3.2795 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=249 \[ -\frac {697995152 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{194481 \sqrt {33}}+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}+\frac {23204503328 \sqrt {5 x+3} \sqrt {1-2 x}}{2139291 \sqrt {3 x+2}}+\frac {333930952 \sqrt {5 x+3} \sqrt {1-2 x}}{305613 (3 x+2)^{3/2}}+\frac {7173272 \sqrt {5 x+3} \sqrt {1-2 x}}{43659 (3 x+2)^{5/2}}+\frac {171004 \sqrt {5 x+3} \sqrt {1-2 x}}{6237 (3 x+2)^{7/2}}+\frac {4508 \sqrt {5 x+3} \sqrt {1-2 x}}{891 (3 x+2)^{9/2}}-\frac {23204503328 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}} \]

[Out]

-23204503328/6417873*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-697995152/6417873*Elliptic
F(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+14/33*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(11/2)+4508/8
91*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)+171004/6237*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+7173272/436
59*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)[phone]/305613*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+232045
03328/2139291*(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 = 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} \[ \frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}+\frac {23204503328 \sqrt {5 x+3} \sqrt {1-2 x}}{2139291 \sqrt {3 x+2}}+\frac {333930952 \sqrt {5 x+3} \sqrt {1-2 x}}{305613 (3 x+2)^{3/2}}+\frac {7173272 \sqrt {5 x+3} \sqrt {1-2 x}}{43659 (3 x+2)^{5/2}}+\frac {171004 \sqrt {5 x+3} \sqrt {1-2 x}}{6237 (3 x+2)^{7/2}}+\frac {4508 \sqrt {5 x+3} \sqrt {1-2 x}}{891 (3 x+2)^{9/2}}-\frac {697995152 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}-\frac {23204503328 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) + (4508*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(891*(2 + 3*x)^(
9/2)) + (171004*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6237*(2 + 3*x)^(7/2)) + (7173272*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4
3659*(2 + 3*x)^(5/2)) + (333930952*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(305613*(2 + 3*x)^(3/2)) + (23204503328*Sqrt[1
- 2*x]*Sqrt[3 + 5*x])/(2139291*Sqrt[2 + 3*x]) - (23204503328*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(194481*Sqrt[33]) - (697995152*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(194481*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 {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {(227-223 x) \sqrt {1-2 x}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {-\frac {49835}{2}+37438 x}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}-\frac {8 \int \frac {-\frac {5473405}{2}+\frac {7481425 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{43659}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}-\frac {16 \int \frac {-\frac {833286615}{4}+\frac {470745975 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1528065}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}-\frac {32 \int \frac {-9037592970+\frac {21914218725 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{32089365}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}-\frac {64 \int \frac {-\frac {964063843575}{8}-\frac {380698882725 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{224625555}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}+\frac {348997576 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{194481}+\frac {23204503328 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2139291}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}-\frac {23204503328 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}-\frac {697995152 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}\\ \end {align*}

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Mathematica [A]  time = 0.42, size = 115, normalized size = 0.46 \[ \frac {16 \sqrt {2} \left (2900562916 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-1460947915 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {12 \sqrt {1-2 x} \sqrt {5 x+3} \left (2819347154352 x^5+9492493272732 x^4+12787628716260 x^3+8615827181322 x^2+2903435279352 x+391506734113\right )}{(3 x+2)^{11/2}}}{12835746} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((12*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(391506734113 + 2903435279352*x + 8615827181322*x^2 + 12787628716260*x^3 + 94
92493272732*x^4 + 2819347154352*x^5))/(2 + 3*x)^(11/2) + 16*Sqrt[2]*(2900562916*EllipticE[ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]], -33/2] - 1460947915*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/12835746

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fricas [F]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(10935*x^8 + 57591*x^7 + 132678*x^6 + 17
4636*x^5 + 143640*x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 599, normalized size = 2.41 \[ \frac {2 \left (84580414630560 x^{7}[phone] x^{6}-2819347154352 \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}-9397823847840 \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}-12530431797120 \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 )-2138118521814 x^{3}-8353621198080 \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 )-57086936770452 x^{2}-2784540399360 \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 )-24956397311829 x -371272053248 \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 )-3523560607017\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{6417873 \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((-2*x+1)^(5/2)/(3*x+2)^(13/2)/(5*x+3)^(1/2),x)

[Out]

2/6417873*(1420041373380*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)-2819347154352*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)-9397823847840*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)-12530431797120*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)-8353621198080*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+1402509998400*2^(1/2)*E
llipticF(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)-2784540399360*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]
16*x^6+187001333120*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*6
6^(1/2))-371272053248*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))[phone]*x^5+211405262133852*x^4-2138118521814*x^3-57086936770452*x^2-24956397311829*x-35235
60607017)*(5*x+3)^(1/2)*(-2*x+1)^(1/2)/(10*x^2+x-3)/(3*x+2)^(11/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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