3.22.87 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx\) [2187]

Optimal. Leaf size=105 \[ \frac {218}{2541 (1-2 x)^{3/2}}+\frac {3274}{65219 \sqrt {1-2 x}}-\frac {5}{11 (1-2 x)^{3/2} (3+5 x)}-\frac {54}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {1400 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]

[Out]

218/2541/(1-2*x)^(3/2)-5/11/(1-2*x)^(3/2)/(3+5*x)-54/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1400/146
41*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+3274/65219/(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 {3274}{65219 \sqrt {1-2 x}}-\frac {5}{11 (1-2 x)^{3/2} (5 x+3)}+\frac {218}{2541 (1-2 x)^{3/2}}-\frac {54}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {1400 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

218/(2541*(1 - 2*x)^(3/2)) + 3274/(65219*Sqrt[1 - 2*x]) - 5/(11*(1 - 2*x)^(3/2)*(3 + 5*x)) - (54*Sqrt[3/7]*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (1400*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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) (3+5 x)^2} \, dx &=-\frac {5}{11 (1-2 x)^{3/2} (3+5 x)}-\frac {1}{11} \int \frac {-17-75 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=\frac {218}{2541 (1-2 x)^{3/2}}-\frac {5}{11 (1-2 x)^{3/2} (3+5 x)}+\frac {2 \int \frac {\frac {3}{2}+\frac {4905 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{2541}\\ &=\frac {218}{2541 (1-2 x)^{3/2}}+\frac {3274}{65219 \sqrt {1-2 x}}-\frac {5}{11 (1-2 x)^{3/2} (3+5 x)}-\frac {4 \int \frac {\frac {58701}{4}-\frac {73665 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{195657}\\ &=\frac {218}{2541 (1-2 x)^{3/2}}+\frac {3274}{65219 \sqrt {1-2 x}}-\frac {5}{11 (1-2 x)^{3/2} (3+5 x)}+\frac {81}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {3500 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac {218}{2541 (1-2 x)^{3/2}}+\frac {3274}{65219 \sqrt {1-2 x}}-\frac {5}{11 (1-2 x)^{3/2} (3+5 x)}-\frac {81}{49} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {3500 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1331}\\ &=\frac {218}{2541 (1-2 x)^{3/2}}+\frac {3274}{65219 \sqrt {1-2 x}}-\frac {5}{11 (1-2 x)^{3/2} (3+5 x)}-\frac {54}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {1400 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 89, normalized size = 0.85 \begin {gather*} -\frac {9111-74108 x+98220 x^2}{195657 (1-2 x)^{3/2} (3+5 x)}-\frac {54}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {1400 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/195657*(9111 - 74108*x + 98220*x^2)/((1 - 2*x)^(3/2)*(3 + 5*x)) - (54*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/49 + (1400*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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

method	result	size
derivativedivides	\(\frac {50 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {1400 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {8}{2541 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {824}{65219 \sqrt {1-2 x}}-\frac {54 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(72\)
default	\(\frac {50 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {1400 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {8}{2541 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {824}{65219 \sqrt {1-2 x}}-\frac {54 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(72\)
trager	\(-\frac {\left (98220 x^{2}-74108 x +9111\right ) \sqrt {1-2 x}}{195657 \left (-1+2 x \right )^{2} \left (3+5 x \right )}-\frac {700 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{14641}+\frac {27 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{343}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/1331*(1-2*x)^(1/2)/(-6/5-2*x)+1400/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+8/2541/(1-2*x)^(3/2)
+824/65219/(1-2*x)^(1/2)-54/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.49, size = 110, normalized size = 1.05 \begin {gather*} -\frac {700}{14641} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {27}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (24555 \, {\left (2 \, x - 1\right )}^{2} + 24112 \, x - 15444\right )}}{195657 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\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)/(3+5*x)^2,x, algorithm="maxima")

[Out]

-700/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 27/343*sqrt(21)*log(-(
sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/195657*(24555*(2*x - 1)^2 + 24112*x - 15444)/(
5*(-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))

