3.22.88 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx\) [2188]
Optimal. Leaf size=132 \[ \frac {13900}{17787 (1-2 x)^{3/2}}+\frac {159800}{456533 \sqrt {1-2 x}}-\frac {340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {4050}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {15250 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]
[Out]
13900/17787/(1-2*x)^(3/2)-340/77/(1-2*x)^(3/2)/(3+5*x)+3/7/(1-2*x)^(3/2)/(2+3*x)/(3+5*x)-4050/2401*arctanh(1/7
*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+15250/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+159800/456533/(1-2
*x)^(1/2)
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 132, 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 {159800}{456533 \sqrt {1-2 x}}+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac {340}{77 (1-2 x)^{3/2} (5 x+3)}+\frac {13900}{17787 (1-2 x)^{3/2}}-\frac {4050}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {15250 \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)^2*(3 + 5*x)^2),x]
[Out]
13900/(17787*(1 - 2*x)^(3/2)) + 159800/(456533*Sqrt[1 - 2*x]) - 340/(77*(1 - 2*x)^(3/2)*(3 + 5*x)) + 3/(7*(1 -
2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)) - (4050*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (15250*Sqrt[5/11]*A
rcTanh[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)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx &=\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac {1}{7} \int \frac {5-105 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {1}{77} \int \frac {-925-5100 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=\frac {13900}{17787 (1-2 x)^{3/2}}-\frac {340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac {2 \int \frac {-\frac {36525}{2}+156375 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{17787}\\ &=\frac {13900}{17787 (1-2 x)^{3/2}}+\frac {159800}{456533 \sqrt {1-2 x}}-\frac {340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {4 \int \frac {\frac {5688825}{4}-898875 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{1369599}\\ &=\frac {13900}{17787 (1-2 x)^{3/2}}+\frac {159800}{456533 \sqrt {1-2 x}}-\frac {340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac {6075}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {38125 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac {13900}{17787 (1-2 x)^{3/2}}+\frac {159800}{456533 \sqrt {1-2 x}}-\frac {340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {6075}{343} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {38125 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1331}\\ &=\frac {13900}{17787 (1-2 x)^{3/2}}+\frac {159800}{456533 \sqrt {1-2 x}}-\frac {340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {4050}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {15250 \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 = 0.24, size = 99, normalized size = 0.75 \begin {gather*} \frac {-2209989+5548760 x+5028300 x^2-14382000 x^3}{1369599 (1-2 x)^{3/2} \left (6+19 x+15 x^2\right )}-\frac {4050}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {15250 \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)^2*(3 + 5*x)^2),x]
[Out]
(-2209989 + 5548760*x + 5028300*x^2 - 14382000*x^3)/(1369599*(1 - 2*x)^(3/2)*(6 + 19*x + 15*x^2)) - (4050*Sqrt
[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (15250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 88, normalized size = 0.67
| | |
| method |
result |
size |
| | |
| risch |
\(\frac {14382000 x^{3}-5028300 x^{2}-5548760 x +2209989}{1369599 \sqrt {1-2 x}\, \left (-1+2 x \right ) \left (15 x^{2}+19 x +6\right )}+\frac {15250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}-\frac {4050 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) |
\(81\) |
| derivativedivides |
\(\frac {250 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {15250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {16}{17787 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2176}{456533 \sqrt {1-2 x}}+\frac {54 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}-\frac {4050 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) |
\(88\) |
| default |
\(\frac {250 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {15250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {16}{17787 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {2176}{456533 \sqrt {1-2 x}}+\frac {54 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}-\frac {4050 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\) |
\(88\) |
| trager |
\(-\frac {\left (14382000 x^{3}-5028300 x^{2}-5548760 x +2209989\right ) \sqrt {1-2 x}}{1369599 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}+\frac {7625 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{14641}-\frac {2025 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2401}\) |
\(134\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1-2*x)^(5/2)/(2+3*x)^2/(3+5*x)^2,x,method=_RETURNVERBOSE)
[Out]
250/1331*(1-2*x)^(1/2)/(-6/5-2*x)+15250/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+16/17787/(1-2*x)^(
3/2)+2176/456533/(1-2*x)^(1/2)+54/343*(1-2*x)^(1/2)/(-4/3-2*x)-4050/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*2
1^(1/2)
________________________________________________________________________________________
Maxima [A]
time = 0.56, size = 128, normalized size = 0.97 \begin {gather*} -\frac {7625}{14641} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2025}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4 \, {\left (1797750 \, {\left (2 \, x - 1\right )}^{3} + 4136175 \, {\left (2 \, x - 1\right )}^{2} + 209440 \, x - 128436\right )}}{1369599 \, {\left (15 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 68 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 77 \, {\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)^2/(3+5*x)^2,x, algorithm="maxima")
[Out]
-7625/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2025/2401*sqrt(21)*lo
g(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 4/1369599*(1797750*(2*x - 1)^3 + 4136175*(2*
x - 1)^2 + 209440*x - 128436)/(15*(-2*x + 1)^(7/2) - 68*(-2*x + 1)^(5/2) + 77*(-2*x + 1)^(3/2))
________________________________________________________________________________________
Fricas [A]
time = 1.20, size = 162, normalized size = 1.23 \begin {gather*} \frac {54922875 \, \sqrt {11} \sqrt {5} {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 88944075 \, \sqrt {7} \sqrt {3} {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (14382000 \, x^{3} - 5028300 \, x^{2} - 5548760 \, x + 2209989\right )} \sqrt {-2 \, x + 1}}{105459123 \, {\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2*x)^(5/2)/(2+3*x)^2/(3+5*x)^2,x, algorithm="fricas")
