\(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\) [2056]

Optimal result

Integrand size = 24, antiderivative size = 133 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {35845 \sqrt {1-2 x}}{1078 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac {162 \sqrt {1-2 x}}{49 (2+3 x) (3+5 x)}-\frac {22479}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {4900}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-22479/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+4900/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
-35845/1078*(1-2*x)^(1/2)/(3+5*x)+3/14*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)+162/49*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {105, 156, 162, 65, 212} \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {22479}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {4900}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-\frac {35845 \sqrt {1-2 x}}{1078 (5 x+3)}+\frac {162 \sqrt {1-2 x}}{49 (3 x+2) (5 x+3)}+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)} \]

[In]

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

[Out]

(-35845*Sqrt[1 - 2*x])/(1078*(3 + 5*x)) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x)) + (162*Sqrt[1 - 2*x])/(
49*(2 + 3*x)*(3 + 5*x)) - (22479*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (4900*Sqrt[5/11]*ArcTanh[Sqr
t[5/11]*Sqrt[1 - 2*x]])/11

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 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*} \text {integral}& = \frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac {1}{14} \int \frac {58-75 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx \\ & = \frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac {162 \sqrt {1-2 x}}{49 (2+3 x) (3+5 x)}+\frac {1}{98} \int \frac {4253-4860 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx \\ & = -\frac {35845 \sqrt {1-2 x}}{1078 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac {162 \sqrt {1-2 x}}{49 (2+3 x) (3+5 x)}-\frac {\int \frac {175579-107535 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{1078} \\ & = -\frac {35845 \sqrt {1-2 x}}{1078 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac {162 \sqrt {1-2 x}}{49 (2+3 x) (3+5 x)}+\frac {67437}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {12250}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx \\ & = -\frac {35845 \sqrt {1-2 x}}{1078 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac {162 \sqrt {1-2 x}}{49 (2+3 x) (3+5 x)}-\frac {67437}{98} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {12250}{11} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right ) \\ & = -\frac {35845 \sqrt {1-2 x}}{1078 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)}+\frac {162 \sqrt {1-2 x}}{49 (2+3 x) (3+5 x)}-\frac {22479}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {4900}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {\sqrt {1-2 x} \left (136021+419448 x+322605 x^2\right )}{1078 (2+3 x)^2 (3+5 x)}-\frac {22479}{49} \sqrt {\frac {3}{7}} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {4900}{11} \sqrt {\frac {5}{11}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[In]

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

[Out]

-1/1078*(Sqrt[1 - 2*x]*(136021 + 419448*x + 322605*x^2))/((2 + 3*x)^2*(3 + 5*x)) - (22479*Sqrt[3/7]*ArcTanh[Sq
rt[3/7]*Sqrt[1 - 2*x]])/49 + (4900*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.57

method	result	size
risch	\(\frac {645210 x^{3}+516291 x^{2}-147406 x -136021}{1078 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{2}}-\frac {22479 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}+\frac {4900 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}\)	\(76\)
derivativedivides	\(\frac {50 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {4900 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {\frac {3861 \left (1-2 x \right )^{\frac {3}{2}}}{49}-\frac {1305 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{2}}-\frac {22479 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(82\)
default	\(\frac {50 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {4900 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}+\frac {\frac {3861 \left (1-2 x \right )^{\frac {3}{2}}}{49}-\frac {1305 \sqrt {1-2 x}}{7}}{\left (-4-6 x \right )^{2}}-\frac {22479 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(82\)
pseudoelliptic	\(\frac {-5439918 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {21}+3361400 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (2+3 x \right )^{2} \left (3+5 x \right ) \sqrt {55}-77 \sqrt {1-2 x}\, \left (322605 x^{2}+419448 x +136021\right )}{83006 \left (2+3 x \right )^{2} \left (3+5 x \right )}\)	\(97\)
trager	\(-\frac {\left (322605 x^{2}+419448 x +136021\right ) \sqrt {1-2 x}}{1078 \left (2+3 x \right )^{2} \left (3+5 x \right )}-\frac {2450 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}+\frac {177 \operatorname {RootOf}\left (\textit {\_Z}^{2}-338709\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-338709\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-338709\right )+2667 \sqrt {1-2 x}}{2+3 x}\right )}{686}\)	\(123\)

[In]

