3.22.77 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx\) [2177]

Optimal. Leaf size=105 \[ -\frac {190}{1617 (1-2 x)^{3/2}}-\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {720}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-190/1617/(1-2*x)^(3/2)+3/7/(1-2*x)^(3/2)/(2+3*x)+720/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-250/13
31*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-1370/41503/(1-2*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {105, 157, 162, 65, 212} \begin {gather*} -\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {190}{1617 (1-2 x)^{3/2}}+\frac {720}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-190/(1617*(1 - 2*x)^(3/2)) - 1370/(41503*Sqrt[1 - 2*x]) + 3/(7*(1 - 2*x)^(3/2)*(2 + 3*x)) + (720*Sqrt[3/7]*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

Rule 65

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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 157

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 162

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 212

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

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx &=\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {1}{7} \int \frac {-10-75 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {190}{1617 (1-2 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}-\frac {2 \int \frac {-555+\frac {4275 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{1617}\\ &=-\frac {190}{1617 (1-2 x)^{3/2}}-\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {4 \int \frac {\frac {55065}{2}-\frac {30825 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{124509}\\ &=-\frac {190}{1617 (1-2 x)^{3/2}}-\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}-\frac {1080}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {625}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {190}{1617 (1-2 x)^{3/2}}-\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {1080}{343} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {625}{121} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {190}{1617 (1-2 x)^{3/2}}-\frac {1370}{41503 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x)}+\frac {720}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 89, normalized size = 0.85 \begin {gather*} \frac {15881-39780 x+24660 x^2}{124509 (1-2 x)^{3/2} (2+3 x)}+\frac {720}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {250}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15881 - 39780*x + 24660*x^2)/(124509*(1 - 2*x)^(3/2)*(2 + 3*x)) + (720*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/343 - (250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]
time = 0.16, size = 72, normalized size = 0.69

method	result	size
derivativedivides	\(-\frac {250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {8}{1617 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {808}{41503 \sqrt {1-2 x}}-\frac {18 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}+\frac {720 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(72\)
default	\(-\frac {250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {8}{1617 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {808}{41503 \sqrt {1-2 x}}-\frac {18 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}+\frac {720 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(72\)
trager	\(\frac {\left (24660 x^{2}-39780 x +15881\right ) \sqrt {1-2 x}}{124509 \left (-1+2 x \right )^{2} \left (2+3 x \right )}+\frac {125 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{1331}-\frac {360 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{2401}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-250/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+8/1617/(1-2*x)^(3/2)+808/41503/(1-2*x)^(1/2)-18/343*(1
-2*x)^(1/2)/(-4/3-2*x)+720/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.53, size = 110, normalized size = 1.05 \begin {gather*} \frac {125}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {360}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (6165 \, {\left (2 \, x - 1\right )}^{2} - 15120 \, x + 9716\right )}}{124509 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 7 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 360/2401*sqrt(21)*log(-(
sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/124509*(6165*(2*x - 1)^2 - 15120*x + 9716)/(3*
(-2*x + 1)^(5/2) - 7*(-2*x + 1)^(3/2))

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Fricas [A]
time = 1.82, size = 142, normalized size = 1.35 \begin {gather*} \frac {900375 \, \sqrt {11} \sqrt {5} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 1437480 \, \sqrt {7} \sqrt {3} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (24660 \, x^{2} - 39780 \, x + 15881\right )} \sqrt {-2 \, x + 1}}{9587193 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9587193*(900375*sqrt(11)*sqrt(5)*(12*x^3 - 4*x^2 - 5*x + 2)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/
(5*x + 3)) + 1437480*sqrt(7)*sqrt(3)*(12*x^3 - 4*x^2 - 5*x + 2)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5
)/(3*x + 2)) + 77*(24660*x^2 - 39780*x + 15881)*sqrt(-2*x + 1))/(12*x^3 - 4*x^2 - 5*x + 2)

