3.22.97 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx\) [2197]

Optimal. Leaf size=125 \[ -\frac {5969}{27951 (1-2 x)^{3/2}}-\frac {65167}{717409 \sqrt {1-2 x}}-\frac {5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac {295}{242 (1-2 x)^{3/2} (3+5 x)}+\frac {162}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {47075 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]

[Out]

-5969/27951/(1-2*x)^(3/2)-5/22/(1-2*x)^(3/2)/(3+5*x)^2+295/242/(1-2*x)^(3/2)/(3+5*x)+162/343*arctanh(1/7*21^(1
/2)*(1-2*x)^(1/2))*21^(1/2)-47075/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-65167/717409/(1-2*x)^(1
/2)

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {105, 156, 157, 162, 65, 212} \begin {gather*} -\frac {65167}{717409 \sqrt {1-2 x}}+\frac {295}{242 (1-2 x)^{3/2} (5 x+3)}-\frac {5969}{27951 (1-2 x)^{3/2}}-\frac {5}{22 (1-2 x)^{3/2} (5 x+3)^2}+\frac {162}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {47075 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-5969/(27951*(1 - 2*x)^(3/2)) - 65167/(717409*Sqrt[1 - 2*x]) - 5/(22*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + 295/(242*(
1 - 2*x)^(3/2)*(3 + 5*x)) + (162*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (47075*Sqrt[5/11]*ArcTanh[Sq
rt[5/11]*Sqrt[1 - 2*x]])/14641

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 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)^3} \, dx &=-\frac {5}{22 (1-2 x)^{3/2} (3+5 x)^2}-\frac {1}{22} \int \frac {-4-105 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac {295}{242 (1-2 x)^{3/2} (3+5 x)}+\frac {1}{242} \int \frac {-772-4425 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {5969}{27951 (1-2 x)^{3/2}}-\frac {5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac {295}{242 (1-2 x)^{3/2} (3+5 x)}-\frac {\int \frac {-18276+\frac {268605 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{27951}\\ &=-\frac {5969}{27951 (1-2 x)^{3/2}}-\frac {65167}{717409 \sqrt {1-2 x}}-\frac {5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac {295}{242 (1-2 x)^{3/2} (3+5 x)}+\frac {2 \int \frac {1290129-\frac {2932515 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{2152227}\\ &=-\frac {5969}{27951 (1-2 x)^{3/2}}-\frac {65167}{717409 \sqrt {1-2 x}}-\frac {5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac {295}{242 (1-2 x)^{3/2} (3+5 x)}-\frac {243}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {235375 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{29282}\\ &=-\frac {5969}{27951 (1-2 x)^{3/2}}-\frac {65167}{717409 \sqrt {1-2 x}}-\frac {5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac {295}{242 (1-2 x)^{3/2} (3+5 x)}+\frac {243}{49} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {235375 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{29282}\\ &=-\frac {5969}{27951 (1-2 x)^{3/2}}-\frac {65167}{717409 \sqrt {1-2 x}}-\frac {5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac {295}{242 (1-2 x)^{3/2} (3+5 x)}+\frac {162}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {47075 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.29, size = 93, normalized size = 0.74 \begin {gather*} \frac {162}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {\frac {11 \left (2971158-6032979 x-9295580 x^2+19550100 x^3\right )}{(1-2 x)^{3/2} (3+5 x)^2}-13840050 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{47348994} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(162*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + ((11*(2971158 - 6032979*x - 9295580*x^2 + 19550100*x^3))
/((1 - 2*x)^(3/2)*(3 + 5*x)^2) - 13840050*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/47348994

________________________________________________________________________________________

Maple [A]
time = 0.17, size = 84, normalized size = 0.67

method	result	size
derivativedivides	\(\frac {-\frac {3125 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {6625 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {47075 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {16}{27951 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2208}{717409 \sqrt {1-2 x}}+\frac {162 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(84\)
default	\(\frac {-\frac {3125 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {6625 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {47075 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {16}{27951 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2208}{717409 \sqrt {1-2 x}}+\frac {162 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(84\)
trager	\(\frac {\left (19550100 x^{3}-9295580 x^{2}-6032979 x +2971158\right ) \sqrt {1-2 x}}{4304454 \left (10 x^{2}+x -3\right )^{2}}-\frac {81 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{343}-\frac {175 \RootOf \left (\textit {\_Z}^{2}-3979855\right ) \ln \left (\frac {-5 \RootOf \left (\textit {\_Z}^{2}-3979855\right ) x +14795 \sqrt {1-2 x}+8 \RootOf \left (\textit {\_Z}^{2}-3979855\right )}{3+5 x}\right )}{322102}\)	\(124\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

31250/14641*(-11/10*(1-2*x)^(3/2)+583/250*(1-2*x)^(1/2))/(-6-10*x)^2-47075/161051*arctanh(1/11*55^(1/2)*(1-2*x
)^(1/2))*55^(1/2)+16/27951/(1-2*x)^(3/2)+2208/717409/(1-2*x)^(1/2)+162/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
*21^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.52, size = 128, normalized size = 1.02 \begin {gather*} \frac {47075}{322102} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {81}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4887525 \, {\left (2 \, x - 1\right )}^{3} + 10014785 \, {\left (2 \, x - 1\right )}^{2} - 1331968 \, x + 815056}{2152227 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\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)^3,x, algorithm="maxima")

