3.1979 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx\)

Optimal. Leaf size=173 \[ \frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac {736065535 \sqrt {1-2 x}}{49392 (3 x+2)}+\frac {31700335 \sqrt {1-2 x}}{21168 (3 x+2)^2}+\frac {302651 \sqrt {1-2 x}}{1512 (3 x+2)^3}+\frac {2165 \sqrt {1-2 x}}{72 (3 x+2)^4}+\frac {91 \sqrt {1-2 x}}{18 (3 x+2)^5}+\frac {25388847535 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24696 \sqrt {21}}-30250 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

7/18*(1-2*x)^(3/2)/(2+3*x)^6+25388847535/518616*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-30250*arctanh(1/1
1*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+91/18*(1-2*x)^(1/2)/(2+3*x)^5+2165/72*(1-2*x)^(1/2)/(2+3*x)^4+302651/1512*(
1-2*x)^(1/2)/(2+3*x)^3+31700335/21168*(1-2*x)^(1/2)/(2+3*x)^2+736065535/49392*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.08, antiderivative size = 173, 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} \[ \frac {7 (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac {736065535 \sqrt {1-2 x}}{49392 (3 x+2)}+\frac {31700335 \sqrt {1-2 x}}{21168 (3 x+2)^2}+\frac {302651 \sqrt {1-2 x}}{1512 (3 x+2)^3}+\frac {2165 \sqrt {1-2 x}}{72 (3 x+2)^4}+\frac {91 \sqrt {1-2 x}}{18 (3 x+2)^5}+\frac {25388847535 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24696 \sqrt {21}}-30250 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^7*(3 + 5*x)),x]

[Out]

(7*(1 - 2*x)^(3/2))/(18*(2 + 3*x)^6) + (91*Sqrt[1 - 2*x])/(18*(2 + 3*x)^5) + (2165*Sqrt[1 - 2*x])/(72*(2 + 3*x
)^4) + (302651*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^3) + (31700335*Sqrt[1 - 2*x])/(21168*(2 + 3*x)^2) + (736065535*S
qrt[1 - 2*x])/(49392*(2 + 3*x)) + (25388847535*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(24696*Sqrt[21]) - 30250*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)^7 (3+5 x)} \, dx &=\frac {7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac {1}{18} \int \frac {(261-291 x) \sqrt {1-2 x}}{(2+3 x)^6 (3+5 x)} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac {91 \sqrt {1-2 x}}{18 (2+3 x)^5}-\frac {1}{270} \int \frac {-36765+58515 x}{\sqrt {1-2 x} (2+3 x)^5 (3+5 x)} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac {91 \sqrt {1-2 x}}{18 (2+3 x)^5}+\frac {2165 \sqrt {1-2 x}}{72 (2+3 x)^4}-\frac {\int \frac {-5288535+7956375 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx}{7560}\\ &=\frac {7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac {91 \sqrt {1-2 x}}{18 (2+3 x)^5}+\frac {2165 \sqrt {1-2 x}}{72 (2+3 x)^4}+\frac {302651 \sqrt {1-2 x}}{1512 (2+3 x)^3}-\frac {\int \frac {-579872475+794458875 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx}{158760}\\ &=\frac {7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac {91 \sqrt {1-2 x}}{18 (2+3 x)^5}+\frac {2165 \sqrt {1-2 x}}{72 (2+3 x)^4}+\frac {302651 \sqrt {1-2 x}}{1512 (2+3 x)^3}+\frac {31700335 \sqrt {1-2 x}}{21168 (2+3 x)^2}-\frac {\int \frac {-44001529425+49928027625 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{2222640}\\ &=\frac {7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac {91 \sqrt {1-2 x}}{18 (2+3 x)^5}+\frac {2165 \sqrt {1-2 x}}{72 (2+3 x)^4}+\frac {302651 \sqrt {1-2 x}}{1512 (2+3 x)^3}+\frac {31700335 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {736065535 \sqrt {1-2 x}}{49392 (2+3 x)}-\frac {\int \frac {-1892960179425+1159303217625 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{15558480}\\ &=\frac {7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac {91 \sqrt {1-2 x}}{18 (2+3 x)^5}+\frac {2165 \sqrt {1-2 x}}{72 (2+3 x)^4}+\frac {302651 \sqrt {1-2 x}}{1512 (2+3 x)^3}+\frac {31700335 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {736065535 \sqrt {1-2 x}}{49392 (2+3 x)}-\frac {25388847535 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{49392}+831875 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac {91 \sqrt {1-2 x}}{18 (2+3 x)^5}+\frac {2165 \sqrt {1-2 x}}{72 (2+3 x)^4}+\frac {302651 \sqrt {1-2 x}}{1512 (2+3 x)^3}+\frac {31700335 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {736065535 \sqrt {1-2 x}}{49392 (2+3 x)}+\frac {25388847535 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{49392}-831875 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac {91 \sqrt {1-2 x}}{18 (2+3 x)^5}+\frac {2165 \sqrt {1-2 x}}{72 (2+3 x)^4}+\frac {302651 \sqrt {1-2 x}}{1512 (2+3 x)^3}+\frac {31700335 \sqrt {1-2 x}}{21168 (2+3 x)^2}+\frac {736065535 \sqrt {1-2 x}}{49392 (2+3 x)}+\frac {25388847535 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24696 \sqrt {21}}-30250 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 98, normalized size = 0.57 \[ \frac {\sqrt {1-2 x} \left (178863925005 x^5+602204446665 x^4+811194684822 x^3+546491397114 x^2+184131053992 x+24823128464\right )}{49392 (3 x+2)^6}+\frac {25388847535 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{24696 \sqrt {21}}-30250 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^7*(3 + 5*x)),x]

