac {-\frac {185288025}{16}-\frac {113175525 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{333534915} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}+\frac {6036028 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{22235661}+\frac {6596986 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{22235661} \\ & = \frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {6036028 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661}-\frac {1199452 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{22235661} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.44 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\frac {4 \left (-\frac {3 \sqrt {3+5 x} \left (-52688263-243200677 x-227945505 x^2+466728543 x^3+985046292 x^4+488918268 x^5\right )}{2 \sqrt {1-2 x} (2+3 x)^{9/2}}+i \sqrt {33} \left (1509007 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1808870 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{66706983} \]

[In]

Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(11/2)),x]

[Out]

(4*((-3*Sqrt[3 + 5*x]*(-52688263 - 243200677*x - 227945505*x^2 + 466728543*x^3 + 985046292*x^4 + 488918268*x^5
))/(2*Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)) + I*Sqrt[33]*(1509007*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 18088
70*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/66706983

Maple [A] (verified)

Time = 1.37 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.26

method	result	size
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{321489 \left (\frac {2}{3}+x \right )^{5}}+\frac {1262 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5250987 \left (\frac {2}{3}+x \right )^{4}}-\frac {30014 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{12252303 \left (\frac {2}{3}+x \right )^{3}}-\frac {118570 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{28588707 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {65848840}{22235661} x^{2}-\frac {6584884}{22235661} x +\frac {6584884}{7411887}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {9882028 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{466948881 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {12072056 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{466948881 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {968 \left (-30 x^{2}-38 x -12\right )}{117649 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(319\)
default	\(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (276291972 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-244459134 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-651891024 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-651891024 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-289729344 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-48288224 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-7333774020 x^{6}-19175958792 x^{5}-15866344773 x^{4}-781374312 x^{3}[phone] x^{2}[phone] x [phone]\right )}{66706983 \left (2+3 x \right )^{\frac {9}{2}} \left (10 x^{2}+x -3\right )}\)	\(504\)

[In]

int((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(11/2),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)*(-2/321489*(-30*x^3-23*x^2+7*x+6)^
(1/2)/(2/3+x)^5+1262/5250987*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4-30014/12252303*(-30*x^3-23*x^2+7*x+6)^(1/2
)/(2/3+x)^3-118570/28588707*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+6584884/66706983*(-30*x^2-3*x+9)/((2/3+x)*(
-30*x^2-3*x+9))^(1/2)+9882028/466948881*(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)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))[phone]/466948881*(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)))-968/117649*(-30*x^2-38*x-12)/((x-1/2)*(-30*x^2-38*x-12))^(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 = 0.66 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\frac {2 \, {\left (135 \, {\left (488918268 \, x^{5} + 985046292 \, x^{4} + 466728543 \, x^{3} - 227945505 \, x^{2} - 243200677 \, x - 52688263\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 76465654 \, \sqrt {-30} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 135810630 \, \sqrt {-30} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{3001814235 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \]

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(11/2),x, algorithm="fricas")

[Out]

2/3001814235*(135*(488918268*x^5 + 985046292*x^4 + 466728543*x^3 - 227945505*x^2 - 243200677*x - 52688263)*sqr
t(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 76465654*sqrt(-30)*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^
2 - 176*x - 32)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 135810630*sqrt(-30)*(486*x^6 + 1377*x^
5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159
/675, 38998/91125, x + 23/90)))/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\text {Timed out} \]

[In]

integrate((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(11/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(11/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*(-2*x + 1)^(3/2)), x)

Giac [F]

\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(11/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*(-2*x + 1)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{11/2}} \,d x \]

[In]

int((5*x + 3)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^(11/2)),x)

[Out]

int((5*x + 3)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^(11/2)), x)