[Out]

47075/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 81/343*sqrt(21)*log(
-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/2152227*(4887525*(2*x - 1)^3 + 10014785*(2*x
- 1)^2 - 1331968*x + 815056)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2))

________________________________________________________________________________________

Fricas [A]
time = 0.73, size = 162, normalized size = 1.30 \begin {gather*} \frac {48440175 \, \sqrt {11} \sqrt {5} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 78270786 \, \sqrt {7} \sqrt {3} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (19550100 \, x^{3} - 9295580 \, x^{2} - 6032979 \, x + 2971158\right )} \sqrt {-2 \, x + 1}}{331442958 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

1/331442958*(48440175*sqrt(11)*sqrt(5)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x +
1) + 5*x - 8)/(5*x + 3)) + 78270786*sqrt(7)*sqrt(3)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(-(sqrt(7)*sqrt(
3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(19550100*x^3 - 9295580*x^2 - 6032979*x + 2971158)*sqrt(-2*x + 1)
)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 10.59, size = 1352, normalized size = 10.82 \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)**3,x)

[Out]

-96880350000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(331442958000*(x - 1/2)**(11/2) + 1
093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) + 156541572000
*sqrt(21)*I*(x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(
x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) - 78270786000*sqrt(21)*I*pi*
(x - 1/2)**(11/2)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**
(7/2) + 441150577098*(x - 1/2)**(5/2)) + 48440175000*sqrt(55)*I*pi*(x - 1/2)**(11/2)/(331442958000*(x - 1/2)**
(11/2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) - 31
9705155000*sqrt(55)*I*(x - 1/2)**(9/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(331442958000*(x - 1/2)**(11/2) + 1093
761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) + 516587187600*sq
rt(21)*I*(x - 1/2)**(9/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x -
1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) - 258293593800*sqrt(21)*I*pi*(x
- 1/2)**(9/2)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2
) + 441150577098*(x - 1/2)**(5/2)) + 159852577500*sqrt(55)*I*pi*(x - 1/2)**(9/2)/(331442958000*(x - 1/2)**(11/
2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) - 351675
670500*sqrt(55)*I*(x - 1/2)**(7/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(331442958000*(x - 1/2)**(11/2) + 10937617
61400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) + 568245906360*sqrt(2
1)*I*(x - 1/2)**(7/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)
**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) - 284122953180*sqrt(21)*I*pi*(x - 1/
2)**(7/2)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) +
441150577098*(x - 1/2)**(5/2)) + 175837835250*sqrt(55)*I*pi*(x - 1/2)**(7/2)/(331442958000*(x - 1/2)**(11/2) +
1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) - 1289477458
50*sqrt(55)*I*(x - 1/2)**(5/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(331442958000*(x - 1/2)**(11/2) + 109376176140
0*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) + 208356832332*sqrt(21)*I
*(x - 1/2)**(5/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9
/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) - 104178416166*sqrt(21)*I*pi*(x - 1/2)**
(5/2)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 4411
50577098*(x - 1/2)**(5/2)) + 64473872925*sqrt(55)*I*pi*(x - 1/2)**(5/2)/(331442958000*(x - 1/2)**(11/2) + 1093
761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) + 15053577000*sqr
t(2)*I*(x - 1/2)**5/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)
**(7/2) + 441150577098*(x - 1/2)**(5/2)) + 31981703600*sqrt(2)*I*(x - 1/2)**4/(331442958000*(x - 1/2)**(11/2)
+ 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) + 164522381
10*sqrt(2)*I*(x - 1/2)**3/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x
- 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) - 506695728*sqrt(2)*I*(x - 1/2)**2/(331442958000*(x - 1/2)**(11
/2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x - 1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2)) + 63131
992*sqrt(2)*I*(x - 1/2)/(331442958000*(x - 1/2)**(11/2) + 1093761761400*(x - 1/2)**(9/2) + 1203137937540*(x -
1/2)**(7/2) + 441150577098*(x - 1/2)**(5/2))

________________________________________________________________________________________

Giac [A]
time = 1.28, size = 128, normalized size = 1.02 \begin {gather*} \frac {47075}{322102} \, \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 {81}{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 (828 \, x - 491\right )}}{2152227 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {125 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 53 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

47075/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 81/343*sqr
t(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/2152227*(828*x - 491)/((
2*x - 1)*sqrt(-2*x + 1)) - 125/5324*(25*(-2*x + 1)^(3/2) - 53*sqrt(-2*x + 1))/(5*x + 3)^2

________________________________________________________________________________________

Mupad [B]
time = 0.10, size = 89, normalized size = 0.71 \begin {gather*} \frac {162\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {47075\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{161051}+\frac {\frac {182087\,{\left (2\,x-1\right )}^2}{978285}-\frac {11008\,x}{444675}+\frac {65167\,{\left (2\,x-1\right )}^3}{717409}+\frac {6736}{444675}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}} \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)^3),x)

[Out]

(162*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - (47075*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))
/161051 + ((182087*(2*x - 1)^2)/978285 - (11008*x)/444675 + (65167*(2*x - 1)^3)/717409 + 6736/444675)/((121*(1
- 2*x)^(3/2))/25 - (22*(1 - 2*x)^(5/2))/5 + (1 - 2*x)^(7/2))

________________________________________________________________________________________