3.20.27 \(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx\) [1927]

Optimal. Leaf size=178 \[ -\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}+\frac {813716}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-112875 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

813716/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-112875/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
-176065/126*(1-2*x)^(1/2)/(3+5*x)^2+7/9*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^2+28/3*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)
^2+1301/7*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+117955/14*(1-2*x)^(1/2)/(3+5*x)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 156, 162, 65, 212} \begin {gather*} \frac {117955 \sqrt {1-2 x}}{14 (5 x+3)}-\frac {176065 \sqrt {1-2 x}}{126 (5 x+3)^2}+\frac {1301 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {28 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}+\frac {813716}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-112875 \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 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

(-176065*Sqrt[1 - 2*x])/(126*(3 + 5*x)^2) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*x)^2) + (28*Sqrt[1 - 2*x])
/(3*(2 + 3*x)^2*(3 + 5*x)^2) + (1301*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^2) + (117955*Sqrt[1 - 2*x])/(14*(3
+ 5*x)) + (813716*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - 112875*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]]

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 100

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 156

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] && ILtQ[m, -1]

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 {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {1}{9} \int \frac {190-303 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1}{126} \int \frac {27202-41160 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {1}{882} \int \frac {2963912-4098150 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}-\frac {\int \frac {213252732-244026090 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{19404}\\ &=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}+\frac {\int \frac {8809230276-5395025790 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{213444}\\ &=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}-\frac {1220574}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {564375}{2} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}+\frac {1220574}{7} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {564375}{2} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}+\frac {813716}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-112875 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.40, size = 104, normalized size = 0.58 \begin {gather*} \frac {\sqrt {1-2 x} \left (2685098+16784696 x+39307638 x^2+40874010 x^3+15923925 x^4\right )}{14 (2+3 x)^3 (3+5 x)^2}+\frac {813716}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-112875 \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 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

(Sqrt[1 - 2*x]*(2685098 + 16784696*x + 39307638*x^2 + 40874010*x^3 + 15923925*x^4))/(14*(2 + 3*x)^3*(3 + 5*x)^
2) + (813716*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - 112875*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]]

________________________________________________________________________________________

Maple [A]
time = 0.17, size = 103, normalized size = 0.58

method	result	size
risch	\(-\frac {31847850 x^{5}[phone] x^{4}[phone] x^{3}-5738246 x^{2}-11414500 x -2685098}{14 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (2+3 x \right )^{3}}-\frac {112875 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {813716 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\)	\(86\)
derivativedivides	\(\frac {-33625 \left (1-2 x \right )^{\frac {3}{2}}+73425 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {112875 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {324 \left (\frac {3544 \left (1-2 x \right )^{\frac {5}{2}}}{21}-\frac {21418 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {25172 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {813716 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\)	\(103\)
default	\(\frac {-33625 \left (1-2 x \right )^{\frac {3}{2}}+73425 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {112875 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {324 \left (\frac {3544 \left (1-2 x \right )^{\frac {5}{2}}}{21}-\frac {21418 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {25172 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {813716 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\)	\(103\)
trager	\(\frac {\left (15923925 x^{4}[phone] x^{3}[phone] x^{2}[phone] x +2685098\right ) \sqrt {1-2 x}}{14 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}+\frac {406858 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{49}-\frac {112875 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{22}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^3,x,method=_RETURNVERBOSE)

[Out]

2500*(-269/20*(1-2*x)^(3/2)+2937/100*(1-2*x)^(1/2))/(-6-10*x)^2-112875/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
*55^(1/2)-324*(3544/21*(1-2*x)^(5/2)-21418/27*(1-2*x)^(3/2)+25172/27*(1-2*x)^(1/2))/(-4-6*x)^3+813716/49*arcta
nh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 164, normalized size = 0.92 \begin {gather*} \frac {112875}{22} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {406858}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {15923925 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 145443720 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 498018162 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 757678432 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 432141633 \, \sqrt {-2 \, x + 1}}{7 \, {\left (675 \, {\left (2 \, x - 1\right )}^{5} + 7695 \, {\left (2 \, x - 1\right )}^{4} + 35082 \, {\left (2 \, x - 1\right )}^{3} + 79954 \, {\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

