3.30.16 \(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx\) [2916]

Optimal. Leaf size=222 \[ \frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}-\frac {106772 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 (2+3 x)^{3/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {3+5 x}}{1764735 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}+\frac {106558 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735}-\frac {220028 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735} \]

[Out]

106558/5294205*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-220028/5294205*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)^(7/2)/(1-2*x)^(1/2)+229/1029*(1-2*x)^
(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-37117/36015*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)-106772/252105*(1-2*x)^
(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)-106558/1764735*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157, 164, 114, 120} \begin {gather*} -\frac {220028 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735}+\frac {106558 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735}+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {5 x+3}}{1764735 \sqrt {3 x+2}}-\frac {106772 \sqrt {1-2 x} \sqrt {5 x+3}}{252105 (3 x+2)^{3/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {5 x+3}}{36015 (3 x+2)^{5/2}}+\frac {229 \sqrt {1-2 x} \sqrt {5 x+3}}{1029 (3 x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(9/2)),x]

[Out]

(229*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1029*(2 + 3*x)^(7/2)) - (37117*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36015*(2 + 3*x
)^(5/2)) - (106772*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(252105*(2 + 3*x)^(3/2)) - (106558*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(1764735*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + (106558*Sqrt[11/3]*Ellipt
icE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1764735 - (220028*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/1764735

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*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx &=\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {1}{7} \int \frac {\left (-\frac {357}{2}-325 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {2 \int \frac {-\frac {86161}{4}-36950 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{1029}\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4 \int \frac {-79446-\frac {556755 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{36015}\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}-\frac {106772 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 (2+3 x)^{3/2}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {8 \int \frac {-\frac {1014441}{8}-\frac {400395 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{756315}\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}-\frac {106772 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 (2+3 x)^{3/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {3+5 x}}{1764735 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {16 \int \frac {-166965+\frac {799185 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5294205}\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}-\frac {106772 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 (2+3 x)^{3/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {3+5 x}}{1764735 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {106558 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1764735}+\frac {1210154 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1764735}\\ &=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}-\frac {106772 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 (2+3 x)^{3/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {3+5 x}}{1764735 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}+\frac {106558 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735}-\frac {220028 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1764735}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 8.11, size = 109, normalized size = 0.49 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {3+5 x} \left (616327+4889131 x+12020751 x^2+11042235 x^3+2877066 x^4\right )}{\sqrt {1-2 x} (2+3 x)^{7/2}}+\sqrt {2} \left (-53279 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+1868510 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{5294205} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(9/2)),x]

[Out]

(2*((3*Sqrt[3 + 5*x]*(616327 + 4889131*x + 12020751*x^2 + 11042235*x^3 + 2877066*x^4))/(Sqrt[1 - 2*x]*(2 + 3*x
)^(7/2)) + Sqrt[2]*(-53279*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 1868510*EllipticF[ArcSin[Sqrt[
2/11]*Sqrt[3 + 5*x]], -33/2])))/5294205

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(162)=324\).
time = 0.11, size = 401, normalized size = 1.81

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {484 \left (-30 x^{2}-38 x -12\right )}{16807 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {998 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{972405 \left (\frac {2}{3}+x \right )^{3}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{83349 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {91804}{352947} x^{2}-\frac {45902}{1764735} x +\frac {45902}{588245}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {30542 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2268945 \left (\frac {2}{3}+x \right )^{2}}+\frac {178096 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{7411887 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {106558 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7411887 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(301\)
default	\(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (49011237 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1438533 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2877066 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1918044 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+426232 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +5546943\right )}{5294205 \left (2+3 x \right )^{\frac {7}{2}} \left (10 x^{2}+x -3\right )}\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/5294205*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(49011237*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*
x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+1438533*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^
(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1
/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+2877066*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)
*(-3-5*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3
-5*x)^(1/2)*(1-2*x)^(1/2)+1918044*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)
^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2)
,1/2*70^(1/2))+426232*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^
(1/2))[phone]*x^5+191527119*x^4+279691380*x^3+181523724*x^2+53247084*x+5546943)/(2+3*x)^(7/2)/(10*x^2+x-3)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(9/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(3/2)), x)

________________________________________________________________________________________

Fricas [A]
time = 0.24, size = 70, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (2877066 \, x^{4} + 11042235 \, x^{3} + 12020751 \, x^{2} + 4889131 \, x + 616327\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1764735 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(9/2),x, algorithm="fricas")

[Out]

-2/1764735*(2877066*x^4 + 11042235*x^3 + 12020751*x^2 + 4889131*x + 616327)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-
2*x + 1)/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(9/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7317 deep

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(9/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(3/2)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^(9/2)),x)

[Out]

int((5*x + 3)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^(9/2)), x)

________________________________________________________________________________________