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Sympy [C] Result contains complex when optimal does not.
time = 7.17, size = 1352, normalized size = 12.88 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-388962000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(2070833688*(x - 1/2)**(11/2) + 72479
17908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) + 620991360*sqrt(21)*I*(x
- 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 84
55904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) - 310495680*sqrt(21)*I*pi*(x - 1/2)**(11/2)/(20708336
88*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)
) + 194481000*sqrt(55)*I*pi*(x - 1/2)**(11/2)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 84
55904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) - 1361367000*sqrt(55)*I*(x - 1/2)**(9/2)*atan(sqrt(11
0)*sqrt(x - 1/2)/11)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2)
+ 3288407199*(x - 1/2)**(5/2)) + 2173469760*sqrt(21)*I*(x - 1/2)**(9/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(20708
33688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5
/2)) - 1086734880*sqrt(21)*I*pi*(x - 1/2)**(9/2)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) +
8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) + 680683500*sqrt(55)*I*pi*(x - 1/2)**(9/2)/(207083
3688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/
2)) - 1588261500*sqrt(55)*I*(x - 1/2)**(7/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(2070833688*(x - 1/2)**(11/2) +
7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) + 2535714720*sqrt(21)
*I*(x - 1/2)**(7/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2)
+ 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) - 1267857360*sqrt(21)*I*pi*(x - 1/2)**(7/2)/(207
0833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**
(5/2)) + 794130750*sqrt(55)*I*pi*(x - 1/2)**(7/2)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2)
+ 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) - 617657250*sqrt(55)*I*(x - 1/2)**(5/2)*atan(sqrt
(110)*sqrt(x - 1/2)/11)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7
/2) + 3288407199*(x - 1/2)**(5/2)) + 986111280*sqrt(21)*I*(x - 1/2)**(5/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(207
0833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**
(5/2)) - 493055640*sqrt(21)*I*pi*(x - 1/2)**(5/2)/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2)
+ 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) + 308828625*sqrt(55)*I*pi*(x - 1/2)**(5/2)/(20708
33688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5
/2)) + 34178760*sqrt(2)*I*(x - 1/2)**5/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 845590422
6*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) + 58794120*sqrt(2)*I*(x - 1/2)**4/(2070833688*(x - 1/2)**(11
/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) + 611226*sqrt(2
)*I*(x - 1/2)**3/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3
288407199*(x - 1/2)**(5/2)) - 21551376*sqrt(2)*I*(x - 1/2)**2/(2070833688*(x - 1/2)**(11/2) + 7247917908*(x -
1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/2)) + 4067294*sqrt(2)*I*(x - 1/2)/(207083
3688*(x - 1/2)**(11/2) + 7247917908*(x - 1/2)**(9/2) + 8455904226*(x - 1/2)**(7/2) + 3288407199*(x - 1/2)**(5/
2))

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Giac [A]
time = 1.93, size = 116, normalized size = 1.10 \begin {gather*} \frac {125}{1331} \, \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 {360}{2401} \, \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 {16 \, {\left (303 \, x - 190\right )}}{124509 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {27 \, \sqrt {-2 \, x + 1}}{343 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 360/2401*sqrt(
21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/124509*(303*x - 190)/((2*x
- 1)*sqrt(-2*x + 1)) + 27/343*sqrt(-2*x + 1)/(3*x + 2)

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Mupad [B]
time = 0.10, size = 73, normalized size = 0.70 \begin {gather*} \frac {\frac {1370\,{\left (2\,x-1\right )}^2}{41503}-\frac {480\,x}{5929}+\frac {2776}{53361}}{\frac {7\,{\left (1-2\,x\right )}^{3/2}}{3}-{\left (1-2\,x\right )}^{5/2}}+\frac {720\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1370*(2*x - 1)^2)/41503 - (480*x)/5929 + 2776/53361)/((7*(1 - 2*x)^(3/2))/3 - (1 - 2*x)^(5/2)) + (720*21^(1/
2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (250*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1331

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