112875/22*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 406858/49*sqrt(21)*log(
-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/7*(15923925*(-2*x + 1)^(9/2) - 145443720*(-2
*x + 1)^(7/2) + 498018162*(-2*x + 1)^(5/2) - 757678432*(-2*x + 1)^(3/2) + 432141633*sqrt(-2*x + 1))/(675*(2*x
- 1)^5 + 7695*(2*x - 1)^4 + 35082*(2*x - 1)^3 + 79954*(2*x - 1)^2 + 182182*x - 49588)

________________________________________________________________________________________

Fricas [A]
time = 1.02, size = 182, normalized size = 1.02 \begin {gather*} \frac {5530875 \, \sqrt {11} \sqrt {5} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 8950876 \, \sqrt {7} \sqrt {3} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (15923925 \, x^{4} + 40874010 \, x^{3} + 39307638 \, x^{2} + 16784696 \, x + 2685098\right )} \sqrt {-2 \, x + 1}}{1078 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/1078*(5530875*sqrt(11)*sqrt(5)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log((sqrt(11)*sqrt(5)
*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 8950876*sqrt(7)*sqrt(3)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 56
4*x + 72)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(15923925*x^4 + 40874010*x^3 + 39307
638*x^2 + 16784696*x + 2685098)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(3/2)/(2+3*x)**4/(3+5*x)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.63, size = 151, normalized size = 0.85 \begin {gather*} \frac {112875}{22} \, \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 {406858}{49} \, \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 {25 \, {\left (1345 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2937 \, \sqrt {-2 \, x + 1}\right )}}{4 \, {\left (5 \, x + 3\right )}^{2}} + \frac {3 \, {\left (15948 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 74963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 88102 \, \sqrt {-2 \, x + 1}\right )}}{7 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")

[Out]

112875/22*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 406858/49*sqr
t(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 25/4*(1345*(-2*x + 1)^(3/2)
- 2937*sqrt(-2*x + 1))/(5*x + 3)^2 + 3/7*(15948*(2*x - 1)^2*sqrt(-2*x + 1) - 74963*(-2*x + 1)^(3/2) + 88102*s
qrt(-2*x + 1))/(3*x + 2)^3

________________________________________________________________________________________

Mupad [B]
time = 0.10, size = 125, normalized size = 0.70 \begin {gather*} \frac {813716\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {112875\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {6859391\,\sqrt {1-2\,x}}{75}-\frac {108239776\,{\left (1-2\,x\right )}^{3/2}}{675}+\frac {166006054\,{\left (1-2\,x\right )}^{5/2}}{1575}-\frac {9696248\,{\left (1-2\,x\right )}^{7/2}}{315}+\frac {23591\,{\left (1-2\,x\right )}^{9/2}}{7}}{\frac {182182\,x}{675}+\frac {79954\,{\left (2\,x-1\right )}^2}{675}+\frac {3898\,{\left (2\,x-1\right )}^3}{75}+\frac {57\,{\left (2\,x-1\right )}^4}{5}+{\left (2\,x-1\right )}^5-\frac {49588}{675}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(3/2)/((3*x + 2)^4*(5*x + 3)^3),x)

[Out]

(813716*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - (112875*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/1
1))/11 + ((6859391*(1 - 2*x)^(1/2))/75 - (108239776*(1 - 2*x)^(3/2))/675 + (166006054*(1 - 2*x)^(5/2))/1575 -
(9696248*(1 - 2*x)^(7/2))/315 + (23591*(1 - 2*x)^(9/2))/7)/((182182*x)/675 + (79954*(2*x - 1)^2)/675 + (3898*(
2*x - 1)^3)/75 + (57*(2*x - 1)^4)/5 + (2*x - 1)^5 - 49588/675)

________________________________________________________________________________________