3.21.66 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\) [2066]
Optimal. Leaf size=126 \[ -\frac {505 \sqrt {1-2 x}}{154 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {33465 \sqrt {1-2 x}}{1694 (3+5 x)}+\frac {1908}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {32025}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
[Out]
1908/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-32025/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-
505/154*(1-2*x)^(1/2)/(3+5*x)^2+3/7*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)^2+33465/1694*(1-2*x)^(1/2)/(3+5*x)
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Rubi [A]
time = 0.03, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {105, 156, 162,
65, 212} \begin {gather*} \frac {33465 \sqrt {1-2 x}}{1694 (5 x+3)}-\frac {505 \sqrt {1-2 x}}{154 (5 x+3)^2}+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {1908}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {32025}{121} \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/(Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^3),x]
[Out]
(-505*Sqrt[1 - 2*x])/(154*(3 + 5*x)^2) + (3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)^2) + (33465*Sqrt[1 - 2*x])/(
1694*(3 + 5*x)) + (1908*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (32025*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]])/121
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*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx &=\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {1}{7} \int \frac {56-75 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {505 \sqrt {1-2 x}}{154 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}-\frac {1}{154} \int \frac {3966-4545 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {505 \sqrt {1-2 x}}{154 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {33465 \sqrt {1-2 x}}{1694 (3+5 x)}+\frac {\int \frac {163938-100395 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{1694}\\ &=-\frac {505 \sqrt {1-2 x}}{154 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {33465 \sqrt {1-2 x}}{1694 (3+5 x)}-\frac {2862}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {160125}{242} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {505 \sqrt {1-2 x}}{154 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {33465 \sqrt {1-2 x}}{1694 (3+5 x)}+\frac {2862}{7} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {160125}{242} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {505 \sqrt {1-2 x}}{154 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {33465 \sqrt {1-2 x}}{1694 (3+5 x)}+\frac {1908}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {32025}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 95, normalized size = 0.75 \begin {gather*} \frac {1908}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {\frac {11 \sqrt {1-2 x} \left (190406+619170 x+501975 x^2\right )}{(2+3 x) (3+5 x)^2}-448350 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{18634} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^3),x]
[Out]
(1908*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + ((11*Sqrt[1 - 2*x]*(190406 + 619170*x + 501975*x^2))/((2
+ 3*x)*(3 + 5*x)^2) - 448350*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/18634
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Maple [A]
time = 0.16, size = 82, normalized size = 0.65
| | |
| method |
result |
size |
| | |
| risch |
\(-\frac {1003950 x^{3}+736365 x^{2}-238358 x -190406}{1694 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (2+3 x \right )}-\frac {32025 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {1908 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) |
\(76\) |
| derivativedivides |
\(\frac {-\frac {16125 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {3175 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {32025 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {18 \sqrt {1-2 x}}{7 \left (-\frac {4}{3}-2 x \right )}+\frac {1908 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) |
\(82\) |
| default |
\(\frac {-\frac {16125 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {3175 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {32025 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {18 \sqrt {1-2 x}}{7 \left (-\frac {4}{3}-2 x \right )}+\frac {1908 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) |
\(82\) |
| trager |
\(\frac {\left (501975 x^{2}+619170 x +190406\right ) \sqrt {1-2 x}}{1694 \left (3+5 x \right )^{2} \left (2+3 x \right )}+\frac {32025 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{2662}+\frac {954 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{49}\) |
\(123\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(2+3*x)^2/(3+5*x)^3/(1-2*x)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1250*(-129/1210*(1-2*x)^(3/2)+127/550*(1-2*x)^(1/2))/(-6-10*x)^2-32025/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2
))*55^(1/2)-18/7*(1-2*x)^(1/2)/(-4/3-2*x)+1908/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)
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Maxima [A]
time = 0.50, size = 128, normalized size = 1.02 \begin {gather*} \frac {32025}{2662} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {954}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {501975 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 2242290 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2501939 \, \sqrt {-2 \, x + 1}}{847 \, {\left (75 \, {\left (2 \, x - 1\right )}^{3} + 505 \, {\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)^2/(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="maxima")
[Out]
32025/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 954/49*sqrt(21)*log(-(
sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/847*(501975*(-2*x + 1)^(5/2) - 2242290*(-2*x +
1)^(3/2) + 2501939*sqrt(-2*x + 1))/(75*(2*x - 1)^3 + 505*(2*x - 1)^2 + 2266*x - 286)
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Fricas [A]
time = 0.90, size = 142, normalized size = 1.13 \begin {gather*} \frac {1569225 \, \sqrt {11} \sqrt {5} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 2539548 \, \sqrt {7} \sqrt {3} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (501975 \, x^{2} + 619170 \, x + 190406\right )} \sqrt {-2 \, x + 1}}{130438 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)^2/(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="fricas")
[Out]
1/130438*(1569225*sqrt(11)*sqrt(5)*(75*x^3 + 140*x^2 + 87*x + 18)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x -
8)/(5*x + 3)) + 2539548*sqrt(7)*sqrt(3)*(75*x^3 + 140*x^2 + 87*x + 18)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) -
3*x + 5)/(3*x + 2)) + 77*(501975*x^2 + 619170*x + 190406)*sqrt(-2*x + 1))/(75*x^3 + 140*x^2 + 87*x + 18)
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Sympy [C] Result contains complex when optimal does not.