[Out]

(Sqrt[1 - 2*x]*(24823128464 + 184131053992*x + 546491397114*x^2 + 811194684822*x^3 + 602204446665*x^4 + 178863
925005*x^5))/(49392*(2 + 3*x)^6) + (25388847535*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(24696*Sqrt[21]) - 30250*Sqr
t[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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fricas [A]  time = 0.80, size = 190, normalized size = 1.10 \[ \frac {15688134000 \, \sqrt {55} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 25388847535 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (178863925005 \, x^{5} + 602204446665 \, x^{4} + 811194684822 \, x^{3} + 546491397114 \, x^{2} + 184131053992 \, x + 24823128464\right )} \sqrt {-2 \, x + 1}}{1037232 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x),x, algorithm="fricas")

[Out]

1/1037232*(15688134000*sqrt(55)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((5*x +
sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 25388847535*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 216
0*x^2 + 576*x + 64)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(178863925005*x^5 + 602204446665*x
^4 + 811194684822*x^3 + 546491397114*x^2 + 184131053992*x + 24823128464)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 +
4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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giac [A]  time = 0.94, size = 171, normalized size = 0.99 \[ 15125 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {25388847535}{1037232} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {178863925005 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 2098728518355 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 9851053562658 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 23121360004806 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 27136250633905 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 12740419709255 \, \sqrt {-2 \, x + 1}}{1580544 \, {\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x),x, algorithm="giac")

[Out]

15125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 25388847535/10372
32*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1580544*(1788639250
05*(2*x - 1)^5*sqrt(-2*x + 1) + 2098728518355*(2*x - 1)^4*sqrt(-2*x + 1) + 9851053562658*(2*x - 1)^3*sqrt(-2*x
+ 1) + 23121360004806*(2*x - 1)^2*sqrt(-2*x + 1) - 27136250633905*(-2*x + 1)^(3/2) + 12740419709255*sqrt(-2*x
+ 1))/(3*x + 2)^6

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maple [A]  time = 0.01, size = 102, normalized size = 0.59 \[ \frac {25388847535 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{518616}-30250 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )-\frac {1458 \left (\frac {736065535 \left (-2 x +1\right )^{\frac {11}{2}}}{148176}-\frac {11104383695 \left (-2 x +1\right )^{\frac {9}{2}}}{190512}+\frac {1240999441 \left (-2 x +1\right )^{\frac {7}{2}}}{4536}-\frac {3744956269 \left (-2 x +1\right )^{\frac {5}{2}}}{5832}+\frac {79114433335 \left (-2 x +1\right )^{\frac {3}{2}}}{104976}-\frac {37144080785 \sqrt {-2 x +1}}{104976}\right )}{\left (-6 x -4\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)/(3*x+2)^7/(5*x+3),x)

[Out]

-30250*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)-1458*(736065535/148176*(-2*x+1)^(11/2)-11104383695/19051
2*(-2*x+1)^(9/2)[phone]/4536*(-2*x+1)^(7/2)-3744956269/5832*(-2*x+1)^(5/2)[phone]/104976*(-2*x+1)^(3/
2)-37144080785/104976*(-2*x+1)^(1/2))/(-6*x-4)^6+25388847535/518616*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1
/2)

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maxima [A]  time = 1.29, size = 182, normalized size = 1.05 \[ 15125 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {25388847535}{1037232} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {178863925005 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2098728518355 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 9851053562658 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 23121360004806 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 27136250633905 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 12740419709255 \, \sqrt {-2 \, x + 1}}{24696 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^7/(3+5*x),x, algorithm="maxima")

[Out]

15125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 25388847535/1037232*sqrt(21
)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/24696*(178863925005*(-2*x + 1)^(11/2)
- 2098728518355*(-2*x + 1)^(9/2) + 9851053562658*(-2*x + 1)^(7/2) - 23121360004806*(-2*x + 1)^(5/2) + 27136250
633905*(-2*x + 1)^(3/2) - 12740419709255*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1
)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

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mupad [B]  time = 1.22, size = 143, normalized size = 0.83 \[ \frac {25388847535\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{518616}-30250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {37144080785\,\sqrt {1-2\,x}}{52488}-\frac {79114433335\,{\left (1-2\,x\right )}^{3/2}}{52488}+\frac {3744956269\,{\left (1-2\,x\right )}^{5/2}}{2916}-\frac {1240999441\,{\left (1-2\,x\right )}^{7/2}}{2268}+\frac {11104383695\,{\left (1-2\,x\right )}^{9/2}}{95256}-\frac {736065535\,{\left (1-2\,x\right )}^{11/2}}{74088}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)/((3*x + 2)^7*(5*x + 3)),x)

[Out]

(25388847535*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/518616 - 30250*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(
1/2))/11) + ((37144080785*(1 - 2*x)^(1/2))/52488 - (79114433335*(1 - 2*x)^(3/2))/52488 + (3744956269*(1 - 2*x)
^(5/2))/2916 - (1240999441*(1 - 2*x)^(7/2))/2268 + (11104383695*(1 - 2*x)^(9/2))/95256 - (736065535*(1 - 2*x)^
(11/2))/74088)/((67228*x)/81 + (12005*(2*x - 1)^2)/27 + (6860*(2*x - 1)^3)/27 + (245*(2*x - 1)^4)/3 + 14*(2*x
- 1)^5 + (2*x - 1)^6 - 184877/729)

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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)**7/(3+5*x),x)

[Out]

Timed out

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