3.22.32 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx\) [2132]

Optimal. Leaf size=112 \[ -\frac {2049}{9317 \sqrt {1-2 x}}-\frac {5}{22 \sqrt {1-2 x} (3+5 x)^2}+\frac {305}{242 \sqrt {1-2 x} (3+5 x)}+\frac {54}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {9975 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]

[Out]

54/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-9975/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-20
49/9317/(1-2*x)^(1/2)-5/22/(3+5*x)^2/(1-2*x)^(1/2)+305/242/(3+5*x)/(1-2*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {2049}{9317 \sqrt {1-2 x}}+\frac {305}{242 \sqrt {1-2 x} (5 x+3)}-\frac {5}{22 \sqrt {1-2 x} (5 x+3)^2}+\frac {54}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {9975 \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)^(3/2)*(2 + 3*x)*(3 + 5*x)^3),x]

[Out]

-2049/(9317*Sqrt[1 - 2*x]) - 5/(22*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 305/(242*Sqrt[1 - 2*x]*(3 + 5*x)) + (54*Sqrt[3
/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (9975*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 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)^{3/2} (2+3 x) (3+5 x)^3} \, dx &=-\frac {5}{22 \sqrt {1-2 x} (3+5 x)^2}-\frac {1}{22} \int \frac {16-75 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {5}{22 \sqrt {1-2 x} (3+5 x)^2}+\frac {305}{242 \sqrt {1-2 x} (3+5 x)}+\frac {1}{242} \int \frac {348-2745 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {2049}{9317 \sqrt {1-2 x}}-\frac {5}{22 \sqrt {1-2 x} (3+5 x)^2}+\frac {305}{242 \sqrt {1-2 x} (3+5 x)}-\frac {\int \frac {-25692+\frac {30735 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{9317}\\ &=-\frac {2049}{9317 \sqrt {1-2 x}}-\frac {5}{22 \sqrt {1-2 x} (3+5 x)^2}+\frac {305}{242 \sqrt {1-2 x} (3+5 x)}-\frac {81}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {49875 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{2662}\\ &=-\frac {2049}{9317 \sqrt {1-2 x}}-\frac {5}{22 \sqrt {1-2 x} (3+5 x)^2}+\frac {305}{242 \sqrt {1-2 x} (3+5 x)}+\frac {81}{7} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {49875 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2662}\\ &=-\frac {2049}{9317 \sqrt {1-2 x}}-\frac {5}{22 \sqrt {1-2 x} (3+5 x)^2}+\frac {305}{242 \sqrt {1-2 x} (3+5 x)}+\frac {54}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {9975 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.25, size = 88, normalized size = 0.79 \begin {gather*} \frac {54}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {-\frac {11 \left (-29338+5515 x+102450 x^2\right )}{\sqrt {1-2 x} (3+5 x)^2}-139650 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{204974} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(54*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + ((-11*(-29338 + 5515*x + 102450*x^2))/(Sqrt[1 - 2*x]*(3 +
5*x)^2) - 139650*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/204974

________________________________________________________________________________________

Maple [A]
time = 0.16, size = 75, normalized size = 0.67

method	result	size
risch	\(-\frac {102450 x^{2}+5515 x -29338}{18634 \left (3+5 x \right )^{2} \sqrt {1-2 x}}-\frac {9975 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {54 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\)	\(64\)
derivativedivides	\(\frac {-\frac {7375 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {1425 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {9975 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {16}{9317 \sqrt {1-2 x}}+\frac {54 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\)	\(75\)
default	\(\frac {-\frac {7375 \left (1-2 x \right )^{\frac {3}{2}}}{1331}+\frac {1425 \sqrt {1-2 x}}{121}}{\left (-6-10 x \right )^{2}}-\frac {9975 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {16}{9317 \sqrt {1-2 x}}+\frac {54 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\)	\(75\)
trager	\(\frac {\left (102450 x^{2}+5515 x -29338\right ) \sqrt {1-2 x}}{18634 \left (3+5 x \right )^{2} \left (-1+2 x \right )}-\frac {27 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{49}+\frac {9975 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{29282}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1250/1331*(-59/10*(1-2*x)^(3/2)+627/50*(1-2*x)^(1/2))/(-6-10*x)^2-9975/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/
2))*55^(1/2)+16/9317/(1-2*x)^(1/2)+54/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 119, normalized size = 1.06 \begin {gather*} \frac {9975}{29282} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {27}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {51225 \, {\left (2 \, x - 1\right )}^{2} + 215930 \, x - 109901}{9317 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 121 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9975/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 27/49*sqrt(21)*log(-(s
qrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/9317*(51225*(2*x - 1)^2 + 215930*x - 109901)/(2
5*(-2*x + 1)^(5/2) - 110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))

