3.2000 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^3} \, dx\)

Optimal. Leaf size=201 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}+\frac{23680975 \sqrt{1-2 x}}{168 (5 x+3)}+\frac{522385 \sqrt{1-2 x}}{168 (3 x+2) (5 x+3)^2}+\frac{11243 \sqrt{1-2 x}}{72 (3 x+2)^2 (5 x+3)^2}+\frac{1393 \sqrt{1-2 x}}{108 (3 x+2)^3 (5 x+3)^2}-\frac{8836825 \sqrt{1-2 x}}{378 (5 x+3)^2}+\frac{163363895 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28 \sqrt{21}}-171675 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-8836825*Sqrt[1 - 2*x])/(378*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4*(3 + 5*x)^2) + (1393*Sqrt[1 -
2*x])/(108*(2 + 3*x)^3*(3 + 5*x)^2) + (11243*Sqrt[1 - 2*x])/(72*(2 + 3*x)^2*(3 + 5*x)^2) + (522385*Sqrt[1 - 2
*x])/(168*(2 + 3*x)*(3 + 5*x)^2) + (23680975*Sqrt[1 - 2*x])/(168*(3 + 5*x)) + (163363895*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]])/(28*Sqrt[21]) - 171675*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0898213, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 151, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^2}+\frac{23680975 \sqrt{1-2 x}}{168 (5 x+3)}+\frac{522385 \sqrt{1-2 x}}{168 (3 x+2) (5 x+3)^2}+\frac{11243 \sqrt{1-2 x}}{72 (3 x+2)^2 (5 x+3)^2}+\frac{1393 \sqrt{1-2 x}}{108 (3 x+2)^3 (5 x+3)^2}-\frac{8836825 \sqrt{1-2 x}}{378 (5 x+3)^2}+\frac{163363895 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28 \sqrt{21}}-171675 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8836825*Sqrt[1 - 2*x])/(378*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4*(3 + 5*x)^2) + (1393*Sqrt[1 -
2*x])/(108*(2 + 3*x)^3*(3 + 5*x)^2) + (11243*Sqrt[1 - 2*x])/(72*(2 + 3*x)^2*(3 + 5*x)^2) + (522385*Sqrt[1 - 2
*x])/(168*(2 + 3*x)*(3 + 5*x)^2) + (23680975*Sqrt[1 - 2*x])/(168*(3 + 5*x)) + (163363895*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]])/(28*Sqrt[21]) - 171675*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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 149

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] && IntegerQ[m]

