3.21.45 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx\) [2045]

Optimal. Leaf size=117 \[ \frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {3840 \sqrt {1-2 x}}{343 (2+3 x)}+\frac {88310}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-250 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

88310/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-250/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+1
/7*(1-2*x)^(1/2)/(2+3*x)^3+55/49*(1-2*x)^(1/2)/(2+3*x)^2+3840/343*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.03, antiderivative size = 117, 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 {3840 \sqrt {1-2 x}}{343 (3 x+2)}+\frac {55 \sqrt {1-2 x}}{49 (3 x+2)^2}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3}+\frac {88310}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-250 \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)^4*(3 + 5*x)),x]

[Out]

Sqrt[1 - 2*x]/(7*(2 + 3*x)^3) + (55*Sqrt[1 - 2*x])/(49*(2 + 3*x)^2) + (3840*Sqrt[1 - 2*x])/(343*(2 + 3*x)) + (
88310*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - 250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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)^4 (3+5 x)} \, dx &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {1}{21} \int \frac {60-75 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {1}{294} \int \frac {4380-4950 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {3840 \sqrt {1-2 x}}{343 (2+3 x)}+\frac {\int \frac {188130-115200 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{2058}\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {3840 \sqrt {1-2 x}}{343 (2+3 x)}-\frac {132465}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+625 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {3840 \sqrt {1-2 x}}{343 (2+3 x)}+\frac {132465}{343} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-625 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3}+\frac {55 \sqrt {1-2 x}}{49 (2+3 x)^2}+\frac {3840 \sqrt {1-2 x}}{343 (2+3 x)}+\frac {88310}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-250 \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 = 0.41, size = 87, normalized size = 0.74 \begin {gather*} \frac {3 \sqrt {1-2 x} \left (5393+15745 x+11520 x^2\right )}{343 (2+3 x)^3}+\frac {88310}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-250 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

(3*Sqrt[1 - 2*x]*(5393 + 15745*x + 11520*x^2))/(343*(2 + 3*x)^3) + (88310*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/343 - 250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]
time = 0.16, size = 75, normalized size = 0.64

method	result	size
risch	\(-\frac {3 \left (23040 x^{3}+19970 x^{2}-4959 x -5393\right )}{343 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {88310 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(69\)
derivativedivides	\(-\frac {250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {162 \left (\frac {1280 \left (1-2 x \right )^{\frac {5}{2}}}{1029}-\frac {7790 \left (1-2 x \right )^{\frac {3}{2}}}{1323}+\frac {1318 \sqrt {1-2 x}}{189}\right )}{\left (-4-6 x \right )^{3}}+\frac {88310 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(75\)
default	\(-\frac {250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {162 \left (\frac {1280 \left (1-2 x \right )^{\frac {5}{2}}}{1029}-\frac {7790 \left (1-2 x \right )^{\frac {3}{2}}}{1323}+\frac {1318 \sqrt {1-2 x}}{189}\right )}{\left (-4-6 x \right )^{3}}+\frac {88310 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(75\)
trager	\(\frac {3 \left (11520 x^{2}+15745 x +5393\right ) \sqrt {1-2 x}}{343 \left (2+3 x \right )^{3}}-\frac {5 \RootOf \left (\textit {\_Z}^{2}-1637717781\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-1637717781\right ) x +185451 \sqrt {1-2 x}-5 \RootOf \left (\textit {\_Z}^{2}-1637717781\right )}{2+3 x}\right )}{2401}-\frac {125 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{11}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+3*x)^4/(3+5*x)/(1-2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-250/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-162*(1280/1029*(1-2*x)^(5/2)-7790/1323*(1-2*x)^(3/2)+131
8/189*(1-2*x)^(1/2))/(-4-6*x)^3+88310/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.55, size = 128, normalized size = 1.09 \begin {gather*} \frac {125}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {44155}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {12 \, {\left (5760 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 27265 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 32291 \, \sqrt {-2 \, x + 1}\right )}}{343 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)^4/(3+5*x)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

125/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 44155/2401*sqrt(21)*log(-(
sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 12/343*(5760*(-2*x + 1)^(5/2) - 27265*(-2*x + 1)
^(3/2) + 32291*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]
time = 1.09, size = 142, normalized size = 1.21 \begin {gather*} \frac {300125 \, \sqrt {11} \sqrt {5} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 485705 \, \sqrt {7} \sqrt {3} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 231 \, {\left (11520 \, x^{2} + 15745 \, x + 5393\right )} \sqrt {-2 \, x + 1}}{26411 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)^4/(3+5*x)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/26411*(300125*sqrt(11)*sqrt(5)*(27*x^3 + 54*x^2 + 36*x + 8)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/
(5*x + 3)) + 485705*sqrt(7)*sqrt(3)*(27*x^3 + 54*x^2 + 36*x + 8)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x +
5)/(3*x + 2)) + 231*(11520*x^2 + 15745*x + 5393)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [C] Result contains complex when optimal does not.
time = 12.67, size = 8978, normalized size = 76.74 \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)**4/(3+5*x)/(1-2*x)**(1/2),x)

[Out]

