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]

7/12*(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^2+163363895/588*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-171675*arcta
nh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-8836825/378*(1-2*x)^(1/2)/(3+5*x)^2+1393/108*(1-2*x)^(1/2)/(2+3*x)^3/
(3+5*x)^2+11243/72*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^2+522385/168*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+23680975/168*(
1-2*x)^(1/2)/(3+5*x)

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Rubi [A]  time = 0.09, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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 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 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*}

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Mathematica [A]  time = 0.21, 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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fricas [A]  time = 1.39, size = 190, normalized size = 0.95 \[ \frac {100944900 \, \sqrt {55} {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 163363895 \, \sqrt {21} {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (3196931625 \, x^{5} + 10337268075 \, x^{4} + 13362164665 \, x^{3} + 8630749831 \, x^{2} + 2785562634 \, x + 359378534\right )} \sqrt {-2 \, x + 1}}{1176 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]

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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giac [A]  time = 1.02, size = 167, normalized size = 0.83 \[ \frac {171675}{2} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {163363895}{1176} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {275 \, {\left (1695 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 3707 \, \sqrt {-2 \, x + 1}\right )}}{4 \, {\left (5 \, x + 3\right )}^{2}} + \frac {85590405 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 602610393 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 1414363195 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1106622615 \, \sqrt {-2 \, x + 1}}{448 \, {\left (3 \, x + 2\right )}^{4}} \]

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

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maple [A]  time = 0.01, size = 112, normalized size = 0.56 \[ \frac {163363895 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{588}-171675 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )+\frac {-466125 \left (-2 x +1\right )^{\frac {3}{2}}+1019425 \sqrt {-2 x +1}}{\left (-10 x -6\right )^{2}}-\frac {162 \left (\frac {3170015 \left (-2 x +1\right )^{\frac {7}{2}}}{168}-\frac {28695733 \left (-2 x +1\right )^{\frac {5}{2}}}{216}+\frac {202051885 \left (-2 x +1\right )^{\frac {3}{2}}}{648}-\frac {52696315 \sqrt {-2 x +1}}{216}\right )}{\left (-6 x -4\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13750*(-339/10*(-2*x+1)^(3/2)+3707/50*(-2*x+1)^(1/2))/(-10*x-6)^2-171675*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))
*55^(1/2)-162*(3170015/168*(-2*x+1)^(7/2)-28695733/216*(-2*x+1)^(5/2)[phone]/648*(-2*x+1)^(3/2)-52696315/21
6*(-2*x+1)^(1/2))/(-6*x-4)^4+163363895/588*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.24, size = 182, normalized size = 0.91 \[ \frac {171675}{2} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {163363895}{1176} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3196931625 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 36659194275 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 168116119510 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 385408507778 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 441689778145 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 202435240315 \, \sqrt {-2 \, x + 1}}{28 \, {\left (2025 \, {\left (2 \, x - 1\right )}^{6} + 27810 \, {\left (2 \, x - 1\right )}^{5} + 159111 \, {\left (2 \, x - 1\right )}^{4} + 485436 \, {\left (2 \, x - 1\right )}^{3} + 832951 \, {\left (2 \, x - 1\right )}^{2} + 1524292 \, x - 471625\right )}} \]

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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mupad [B]  time = 1.22, size = 143, normalized size = 0.71 \[ \frac {\frac {5783864009\,\sqrt {1-2\,x}}{1620}-\frac {12619707947\,{\left (1-2\,x\right )}^{3/2}}{1620}+\frac {27529179127\,{\left (1-2\,x\right )}^{5/2}}{4050}-\frac {16811611951\,{\left (1-2\,x\right )}^{7/2}}{5670}+\frac {488789257\,{\left (1-2\,x\right )}^{9/2}}{756}-\frac {4736195\,{\left (1-2\,x\right )}^{11/2}}{84}}{\frac {1524292\,x}{2025}+\frac {832951\,{\left (2\,x-1\right )}^2}{2025}+\frac {161812\,{\left (2\,x-1\right )}^3}{675}+\frac {5893\,{\left (2\,x-1\right )}^4}{75}+\frac {206\,{\left (2\,x-1\right )}^5}{15}+{\left (2\,x-1\right )}^6-\frac {18865}{81}}+\frac {163363895\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{588}-171675\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5783864009*(1 - 2*x)^(1/2))/1620 - (12619707947*(1 - 2*x)^(3/2))/1620 + (27529179127*(1 - 2*x)^(5/2))/4050 -
(16811611951*(1 - 2*x)^(7/2))/5670 + (488789257*(1 - 2*x)^(9/2))/756 - (4736195*(1 - 2*x)^(11/2))/84)/((15242
92*x)/2025 + (832951*(2*x - 1)^2)/2025 + (161812*(2*x - 1)^3)/675 + (5893*(2*x - 1)^4)/75 + (206*(2*x - 1)^5)/
15 + (2*x - 1)^6 - 18865/81) + (163363895*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/588 - 171675*55^(1/2)*
atanh((55^(1/2)*(1 - 2*x)^(1/2))/11)

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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)**5/(3+5*x)**3,x)

[Out]

Timed out

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