[Out]
1/105459123*(54922875*sqrt(11)*sqrt(5)*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x +
1) - 5*x + 8)/(5*x + 3)) + 88944075*sqrt(7)*sqrt(3)*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log((sqrt(7)*sqrt(3)
*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(14382000*x^3 - 5028300*x^2 - 5548760*x + 2209989)*sqrt(-2*x + 1))/
(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 9.95, size = 2966, normalized size = 22.47 \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)**2/(3+5*x)**2,x)
[Out]
-3986690400000*sqrt(2)*I*(x - 1/2)**(17/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 43880
2755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1
/2)**4 + 48145569800559*(x - 1/2)**3) - 22659187320000*sqrt(2)*I*(x - 1/2)**(15/2)/(22779170568000*(x - 1/2)**
9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 5631302031589
08*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) - 51564023280000*sqrt(2)*I*(x -
1/2)**(13/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 6628
50240685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1
/2)**3) - 58784347960800*sqrt(2)*I*(x - 1/2)**(11/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)*
*8 + 438802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228
936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) - 33664789429040*sqrt(2)*I*(x - 1/2)**(9/2)/(22779170568000*(x
- 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 5631
30203158908*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) - 7840313302668*sqrt(2)
*I*(x - 1/2)**(7/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7
+ 662850240685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559
*(x - 1/2)**3) - 57369762912*sqrt(2)*I*(x - 1/2)**(5/2)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/
2)**8 + 438802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993
228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) + 10827136628*sqrt(2)*I*(x - 1/2)**(3/2)/(22779170568000*(x
- 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 5631
30203158908*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) + 23726682000000*sqrt(5
5)*I*(x - 1/2)**9*atan(sqrt(110)*sqrt(x - 1/2)/11)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8
+ 438802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 25510899322893
6*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) - 38423840400000*sqrt(21)*I*(x - 1/2)**9*atan(sqrt(42)*sqrt(x -
1/2)/7)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 662850240
685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**
3) - 11863341000000*sqrt(55)*I*pi*(x - 1/2)**9/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 4
38802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x
- 1/2)**4 + 48145569800559*(x - 1/2)**3) + 19211920200000*sqrt(21)*I*pi*(x - 1/2)**9/(22779170568000*(x - 1/2
)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 5631302031
58908*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) + 161341437600000*sqrt(55)*I*
(x - 1/2)**8*atan(sqrt(110)*sqrt(x - 1/2)/11)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 43
8802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x
- 1/2)**4 + 48145569800559*(x - 1/2)**3) - 261282114720000*sqrt(21)*I*(x - 1/2)**8*atan(sqrt(42)*sqrt(x - 1/2)
/7)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 6628502406852
48*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) -
80670718800000*sqrt(55)*I*pi*(x - 1/2)**8/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 43880
2755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1
/2)**4 + 48145569800559*(x - 1/2)**3) + 130641057360000*sqrt(21)*I*pi*(x - 1/2)**8/(22779170568000*(x - 1/2)**
9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 5631302031589
08*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) + 457054984260000*sqrt(55)*I*(x
- 1/2)**7*atan(sqrt(110)*sqrt(x - 1/2)/11)/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 43880
2755708240*(x - 1/2)**7 + 662850240685248*(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1
/2)**4 + 48145569800559*(x - 1/2)**3) - 740171245572000*sqrt(21)*I*(x - 1/2)**7*atan(sqrt(42)*sqrt(x - 1/2)/7)
/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 438802755708240*(x - 1/2)**7 + 662850240685248*
(x - 1/2)**6 + 563130203158908*(x - 1/2)**5 + 255108993228936*(x - 1/2)**4 + 48145569800559*(x - 1/2)**3) - 22
8527492130000*sqrt(55)*I*pi*(x - 1/2)**7/(22779170568000*(x - 1/2)**9 + 154898359862400*(x - 1/2)**8 + 4388027
55708240*(x - 1/2)**7 + 662850240685248*(x - 1/...
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Giac [A]
time = 1.08, size = 137, normalized size = 1.04 \begin {gather*} -\frac {7625}{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 {2025}{2401} \, \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 {4 \, {\left (591090 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1343273 \, \sqrt {-2 \, x + 1}\right )}}{456533 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} + \frac {16 \, {\left (816 \, x - 485\right )}}{1369599 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(1-2*x)^(5/2)/(2+3*x)^2/(3+5*x)^2,x, algorithm="giac")
[Out]
-7625/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2025/2401*s
qrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/456533*(591090*(-2*x +
1)^(3/2) - 1343273*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9) + 16/1369599*(816*x - 485)/((2*x - 1)*sqrt(-2*
x + 1))
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Mupad [B]
time = 1.27, size = 90, normalized size = 0.68 \begin {gather*} \frac {15250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641}-\frac {4050\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {\frac {2176\,x}{53361}+\frac {367660\,{\left (2\,x-1\right )}^2}{456533}+\frac {159800\,{\left (2\,x-1\right )}^3}{456533}-\frac {2224}{88935}}{\frac {77\,{\left (1-2\,x\right )}^{3/2}}{15}-\frac {68\,{\left (1-2\,x\right )}^{5/2}}{15}+{\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)^2*(5*x + 3)^2),x)
[Out]
(15250*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641 - (4050*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/
7))/2401 - ((2176*x)/53361 + (367660*(2*x - 1)^2)/456533 + (159800*(2*x - 1)^3)/456533 - 2224/88935)/((77*(1 -
2*x)^(3/2))/15 - (68*(1 - 2*x)^(5/2))/15 + (1 - 2*x)^(7/2))
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