time = 10.43, size = 3624, normalized size = 28.76 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)**2/(3+5*x)**3/(1-2*x)**(1/2),x)
[Out]
1866900000*sqrt(55)*I*(x - 1/2)**(15/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(3913140000*(x - 1/2)**(15/2) + 217
83146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1
/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) - 92286600000*sqrt(55)*I*(x - 1/2)**(15/2)*atan(sqrt(110)*sqrt(x - 1
/2)/11)/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 539778
53160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 1437480000*sqrt(21)*I*(
x - 1/2)**(15/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2
) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(
x - 1/2)**(5/2)) + 153810360000*sqrt(21)*I*(x - 1/2)**(15/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(3913140000*(x - 1
/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 3
0035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) - 76905180000*sqrt(21)*I*pi*(x - 1/2)**(15/2)/(3913
140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/
2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 46143300000*sqrt(55)*I*pi*(x - 1/2)*
*(15/2)/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 539778
53160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 10392410000*sqrt(55)*I*
(x - 1/2)**(13/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(1
3/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 668409965
3*(x - 1/2)**(5/2)) - 513728740000*sqrt(55)*I*(x - 1/2)**(13/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(3913140000*(
x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2
) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 8001972000*sqrt(21)*I*(x - 1/2)**(13/2)*atan
(sqrt(42)/(6*sqrt(x - 1/2)))/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x -
1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 85
6211004000*sqrt(21)*I*(x - 1/2)**(13/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(3913140000*(x - 1/2)**(15/2) + 2178314
6000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)*
*(7/2) + 6684099653*(x - 1/2)**(5/2)) - 428105502000*sqrt(21)*I*pi*(x - 1/2)**(13/2)/(3913140000*(x - 1/2)**(1
5/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045
194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 256864370000*sqrt(55)*I*pi*(x - 1/2)**(13/2)/(3913140000
*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9
/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 23137114000*sqrt(55)*I*(x - 1/2)**(11/2)*a
tan(sqrt(110)/(10*sqrt(x - 1/2)))/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*
(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2))
- 1143738596000*sqrt(55)*I*(x - 1/2)**(11/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(3913140000*(x - 1/2)**(15/2) +
21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x
- 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 17815168800*sqrt(21)*I*(x - 1/2)**(11/2)*atan(sqrt(42)/(6*sqrt
(x - 1/2)))/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53
977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 1906223061600*sqrt(
21)*I*(x - 1/2)**(11/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**
(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099
653*(x - 1/2)**(5/2)) - 953111530800*sqrt(21)*I*pi*(x - 1/2)**(11/2)/(3913140000*(x - 1/2)**(15/2) + 217831460
00*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(
7/2) + 6684099653*(x - 1/2)**(5/2)) + 571869298000*sqrt(55)*I*pi*(x - 1/2)**(11/2)/(3913140000*(x - 1/2)**(15/
2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + 53977853160*(x - 1/2)**(9/2) + 3003504519
4*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) + 25752018600*sqrt(55)*I*(x - 1/2)**(9/2)*atan(sqrt(110)/(10
*sqrt(x - 1/2)))/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x - 1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2)
+ 53977853160*(x - 1/2)**(9/2) + 30035045194*(x - 1/2)**(7/2) + 6684099653*(x - 1/2)**(5/2)) - 1273001360400*
sqrt(55)*I*(x - 1/2)**(9/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(3913140000*(x - 1/2)**(15/2) + 21783146000*(x -
1/2)**(13/2) + 48496848400*(x - 1/2)**(11/2) + ...
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Giac [A]
time = 1.26, size = 123, normalized size = 0.98 \begin {gather*} \frac {32025}{2662} \, \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 {954}{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 {27 \, \sqrt {-2 \, x + 1}}{7 \, {\left (3 \, x + 2\right )}} - \frac {25 \, {\left (645 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1397 \, \sqrt {-2 \, x + 1}\right )}}{484 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)^2/(3+5*x)^3/(1-2*x)^(1/2),x, algorithm="giac")
[Out]
32025/2662*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 954/49*sqrt(
21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 27/7*sqrt(-2*x + 1)/(3*x + 2)
- 25/484*(645*(-2*x + 1)^(3/2) - 1397*sqrt(-2*x + 1))/(5*x + 3)^2
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Mupad [B]
time = 0.10, size = 89, normalized size = 0.71 \begin {gather*} \frac {1908\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {32025\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}+\frac {\frac {227449\,\sqrt {1-2\,x}}{5775}-\frac {149486\,{\left (1-2\,x\right )}^{3/2}}{4235}+\frac {6693\,{\left (1-2\,x\right )}^{5/2}}{847}}{\frac {2266\,x}{75}+\frac {101\,{\left (2\,x-1\right )}^2}{15}+{\left (2\,x-1\right )}^3-\frac {286}{75}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((1 - 2*x)^(1/2)*(3*x + 2)^2*(5*x + 3)^3),x)
[Out]
(1908*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - (32025*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))
/1331 + ((227449*(1 - 2*x)^(1/2))/5775 - (149486*(1 - 2*x)^(3/2))/4235 + (6693*(1 - 2*x)^(5/2))/847)/((2266*x)
/75 + (101*(2*x - 1)^2)/15 + (2*x - 1)^3 - 286/75)
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