-36294822144000*sqrt(55)*I*(x - 1/2)**(35/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(1596972174336*(x - 1/2)**(35/2)
+ 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1/2)**(31/2) + 304311919887360*(x - 1/2)**(29/2) + 6
21303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2)**(25/2) + 845663101446240*(x - 1/2)**(23/2) + 563
775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)**(19/2) + 63946746868620*(x - 1/2)**(17/2) + 746045
3801339*(x - 1/2)**(15/2)) + 1753277239296*sqrt(21)*I*(x - 1/2)**(35/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(1596
972174336*(x - 1/2)**(35/2) + 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1/2)**(31/2) + 3043119198
87360*(x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2)**(25/2) + 845663101446
240*(x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)**(19/2) + 63946746868620
*(x - 1/2)**(17/2) + 7460453801339*(x - 1/2)**(15/2)) + 60490725267456*sqrt(21)*I*(x - 1/2)**(35/2)*atan(sqrt(
42)*sqrt(x - 1/2)/7)/(1596972174336*(x - 1/2)**(35/2) + 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x -
1/2)**(31/2) + 304311919887360*(x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1
/2)**(25/2) + 845663101446240*(x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2
)**(19/2) + 63946746868620*(x - 1/2)**(17/2) + 7460453801339*(x - 1/2)**(15/2)) - 30245362633728*sqrt(21)*I*pi
*(x - 1/2)**(35/2)/(1596972174336*(x - 1/2)**(35/2) + 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1
/2)**(31/2) + 304311919887360*(x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2
)**(25/2) + 845663101446240*(x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)*
*(19/2) + 63946746868620*(x - 1/2)**(17/2) + 7460453801339*(x - 1/2)**(15/2)) + 18147411072000*sqrt(55)*I*pi*(
x - 1/2)**(35/2)/(1596972174336*(x - 1/2)**(35/2) + 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1/2
)**(31/2) + 304311919887360*(x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2)*
*(25/2) + 845663101446240*(x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)**(
19/2) + 63946746868620*(x - 1/2)**(17/2) + 7460453801339*(x - 1/2)**(15/2)) - 423439591680000*sqrt(55)*I*(x -
1/2)**(33/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(1596972174336*(x - 1/2)**(35/2) + 18631342033920*(x - 1/2)**(33
/2) + 97814545678080*(x - 1/2)**(31/2) + 304311919887360*(x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2)
+ 869824904344704*(x - 1/2)**(25/2) + 845663101446240*(x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) +
246651737921820*(x - 1/2)**(19/2) + 63946746868620*(x - 1/2)**(17/2) + 7460453801339*(x - 1/2)**(15/2)) + 204
54901125120*sqrt(21)*I*(x - 1/2)**(33/2)*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(1596972174336*(x - 1/2)**(35/2) + 1
8631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1/2)**(31/2) + 304311919887360*(x - 1/2)**(29/2) + 62130
3503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2)**(25/2) + 845663101446240*(x - 1/2)**(23/2) + 5637754
00964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)**(19/2) + 63946746868620*(x - 1/2)**(17/2) + 7460453801
339*(x - 1/2)**(15/2)) + 705725128120320*sqrt(21)*I*(x - 1/2)**(33/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(15969721
74336*(x - 1/2)**(35/2) + 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1/2)**(31/2) + 30431191988736
0*(x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2)**(25/2) + 845663101446240*
(x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)**(19/2) + 63946746868620*(x
- 1/2)**(17/2) + 7460453801339*(x - 1/2)**(15/2)) - 352862564060160*sqrt(21)*I*pi*(x - 1/2)**(33/2)/(159697217
4336*(x - 1/2)**(35/2) + 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1/2)**(31/2) + 304311919887360
*(x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2)**(25/2) + 845663101446240*(
x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)**(19/2) + 63946746868620*(x -
1/2)**(17/2) + 7460453801339*(x - 1/2)**(15/2)) + 211719795840000*sqrt(55)*I*pi*(x - 1/2)**(33/2)/(1596972174
336*(x - 1/2)**(35/2) + 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1/2)**(31/2) + 304311919887360*
(x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2)**(25/2) + 845663101446240*(x
- 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)**(19/2) + 63946746868620*(x -
1/2)**(17/2) + 7460453801339*(x - 1/2)**(15/2)) - 2223057856320000*sqrt(55)*I*(x - 1/2)**(31/2)*atan(sqrt(110)
*sqrt(x - 1/2)/11)/(1596972174336*(x - 1/2)**(35/2) + 18631342033920*(x - 1/2)**(33/2) + 97814545678080*(x - 1
/2)**(31/2) + 304311919887360*(x - 1/2)**(29/2) + 621303503103360*(x - 1/2)**(27/2) + 869824904344704*(x - 1/2
)**(25/2) + 845663101446240*(x - 1/2)**(23/2) + 563775400964160*(x - 1/2)**(21/2) + 246651737921820*(x - 1/2)*
*(19/2) + 63946746868620*(x - 1/2)**(17/2) + 74...

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Giac [A]
time = 1.03, size = 123, normalized size = 1.05 \begin {gather*} \frac {125}{11} \, \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 {44155}{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 {3 \, {\left (5760 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 27265 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 32291 \, \sqrt {-2 \, x + 1}\right )}}{686 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)^4/(3+5*x)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

125/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 44155/2401*sqrt(
21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 3/686*(5760*(2*x - 1)^2*sqrt(
-2*x + 1) - 27265*(-2*x + 1)^(3/2) + 32291*sqrt(-2*x + 1))/(3*x + 2)^3

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Mupad [B]
time = 0.10, size = 89, normalized size = 0.76 \begin {gather*} \frac {88310\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {2636\,\sqrt {1-2\,x}}{63}-\frac {15580\,{\left (1-2\,x\right )}^{3/2}}{441}+\frac {2560\,{\left (1-2\,x\right )}^{5/2}}{343}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - 2*x)^(1/2)*(3*x + 2)^4*(5*x + 3)),x)

[Out]

(88310*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (250*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11)
)/11 + ((2636*(1 - 2*x)^(1/2))/63 - (15580*(1 - 2*x)^(3/2))/441 + (2560*(1 - 2*x)^(5/2))/343)/((98*x)/3 + 7*(2
*x - 1)^2 + (2*x - 1)^3 - 98/27)

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