3.29.14 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx\) [2814]

Optimal. Leaf size=251 \[ \frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {5544440 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {11171040 \sqrt {1-2 x} \sqrt {2+3 x}}{49 \sqrt {3+5 x}}-\frac {2234208}{49} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {201616}{49} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

2/3*(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2)-201616/147*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))
*33^(1/2)-2234208/49*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+44/3*(1-2*x)^(1/2)/(2+3*x)
^(5/2)/(3+5*x)^(3/2)+11924/63*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2)+2488904/441*(1-2*x)^(1/2)/(3+5*x)^(3/2
)/(2+3*x)^(1/2)-5544440/147*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)[phone]/49*(1-2*x)^(1/2)*(2+3*x)^(1/2)/
(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {201616}{49} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2234208}{49} \sqrt {33} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}+\frac {11171040 \sqrt {3 x+2} \sqrt {1-2 x}}{49 \sqrt {5 x+3}}-\frac {5544440 \sqrt {3 x+2} \sqrt {1-2 x}}{147 (5 x+3)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 - 2*x)^(3/2))/(3*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)) + (44*Sqrt[1 - 2*x])/(3*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2
)) + (11924*Sqrt[1 - 2*x])/(63*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (2488904*Sqrt[1 - 2*x])/(441*Sqrt[2 + 3*x]*(
3 + 5*x)^(3/2)) - (5544440*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(147*(3 + 5*x)^(3/2)) + (11171040*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(49*Sqrt[3 + 5*x]) - (2234208*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/49 - (20161
6*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/49

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 {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{5/2}} \, dx &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {2}{21} \int \frac {(231-231 x) \sqrt {1-2 x}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {4}{315} \int \frac {-\frac {51975}{2}+39270 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac {8 \int \frac {-\frac {5672205}{2}+\frac {7825125 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{6615}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16 \int \frac {-\frac {852857775}{4}+\frac {490002975 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{46305}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {5544440 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {32 \int \frac {-\frac {34932969225}{4}+\frac {21612920175 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{1528065}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {5544440 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {11171040 \sqrt {1-2 x} \sqrt {2+3 x}}{49 \sqrt {3+5 x}}-\frac {64 \int \frac {-\frac {909761408325}{8}-\frac {359255409975 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{16808715}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {5544440 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {11171040 \sqrt {1-2 x} \sqrt {2+3 x}}{49 \sqrt {3+5 x}}+\frac {1108888}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {6702624}{49} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {5544440 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {11171040 \sqrt {1-2 x} \sqrt {2+3 x}}{49 \sqrt {3+5 x}}-\frac {2234208}{49} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {201616}{49} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 8.93, size = 114, normalized size = 0.45 \begin {gather*} \frac {2}{147} \left (\frac {\sqrt {1-2 x} \left (763335749+5915384456 x+18325125498 x^2+28367736228 x^3+21944379060 x^4+6786406800 x^5\right )}{(2+3 x)^{7/2} (3+5 x)^{3/2}}+12 \sqrt {2} \left (279276 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-140665 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[1 - 2*x]*(763335749 + 5915384456*x + 18325125498*x^2 + 28367736228*x^3 + 21944379060*x^4 + 678640680
0*x^5))/((2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)) + 12*Sqrt[2]*(279276*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33
/2] - 140665*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/147

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(490\) vs. \(2(185)=370\).
time = 0.11, size = 491, normalized size = 1.96

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3 \left (x +\frac {3}{5}\right )^{2}}+\frac {-523600 x^{2}-\frac {261800}{3} x +\frac {523600}{3}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {268 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27 \left (\frac {2}{3}+x \right )^{3}}+\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {41369840}{49} x^{2}-\frac {4136984}{49} x +\frac {12410952}{49}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {28234 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{63 \left (\frac {2}{3}+x \right )^{2}}+\frac {21216760 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1029 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {11171040 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{343 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(325\)
default	\(\frac {2 \sqrt {1-2 x}\, \left (452427120 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-224549820 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-583829532 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-568859544 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-246173136 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-39919968 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}[phone] x^{3}-6494356586 x^{2}-4388712958 x -763335749\right )}{147 \left (2+3 x \right )^{\frac {7}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\)	\(491\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/147*(1-2*x)^(1/2)*(452427120*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^
(1/2)*(1-2*x)^(1/2)-224549820*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^(
1/2)*(1-2*x)^(1/2)[phone]*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)-583829532*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)[phone]*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)-568859544*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)[phone]*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)-246173136*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)[phone]*x^6+80431488*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))-39919968*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))[phone]*x^5+34791093396*x^4+8282514768*x^3-6494356586*x^2-4388712958*x-763335749)/(2+3*x)^(
7/2)/(3+5*x)^(3/2)/(-1+2*x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.20, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (6786406800 \, x^{5} + 21944379060 \, x^{4} + 28367736228 \, x^{3} + 18325125498 \, x^{2} + 5915384456 \, x + 763335749\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{147 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/147*(6786406800*x^5 + 21944379060*x^4 + 28367736228*x^3 + 18325125498*x^2 + 5915384456*x + 763335749)*sqrt(5
*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________