Rule 151

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] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^3} \, dx &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^2}+\frac{1}{12} \int \frac{(265-299 x) \sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^3} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^2}+\frac{1393 \sqrt{1-2 x}}{108 (2+3 x)^3 (3+5 x)^2}-\frac{1}{108} \int \frac{-38107+60891 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^2}+\frac{1393 \sqrt{1-2 x}}{108 (2+3 x)^3 (3+5 x)^2}+\frac{11243 \sqrt{1-2 x}}{72 (2+3 x)^2 (3+5 x)^2}-\frac{\int \frac{-5461015+8263605 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx}{1512}\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^2}+\frac{1393 \sqrt{1-2 x}}{108 (2+3 x)^3 (3+5 x)^2}+\frac{11243 \sqrt{1-2 x}}{72 (2+3 x)^2 (3+5 x)^2}+\frac{522385 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^2}-\frac{\int \frac{-595043015+822756375 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx}{10584}\\ &=-\frac{8836825 \sqrt{1-2 x}}{378 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^2}+\frac{1393 \sqrt{1-2 x}}{108 (2+3 x)^3 (3+5 x)^2}+\frac{11243 \sqrt{1-2 x}}{72 (2+3 x)^2 (3+5 x)^2}+\frac{522385 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^2}+\frac{\int \frac{-42813214290+48991357800 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx}{232848}\\ &=-\frac{8836825 \sqrt{1-2 x}}{378 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^2}+\frac{1393 \sqrt{1-2 x}}{108 (2+3 x)^3 (3+5 x)^2}+\frac{11243 \sqrt{1-2 x}}{72 (2+3 x)^2 (3+5 x)^2}+\frac{522385 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^2}+\frac{23680975 \sqrt{1-2 x}}{168 (3+5 x)}-\frac{\int \frac{-1768565653470+1083120434550 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{2561328}\\ &=-\frac{8836825 \sqrt{1-2 x}}{378 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^2}+\frac{1393 \sqrt{1-2 x}}{108 (2+3 x)^3 (3+5 x)^2}+\frac{11243 \sqrt{1-2 x}}{72 (2+3 x)^2 (3+5 x)^2}+\frac{522385 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^2}+\frac{23680975 \sqrt{1-2 x}}{168 (3+5 x)}-\frac{163363895}{56} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{9442125}{2} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{8836825 \sqrt{1-2 x}}{378 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^2}+\frac{1393 \sqrt{1-2 x}}{108 (2+3 x)^3 (3+5 x)^2}+\frac{11243 \sqrt{1-2 x}}{72 (2+3 x)^2 (3+5 x)^2}+\frac{522385 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^2}+\frac{23680975 \sqrt{1-2 x}}{168 (3+5 x)}+\frac{163363895}{56} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{9442125}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{8836825 \sqrt{1-2 x}}{378 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^2}+\frac{1393 \sqrt{1-2 x}}{108 (2+3 x)^3 (3+5 x)^2}+\frac{11243 \sqrt{1-2 x}}{72 (2+3 x)^2 (3+5 x)^2}+\frac{522385 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^2}+\frac{23680975 \sqrt{1-2 x}}{168 (3+5 x)}+\frac{163363895 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28 \sqrt{21}}-171675 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.15427, size = 105, normalized size = 0.52 \[ \frac{\sqrt{1-2 x} \left (3196931625 x^5+10337268075 x^4+13362164665 x^3+8630749831 x^2+2785562634 x+359378534\right )}{56 (3 x+2)^4 (5 x+3)^2}+\frac{163363895 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28 \sqrt{21}}-171675 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(359378534 + 2785562634*x + 8630749831*x^2 + 13362164665*x^3 + 10337268075*x^4 + 3196931625*x^5
))/(56*(2 + 3*x)^4*(3 + 5*x)^2) + (163363895*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(28*Sqrt[21]) - 171675*Sqrt[55]
*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.012, size = 112, normalized size = 0.6 \begin{align*} -162\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{3170015\, \left ( 1-2\,x \right ) ^{7/2}}{168}}-{\frac{28695733\, \left ( 1-2\,x \right ) ^{5/2}}{216}}+{\frac{202051885\, \left ( 1-2\,x \right ) ^{3/2}}{648}}-{\frac{52696315\,\sqrt{1-2\,x}}{216}} \right ) }+{\frac{163363895\,\sqrt{21}}{588}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+13750\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{339\, \left ( 1-2\,x \right ) ^{3/2}}{10}}+{\frac{3707\,\sqrt{1-2\,x}}{50}} \right ) }-171675\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-162*(3170015/168*(1-2*x)^(7/2)-28695733/216*(1-2*x)^(5/2)[phone]/648*(1-2*x)^(3/2)-52696315/216*(1-2*x)^(1
/2))/(-6*x-4)^4+163363895/588*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+13750*(-339/10*(1-2*x)^(3/2)+3707/5
0*(1-2*x)^(1/2))/(-10*x-6)^2-171675*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.67642, size = 246, normalized size = 1.22 \begin{align*} \frac{171675}{2} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{163363895}{1176} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3196931625 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 36659194275 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 168116119510 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 385408507778 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 441689778145 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 202435240315 \, \sqrt{-2 \, x + 1}}{28 \,{\left (2025 \,{\left (2 \, x - 1\right )}^{6} + 27810 \,{\left (2 \, x - 1\right )}^{5} + 159111 \,{\left (2 \, x - 1\right )}^{4} + 485436 \,{\left (2 \, x - 1\right )}^{3} + 832951 \,{\left (2 \, x - 1\right )}^{2} + 1524292 \, x - 471625\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

171675/2*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 163363895/1176*sqrt(21)*
log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/28*(3196931625*(-2*x + 1)^(11/2) - 36659
194275*(-2*x + 1)^(9/2) + 168116119510*(-2*x + 1)^(7/2) - 385408507778*(-2*x + 1)^(5/2) + 441689778145*(-2*x +
1)^(3/2) - 202435240315*sqrt(-2*x + 1))/(2025*(2*x - 1)^6 + 27810*(2*x - 1)^5 + 159111*(2*x - 1)^4 + 485436*(
2*x - 1)^3 + 832951*(2*x - 1)^2 + 1524292*x - 471625)

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Fricas [A]  time = 1.35019, size = 655, normalized size = 3.26 \begin{align*} \frac{100944900 \, \sqrt{55}{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 163363895 \, \sqrt{21}{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (3196931625 \, x^{5} + 10337268075 \, x^{4} + 13362164665 \, x^{3} + 8630749831 \, x^{2} + 2785562634 \, x + 359378534\right )} \sqrt{-2 \, x + 1}}{1176 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1176*(100944900*sqrt(55)*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log((5*x +
sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 163363895*sqrt(21)*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 52
24*x^2 + 1344*x + 144)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(3196931625*x^5 + 10337268075*x
^4 + 13362164665*x^3 + 8630749831*x^2 + 2785562634*x + 359378534)*sqrt(-2*x + 1))/(2025*x^6 + 7830*x^5 + 12609
*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 2.29542, size = 225, normalized size = 1.12 \begin{align*} \frac{171675}{2} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{163363895}{1176} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{275 \,{\left (1695 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3707 \, \sqrt{-2 \, x + 1}\right )}}{4 \,{\left (5 \, x + 3\right )}^{2}} + \frac{85590405 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 602610393 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 1414363195 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1106622615 \, \sqrt{-2 \, x + 1}}{448 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

171675/2*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 163363895/1176
*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 275/4*(1695*(-2*x + 1)^
(3/2) - 3707*sqrt(-2*x + 1))/(5*x + 3)^2 + 1/448*(85590405*(2*x - 1)^3*sqrt(-2*x + 1) + 602610393*(2*x - 1)^2*
sqrt(-2*x + 1) - 1414363195*(-2*x + 1)^(3/2) + 1106622615*sqrt(-2*x + 1))/(3*x + 2)^4