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

[Out]

1/1078*(645210*x^3+516291*x^2-147406*x-136021)/(3+5*x)/(1-2*x)^(1/2)/(2+3*x)^2-22479/343*arctanh(1/7*21^(1/2)*
(1-2*x)^(1/2))*21^(1/2)+4900/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {1680700 \, \sqrt {11} \sqrt {5} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 2719959 \, \sqrt {7} \sqrt {3} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (322605 \, x^{2} + 419448 \, x + 136021\right )} \sqrt {-2 \, x + 1}}{83006 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

[In]

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

[Out]

1/83006*(1680700*sqrt(11)*sqrt(5)*(45*x^3 + 87*x^2 + 56*x + 12)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x +
8)/(5*x + 3)) + 2719959*sqrt(7)*sqrt(3)*(45*x^3 + 87*x^2 + 56*x + 12)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*
x - 5)/(3*x + 2)) - 77*(322605*x^2 + 419448*x + 136021)*sqrt(-2*x + 1))/(45*x^3 + 87*x^2 + 56*x + 12)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.38 (sec) , antiderivative size = 3624, normalized size of antiderivative = 27.25 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-222264000*sqrt(55)*I*(x - 1/2)**(15/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(537878880*(x - 1/2)**(15/2) + 3101
768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**
(7/2) + 1096135733*(x - 1/2)**(5/2)) + 21559608000*sqrt(55)*I*(x - 1/2)**(15/2)*atan(sqrt(110)*sqrt(x - 1/2)/1
1)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x
- 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 682149600*sqrt(21)*I*(x - 1/2)**(
15/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 715378910
4*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2))
- 35932818240*sqrt(21)*I*(x - 1/2)**(15/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(537878880*(x - 1/2)**(15/2) + 3101
768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**
(7/2) + 1096135733*(x - 1/2)**(5/2)) - 10779804000*sqrt(55)*I*pi*(x - 1/2)**(15/2)/(537878880*(x - 1/2)**(15/2
) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x
- 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 17966409120*sqrt(21)*I*pi*(x - 1/2)**(15/2)/(537878880*(x - 1/2
)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 475466
6686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 1281722400*sqrt(55)*I*(x - 1/2)**(13/2)*atan(sqrt(110)/
(10*sqrt(x - 1/2)))/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2)
+ 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 124327072800*sqr
t(55)*I*(x - 1/2)**(13/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)
**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 10961357
33*(x - 1/2)**(5/2)) - 3933729360*sqrt(21)*I*(x - 1/2)**(13/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(537878880*(x
- 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4
754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 207212585184*sqrt(21)*I*(x - 1/2)**(13/2)*atan(sqr
t(42)*sqrt(x - 1/2)/7)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11
/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 62163536400*s
qrt(55)*I*pi*(x - 1/2)**(13/2)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1
/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 10360
6292592*sqrt(21)*I*pi*(x - 1/2)**(13/2)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 71537891
04*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)
) - 2956111200*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(537878880*(x - 1/2)**(15/2) +
3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/
2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 286742786400*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1
/2)/11)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 824847223
2*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 9072589680*sqrt(21)*I*(x - 1
/2)**(11/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 715
3789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**
(5/2)) - 477906482592*sqrt(21)*I*(x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(537878880*(x - 1/2)**(15/2)
+ 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x -
1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 143371393200*sqrt(55)*I*pi*(x - 1/2)**(11/2)/(537878880*(x - 1/2
)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 475466
6686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 238953241296*sqrt(21)*I*pi*(x - 1/2)**(11/2)/(537878880
*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2)
+ 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 3408459600*sqrt(55)*I*(x - 1/2)**(9/2)*atan(sq
rt(110)/(10*sqrt(x - 1/2)))/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)
**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 33062058
1200*sqrt(55)*I*(x - 1/2)**(9/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x
- 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1
096135733*(x - 1/2)**(5/2)) - 10460890440*sqrt(21)*I*(x - 1/2)**(9/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(537878
880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9
/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 551036421936*sqrt(21)*I*(x - 1/2)**(9/2)*at
an(sqrt(42)*sqrt(x - 1/2)/7)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2
)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 1653102
90600*sqrt(55)*I*pi*(x - 1/2)**(9/2)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*
(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) +
275518210968*sqrt(21)*I*pi*(x - 1/2)**(9/2)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 715
3789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**
(5/2)) - 1964738300*sqrt(55)*I*(x - 1/2)**(7/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(537878880*(x - 1/2)**(15/2
) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x
- 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 190579615100*sqrt(55)*I*(x - 1/2)**(7/2)*atan(sqrt(110)*sqrt(x
- 1/2)/11)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 824847
2232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 6029970870*sqrt(21)*I*(x
- 1/2)**(7/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7
153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)
**(5/2)) - 317633913828*sqrt(21)*I*(x - 1/2)**(7/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(537878880*(x - 1/2)**(15/2
) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x
- 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 95289807550*sqrt(55)*I*pi*(x - 1/2)**(7/2)/(537878880*(x - 1/2)
**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666
686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 158816956914*sqrt(21)*I*pi*(x - 1/2)**(7/2)/(537878880*(
x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) +
4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 452948650*sqrt(55)*I*(x - 1/2)**(5/2)*atan(sqrt(
110)/(10*sqrt(x - 1/2)))/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(
11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 43936019050
*sqrt(55)*I*(x - 1/2)**(5/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1
/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 10961
35733*(x - 1/2)**(5/2)) - 1390142985*sqrt(21)*I*(x - 1/2)**(5/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(537878880*(
x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) +
4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 73226980134*sqrt(21)*I*(x - 1/2)**(5/2)*atan(sqr
t(42)*sqrt(x - 1/2)/7)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/2)**(11
/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 21968009525*s
qrt(55)*I*pi*(x - 1/2)**(5/2)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/
2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) + 366134
90067*sqrt(21)*I*pi*(x - 1/2)**(5/2)/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*
(x - 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) -
3577044240*sqrt(2)*I*(x - 1/2)**7/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x
- 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 1
6574320224*sqrt(2)*I*(x - 1/2)**6/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x
- 1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 28
795031496*sqrt(2)*I*(x - 1/2)**5/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x -
1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 222
30787656*sqrt(2)*I*(x - 1/2)**4/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x -
1/2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2)) - 6435
172205*sqrt(2)*I*(x - 1/2)**3/(537878880*(x - 1/2)**(15/2) + 3101768208*(x - 1/2)**(13/2) + 7153789104*(x - 1/
2)**(11/2) + 8248472232*(x - 1/2)**(9/2) + 4754666686*(x - 1/2)**(7/2) + 1096135733*(x - 1/2)**(5/2))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {2450}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22479}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {322605 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 1484106 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 1705585 \, \sqrt {-2 \, x + 1}}{539 \, {\left (45 \, {\left (2 \, x - 1\right )}^{3} + 309 \, {\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \]