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Fricas [A]
time = 1.61, size = 142, normalized size = 1.35 \begin {gather*} \frac {720300 \, \sqrt {11} \sqrt {5} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 1185921 \, \sqrt {7} \sqrt {3} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (98220 \, x^{2} - 74108 \, x + 9111\right )} \sqrt {-2 \, x + 1}}{15065589 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15065589*(720300*sqrt(11)*sqrt(5)*(20*x^3 - 8*x^2 - 7*x + 3)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8
)/(5*x + 3)) + 1185921*sqrt(7)*sqrt(3)*(20*x^3 - 8*x^2 - 7*x + 3)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x -
5)/(3*x + 2)) - 77*(98220*x^2 - 74108*x + 9111)*sqrt(-2*x + 1))/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [C] Result contains complex when optimal does not.
time = 6.80, 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)/(3+5*x)**2,x)

[Out]

1440600000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(15065589000*(x - 1/2)**(11/2) + 4971
6443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) - 2371842000*sqrt(21)*
I*(x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/
2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) - 720300000*sqrt(55)*I*pi*(x - 1/2)**(11/2)/
(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x
- 1/2)**(5/2)) + 1185921000*sqrt(21)*I*pi*(x - 1/2)**(11/2)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x -
1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) + 4753980000*sqrt(55)*I*(x - 1/2)**
(9/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 5468808
8070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) - 7827078600*sqrt(21)*I*(x - 1/2)**(9/2)*atan(sqrt(42)*s
qrt(x - 1/2)/7)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) +
20052298959*(x - 1/2)**(5/2)) - 2376990000*sqrt(55)*I*pi*(x - 1/2)**(9/2)/(15065589000*(x - 1/2)**(11/2) + 49
716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) + 3913539300*sqrt(21
)*I*pi*(x - 1/2)**(9/2)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)*
*(7/2) + 20052298959*(x - 1/2)**(5/2)) + 5229378000*sqrt(55)*I*(x - 1/2)**(7/2)*atan(sqrt(110)*sqrt(x - 1/2)/1
1)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*
(x - 1/2)**(5/2)) - 8609786460*sqrt(21)*I*(x - 1/2)**(7/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(15065589000*(x - 1/
2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) - 261
4689000*sqrt(55)*I*pi*(x - 1/2)**(7/2)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 5468808
8070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) + 4304893230*sqrt(21)*I*pi*(x - 1/2)**(7/2)/(15065589000
*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2
)) + 1917438600*sqrt(55)*I*(x - 1/2)**(5/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(15065589000*(x - 1/2)**(11/2) +
49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) - 3156921702*sqrt(
21)*I*(x - 1/2)**(5/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**
(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) - 958719300*sqrt(55)*I*pi*(x - 1/2)**(5/2
)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(
x - 1/2)**(5/2)) + 1578460851*sqrt(21)*I*pi*(x - 1/2)**(5/2)/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x -
1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) - 378147000*sqrt(2)*I*(x - 1/2)**5
/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x
- 1/2)**(5/2)) - 924754600*sqrt(2)*I*(x - 1/2)**4/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/
2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) - 648742710*sqrt(2)*I*(x - 1/2)**3/(15065589
000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(
5/2)) - 83629392*sqrt(2)*I*(x - 1/2)**2/(15065589000*(x - 1/2)**(11/2) + 49716443700*(x - 1/2)**(9/2) + 546880
88070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2)) + 15782998*sqrt(2)*I*(x - 1/2)/(15065589000*(x - 1/2)**
(11/2) + 49716443700*(x - 1/2)**(9/2) + 54688088070*(x - 1/2)**(7/2) + 20052298959*(x - 1/2)**(5/2))

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Giac [A]
time = 1.59, size = 116, normalized size = 1.10 \begin {gather*} -\frac {700}{14641} \, \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 {27}{343} \, \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 (309 \, x - 193\right )}}{195657 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {125 \, \sqrt {-2 \, x + 1}}{1331 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-700/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 27/343*sqrt(
21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/195657*(309*x - 193)/((2*x
- 1)*sqrt(-2*x + 1)) - 125/1331*sqrt(-2*x + 1)/(5*x + 3)

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Mupad [B]
time = 0.10, size = 74, normalized size = 0.70 \begin {gather*} \frac {1400\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641}-\frac {54\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {4384\,x}{88935}+\frac {3274\,{\left (2\,x-1\right )}^2}{65219}-\frac {936}{29645}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1400*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641 - (54*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))
/343 - ((4384*x)/88935 + (3274*(2*x - 1)^2)/65219 - 936/29645)/((11*(1 - 2*x)^(3/2))/5 - (1 - 2*x)^(5/2))

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