3.2970 \(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=253 \[ -\frac {65672 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{823543}+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {5 x+3}}{823543 \sqrt {3 x+2}}-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{117649 (3 x+2)^{3/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {5 x+3}}{16807 (3 x+2)^{5/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {5 x+3}}{2401 (3 x+2)^{7/2}}+\frac {220 \sqrt {5 x+3}}{49 \sqrt {1-2 x} (3 x+2)^{7/2}}+\frac {98642 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543} \]

[Out]

11/21*(3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^(7/2)+98642/2470629*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(
1/2))*33^(1/2)-65672/2470629*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+220/49*(3+5*x)^(1/
2)/(2+3*x)^(7/2)/(1-2*x)^(1/2)-4545/2401*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-11433/16807*(1-2*x)^(1/2)*(
3+5*x)^(1/2)/(2+3*x)^(5/2)-33778/117649*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)-98642/823543*(1-2*x)^(1/2)*(
3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 253, 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} \[ \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {5 x+3}}{823543 \sqrt {3 x+2}}-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{117649 (3 x+2)^{3/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {5 x+3}}{16807 (3 x+2)^{5/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {5 x+3}}{2401 (3 x+2)^{7/2}}+\frac {220 \sqrt {5 x+3}}{49 \sqrt {1-2 x} (3 x+2)^{7/2}}-\frac {65672 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}+\frac {98642 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(220*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) - (4545*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2401*(2 + 3*x)^(7
/2)) - (11433*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(16807*(2 + 3*x)^(5/2)) - (33778*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1176
49*(2 + 3*x)^(3/2)) - (98642*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(823543*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(21*(1
- 2*x)^(3/2)*(2 + 3*x)^(7/2)) + (98642*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/823543 -
(65672*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/823543

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}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx &=\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {1}{21} \int \frac {\left (-\frac {345}{2}-315 x\right ) \sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {1}{147} \int \frac {-\frac {34305}{2}-\frac {58275 x}{2}}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {2 \int \frac {-\frac {397335}{4}-\frac {340875 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{7203}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {4 \int \frac {-\frac {1461615}{4}-\frac {2572425 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{252105}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {8 \int \frac {-\frac {4326885}{8}-\frac {3800025 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{5294205}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {16 \int \frac {-\frac {1468575}{8}+\frac {11097225 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{37059435}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {98642 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{823543}+\frac {361196 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{823543}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}+\frac {98642 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}-\frac {65672 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 113, normalized size = 0.45 \[ \frac {2 \left (\sqrt {2} \left (591115 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-49321 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )+\frac {\sqrt {5 x+3} \left (-15980004 x^5-28748088 x^4-7681599 x^3+10746933 x^2+6524789 x+866085\right )}{(1-2 x)^{3/2} (3 x+2)^{7/2}}\right )}{2470629} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[3 + 5*x]*(866085 + 6524789*x + 10746933*x^2 - 7681599*x^3 - 28748088*x^4 - 15980004*x^5))/((1 - 2*x)
^(3/2)*(2 + 3*x)^(7/2)) + Sqrt[2]*(-49321*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 591115*Elliptic
F[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/2470629

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fricas [F]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1944 \, x^{8} + 3564 \, x^{7} + 378 \, x^{6} - 2583 \, x^{5} - 1050 \, x^{4} + 616 \, x^{3} + 336 \, x^{2} - 48 \, x - 32}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(1944*x^8 + 3564*x^7 + 378*x^6 - 2583
*x^5 - 1050*x^4 + 616*x^3 + 336*x^2 - 48*x - 32), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 501, normalized size = 1.98 \[ \frac {2 \sqrt {-2 x +1}\, \left (-79900020 x^{6}-191680452 x^{5}+2663334 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-31920210 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-124652259 x^{4}+3995001 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-47880315 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{3}+887778 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-10640070 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{2}-986420 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x -394568 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4728920 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2598255\right )}{2470629 \left (3 x +2\right )^{\frac {7}{2}} \left (2 x -1\right )^{2} \sqrt {5 x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2470629*(-2*x+1)^(1/2)*(2663334*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)-31920210*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)+3995001*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)-47880315*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)+887778*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)-10640070*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)-986420*2^(1/2)*EllipticE(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]*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)-79900020*x^6-394568*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*Ellipti
cE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))+4728920*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))-191680452*x^5-124652259*x^4+30689868*x^3+64864744*x^2+23904792*x+2598255
)/(3*x+2)^(7/2)/(2*x-1)^2/(5*x+3)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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