________________________________________________________________________________________

Fricas [A]
time = 1.51, size = 142, normalized size = 1.27 \begin {gather*} \frac {488775 \, \sqrt {11} \sqrt {5} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 790614 \, \sqrt {7} \sqrt {3} {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (102450 \, x^{2} + 5515 \, x - 29338\right )} \sqrt {-2 \, x + 1}}{1434818 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1434818*(488775*sqrt(11)*sqrt(5)*(50*x^3 + 35*x^2 - 12*x - 9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8
)/(5*x + 3)) + 790614*sqrt(7)*sqrt(3)*(50*x^3 + 35*x^2 - 12*x - 9)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x
+ 5)/(3*x + 2)) + 77*(102450*x^2 + 5515*x - 29338)*sqrt(-2*x + 1))/(50*x^3 + 35*x^2 - 12*x - 9)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 7.67, size = 1875, normalized size = 16.74 \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)**(3/2)/(2+3*x)/(3+5*x)**3,x)

[Out]

-9775500000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(14348180000*(x - 1/2)**(11/2) + 631
31992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/
2)**(3/2)) + 15812280000*sqrt(21)*I*(x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(14348180000*(x - 1/2)**(
11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170
338*(x - 1/2)**(3/2)) - 7906140000*sqrt(21)*I*pi*(x - 1/2)**(11/2)/(14348180000*(x - 1/2)**(11/2) + 6313199200
0*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/
2)) + 4887750000*sqrt(55)*I*pi*(x - 1/2)**(11/2)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2)
+ 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) - 43012200000*
sqrt(55)*I*(x - 1/2)**(9/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x -
1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) +
69574032000*sqrt(21)*I*(x - 1/2)**(9/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(14348180000*(x - 1/2)**(11/2) + 631319
92000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)*
*(3/2)) - 34787016000*sqrt(21)*I*pi*(x - 1/2)**(9/2)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(
9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 21506100
000*sqrt(55)*I*pi*(x - 1/2)**(9/2)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 10416778680
0*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) - 70970130000*sqrt(55)*I*(x
- 1/2)**(7/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) +
104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 114797152800*s
qrt(21)*I*(x - 1/2)**(7/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/
2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) - 573
98576400*sqrt(21)*I*pi*(x - 1/2)**(7/2)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167
786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 35485065000*sqrt(55)*
I*pi*(x - 1/2)**(7/2)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**
(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) - 52044762000*sqrt(55)*I*(x - 1/2)**(5/2)
*atan(sqrt(110)*sqrt(x - 1/2)/11)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800
*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 84184578720*sqrt(21)*I*(x -
1/2)**(5/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 10
4167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) - 42092289360*sqrt(
21)*I*pi*(x - 1/2)**(5/2)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/
2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 26022381000*sqrt(55)*I*pi*(x - 1/2)
**(5/2)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 763897
10320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) - 14312309550*sqrt(55)*I*(x - 1/2)**(3/2)*atan(sqrt(110
)*sqrt(x - 1/2)/11)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7
/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 23150759148*sqrt(21)*I*(x - 1/2)**(3/2)*a
tan(sqrt(42)*sqrt(x - 1/2)/7)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x
- 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) - 11575379574*sqrt(21)*I*pi*(x -
1/2)**(3/2)/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76
389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 7156154775*sqrt(55)*I*pi*(x - 1/2)**(3/2)/(143481
80000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)
**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 1577730000*sqrt(2)*I*(x - 1/2)**5/(14348180000*(x - 1/2)**(11/2) + 6
3131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x -
1/2)**(3/2)) + 5133667000*sqrt(2)*I*(x - 1/2)**4/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2)
+ 104167786800*(x - 1/2)**(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 5552000300*s
qrt(2)*I*(x - 1/2)**3/(14348180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/2)**(9/2) + 104167786800*(x - 1/2)**
(7/2) + 76389710320*(x - 1/2)**(5/2) + 21007170338*(x - 1/2)**(3/2)) + 1979023970*sqrt(2)*I*(x - 1/2)**2/(1434
8180000*(x - 1/2)**(11/2) + 63131992000*(x - 1/...

________________________________________________________________________________________

Giac [A]
time = 1.20, size = 116, normalized size = 1.04 \begin {gather*} \frac {9975}{29282} \, \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}{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 {16}{9317 \, \sqrt {-2 \, x + 1}} - \frac {25 \, {\left (295 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 627 \, \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)^(3/2)/(2+3*x)/(3+5*x)^3,x, algorithm="giac")

[Out]

9975/29282*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 27/49*sqrt(2
1)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/9317/sqrt(-2*x + 1) - 25/53
24*(295*(-2*x + 1)^(3/2) - 627*sqrt(-2*x + 1))/(5*x + 3)^2

________________________________________________________________________________________

Mupad [B]
time = 1.26, size = 81, normalized size = 0.72 \begin {gather*} \frac {54\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {9975\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641}-\frac {\frac {3926\,x}{4235}+\frac {2049\,{\left (2\,x-1\right )}^2}{9317}-\frac {9991}{21175}}{\frac {121\,\sqrt {1-2\,x}}{25}-\frac {22\,{\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)^(3/2)*(3*x + 2)*(5*x + 3)^3),x)

[Out]

(54*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - (9975*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/14
641 - ((3926*x)/4235 + (2049*(2*x - 1)^2)/9317 - 9991/21175)/((121*(1 - 2*x)^(1/2))/25 - (22*(1 - 2*x)^(3/2))/
5 + (1 - 2*x)^(5/2))

________________________________________________________________________________________