[In]

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

[Out]

-2450/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22479/686*sqrt(21)*log(
-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/539*(322605*(-2*x + 1)^(5/2) - 1484106*(-2*x
+ 1)^(3/2) + 1705585*sqrt(-2*x + 1))/(45*(2*x - 1)^3 + 309*(2*x - 1)^2 + 1414*x - 168)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=-\frac {2450}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {22479}{686} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {125 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} + \frac {9 \, {\left (429 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1015 \, \sqrt {-2 \, x + 1}\right )}}{196 \, {\left (3 \, x + 2\right )}^{2}} \]

[In]

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

[Out]

-2450/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22479/686*sqr
t(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/11*sqrt(-2*x + 1)/(5*x
+ 3) + 9/196*(429*(-2*x + 1)^(3/2) - 1015*sqrt(-2*x + 1))/(3*x + 2)^2

Mupad [B] (verification not implemented)

Time = 1.55 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx=\frac {4900\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {22479\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {48731\,\sqrt {1-2\,x}}{693}-\frac {494702\,{\left (1-2\,x\right )}^{3/2}}{8085}+\frac {7169\,{\left (1-2\,x\right )}^{5/2}}{539}}{\frac {1414\,x}{45}+\frac {103\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {56}{15}} \]

[In]

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

[Out]

(4900*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121 - (22479*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7)
)/343 - ((48731*(1 - 2*x)^(1/2))/693 - (494702*(1 - 2*x)^(3/2))/8085 + (7169*(1 - 2*x)^(5/2))/539)/((1414*x)/4
5 + (103*(2*x - 1)^2)/15 + (2*x - 1)^3 - 56/15)