3.2840 \(\int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac {31704544 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{66706983 \sqrt {33}}+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}+\frac {924247516 \sqrt {1-2 x} \sqrt {5 x+3}}{733776813 \sqrt {3 x+2}}+\frac {11460644 \sqrt {1-2 x} \sqrt {5 x+3}}{104825259 (3 x+2)^{3/2}}-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{14975037 (3 x+2)^{5/2}}-\frac {251590 \sqrt {1-2 x} \sqrt {5 x+3}}{2139291 (3 x+2)^{7/2}}+\frac {940 \sqrt {1-2 x} \sqrt {5 x+3}}{43659 (3 x+2)^{9/2}}-\frac {924247516 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{66706983 \sqrt {33}} \]

[Out]

-924247516/2201330439*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-31704544/2201330439*Ellip
ticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/231*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2)+940
/43659*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)-251590/2139291*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-3626
66/14975037*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)[phone]/104825259*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(
3/2)[phone]/733776813*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 249, 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 = {98, 150, 152, 158, 113, 119} \[ \frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}+\frac {924247516 \sqrt {1-2 x} \sqrt {5 x+3}}{733776813 \sqrt {3 x+2}}+\frac {11460644 \sqrt {1-2 x} \sqrt {5 x+3}}{104825259 (3 x+2)^{3/2}}-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{14975037 (3 x+2)^{5/2}}-\frac {251590 \sqrt {1-2 x} \sqrt {5 x+3}}{2139291 (3 x+2)^{7/2}}+\frac {940 \sqrt {1-2 x} \sqrt {5 x+3}}{43659 (3 x+2)^{9/2}}-\frac {31704544 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{66706983 \sqrt {33}}-\frac {924247516 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{66706983 \sqrt {33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(940*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(9/2)) - (251590*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2139291*(2 +
3*x)^(7/2)) - (362666*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14975037*(2 + 3*x)^(5/2)) + (11460644*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(104825259*(2 + 3*x)^(3/2)) + (924247516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(733776813*Sqrt[2 + 3*x]) + (2
*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(231*(2 + 3*x)^(11/2)) - (924247516*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(66706983*Sqrt[33]) - (31704544*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(66706983*Sqrt[33]
)

Rule 98

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 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*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))])/b, 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 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*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))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(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 150

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 152

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 158

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}}{\sqrt {1-2 x} (2+3 x)^{13/2}} \, dx &=\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac {2}{231} \int \frac {\left (-540-\frac {1855 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx\\ &=\frac {940 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{9/2}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac {4 \int \frac {-\frac {325685}{4}-\frac {551425 x}{4}}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx}{43659}\\ &=\frac {940 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{9/2}}-\frac {251590 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 (2+3 x)^{7/2}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac {8 \int \frac {-\frac {3890945}{8}-\frac {3144875 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{2139291}\\ &=\frac {940 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{9/2}}-\frac {251590 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac {362666 \sqrt {1-2 x} \sqrt {3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac {16 \int \frac {-\frac {23392455}{8}-\frac {13599975 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{74875185}\\ &=\frac {940 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{9/2}}-\frac {251590 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac {362666 \sqrt {1-2 x} \sqrt {3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac {11460644 \sqrt {1-2 x} \sqrt {3+5 x}}{104825259 (2+3 x)^{3/2}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac {32 \int \frac {-\frac {868793295}{16}+\frac {214887075 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{1572378885}\\ &=\frac {940 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{9/2}}-\frac {251590 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac {362666 \sqrt {1-2 x} \sqrt {3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac {11460644 \sqrt {1-2 x} \sqrt {3+5 x}}{104825259 (2+3 x)^{3/2}}+\frac {924247516 \sqrt {1-2 x} \sqrt {3+5 x}}{733776813 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac {64 \int \frac {-\frac {11051690775}{16}-\frac {17329640925 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{11006652195}\\ &=\frac {940 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{9/2}}-\frac {251590 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac {362666 \sqrt {1-2 x} \sqrt {3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac {11460644 \sqrt {1-2 x} \sqrt {3+5 x}}{104825259 (2+3 x)^{3/2}}+\frac {924247516 \sqrt {1-2 x} \sqrt {3+5 x}}{733776813 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}+\frac {15852272 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{66706983}+\frac {924247516 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{733776813}\\ &=\frac {940 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{9/2}}-\frac {251590 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac {362666 \sqrt {1-2 x} \sqrt {3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac {11460644 \sqrt {1-2 x} \sqrt {3+5 x}}{104825259 (2+3 x)^{3/2}}+\frac {924247516 \sqrt {1-2 x} \sqrt {3+5 x}}{733776813 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac {924247516 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{66706983 \sqrt {33}}-\frac {31704544 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{66706983 \sqrt {33}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.40, size = 112, normalized size = 0.45 \[ \frac {-6417960640 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {48 \sqrt {2-4 x} \sqrt {5 x+3} \left (112296073194 x^5+377569336554 x^4+507518001945 x^3+340525216341 x^2+113962415157 x+15211411193\right )}{(3 x+2)^{11/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{17610643512 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(15211411193 + 113962415157*x + 340525216341*x^2 + 507518001945*x^3 + 3775693
36554*x^4 + 112296073194*x^5))/(2 + 3*x)^(11/2) + 14787960256*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2] - 6417960640*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(17610643512*Sqrt[2])

________________________________________________________________________________________

fricas [F]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(4374*x^8 + 18225*x^7 + 30618*x^6 + 2
4948*x^5 + 7560*x^4 - 3024*x^3 - 3360*x^2 - 1088*x - 128), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.03, size = 599, normalized size = 2.41 \[ \frac {2 \left (3368882195820 x^{7}[phone] x^{6}-112296073194 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{5} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{5} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{5}-374320243980 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{4}-499093658640 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-127213913772 x^{3}-332729105760 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2266497365808 x^{2}-110909701920 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-980027502834 x -14787960256 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-136902700737\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{2201330439 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2201330439*(48736388610*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^5*(5*x+3)^(1/2)*(3*x+2)^(1
/2)*(-2*x+1)^(1/2)-112296073194*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^5*(5*x+3)^(1/2)*(3*x
+2)^(1/2)*(-2*x+1)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^4*(5*x+3)^(1/2
)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-374320243980*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^4*(5*x+3
)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^3*
(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-499093658640*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2)
)*x^3*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66
^(1/2))*x^2*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-332729105760*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/
2*I*66^(1/2))*x^2*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*x^7+48134704800*2^(1/2)*EllipticF(1
/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-110909701920*2^(1/2)*Ellipti
cE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*x^6+64179
60640*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))-14787
960256*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))+1534
7583409266*x^5+8340186467079*x^4-127213913772*x^3-2266497365808*x^2-980027502834*x-136902700737)*(-2*x+1)^(1/2
)*(5*x+3)^(1/2)/(10*x^2+x-3)/(3*x+2)^(11/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{13/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________