3.19.47 \(\int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx\) [1847]

Optimal. Leaf size=158 \[ -\frac {48645 \sqrt {1-2 x}}{98 (3+5 x)}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}-\frac {335579}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+6650 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-335579/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+6650/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)
-48645/98*(1-2*x)^(1/2)/(3+5*x)+1/3*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)+139/42*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)+726
1/147*(1-2*x)^(1/2)/(2+3*x)/(3+5*x)

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Rubi [A]
time = 0.04, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {101, 156, 162, 65, 212} \begin {gather*} -\frac {48645 \sqrt {1-2 x}}{98 (5 x+3)}+\frac {7261 \sqrt {1-2 x}}{147 (3 x+2) (5 x+3)}+\frac {139 \sqrt {1-2 x}}{42 (3 x+2)^2 (5 x+3)}+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)}-\frac {335579}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+6650 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-48645*Sqrt[1 - 2*x])/(98*(3 + 5*x)) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3 + 5*x)) + (139*Sqrt[1 - 2*x])/(42*(2 +
3*x)^2*(3 + 5*x)) + (7261*Sqrt[1 - 2*x])/(147*(2 + 3*x)*(3 + 5*x)) - (335579*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]])/49 + 6650*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 101

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

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 {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^2} \, dx &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}-\frac {1}{3} \int \frac {-23+35 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}-\frac {1}{42} \int \frac {-2524+3475 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}-\frac {1}{294} \int \frac {-190359+217830 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {48645 \sqrt {1-2 x}}{98 (3+5 x)}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}+\frac {\int \frac {-7863537+4815855 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{3234}\\ &=-\frac {48645 \sqrt {1-2 x}}{98 (3+5 x)}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}+\frac {1006737}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-16625 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {48645 \sqrt {1-2 x}}{98 (3+5 x)}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}-\frac {1006737}{98} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+16625 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {48645 \sqrt {1-2 x}}{98 (3+5 x)}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)}+\frac {139 \sqrt {1-2 x}}{42 (2+3 x)^2 (3+5 x)}+\frac {7261 \sqrt {1-2 x}}{147 (2+3 x) (3+5 x)}-\frac {335579}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+6650 \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 = 0.34, size = 99, normalized size = 0.63 \begin {gather*} -\frac {\sqrt {1-2 x} \left (369116+1692159 x+2583264 x^2+1313415 x^3\right )}{98 (2+3 x)^3 (3+5 x)}-\frac {335579}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+6650 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/98*(Sqrt[1 - 2*x]*(369116 + 1692159*x + 2583264*x^2 + 1313415*x^3))/((2 + 3*x)^3*(3 + 5*x)) - (335579*Sqrt[
3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + 6650*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]
time = 0.15, size = 91, normalized size = 0.58

method	result	size
risch	\(\frac {2626830 x^{4}+3853113 x^{3}+801054 x^{2}-953927 x -369116}{98 \left (2+3 x \right )^{3} \sqrt {1-2 x}\, \left (3+5 x \right )}+\frac {6650 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {335579 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(81\)
derivativedivides	\(\frac {50 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+\frac {6650 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {\frac {196533 \left (1-2 x \right )^{\frac {5}{2}}}{49}-\frac {132276 \left (1-2 x \right )^{\frac {3}{2}}}{7}+22263 \sqrt {1-2 x}}{\left (-4-6 x \right )^{3}}-\frac {335579 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(91\)
default	\(\frac {50 \sqrt {1-2 x}}{-\frac {6}{5}-2 x}+\frac {6650 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {\frac {196533 \left (1-2 x \right )^{\frac {5}{2}}}{49}-\frac {132276 \left (1-2 x \right )^{\frac {3}{2}}}{7}+22263 \sqrt {1-2 x}}{\left (-4-6 x \right )^{3}}-\frac {335579 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(91\)
trager	\(-\frac {\left (1313415 x^{3}+2583264 x^{2}+1692159 x +369116\right ) \sqrt {1-2 x}}{98 \left (2+3 x \right )^{3} \left (3+5 x \right )}-\frac {335579 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{686}+\frac {3325 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{11}\)	\(129\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50*(1-2*x)^(1/2)/(-6/5-2*x)+6650/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+324*(7279/588*(1-2*x)^(5/2)-
11023/189*(1-2*x)^(3/2)+7421/108*(1-2*x)^(1/2))/(-4-6*x)^3-335579/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(
1/2)

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Maxima [A]
time = 0.64, size = 146, normalized size = 0.92 \begin {gather*} -\frac {3325}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {335579}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1313415 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 9106773 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 21041937 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 16201507 \, \sqrt {-2 \, x + 1}}{49 \, {\left (135 \, {\left (2 \, x - 1\right )}^{4} + 1242 \, {\left (2 \, x - 1\right )}^{3} + 4284 \, {\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3325/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 335579/686*sqrt(21)*log(
-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/49*(1313415*(-2*x + 1)^(7/2) - 9106773*(-2*x
+ 1)^(5/2) + 21041937*(-2*x + 1)^(3/2) - 16201507*sqrt(-2*x + 1))/(135*(2*x - 1)^4 + 1242*(2*x - 1)^3 + 4284*
(2*x - 1)^2 + 13132*x - 2793)

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Fricas [A]
time = 0.87, size = 162, normalized size = 1.03 \begin {gather*} \frac {2280950 \, \sqrt {11} \sqrt {5} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 3691369 \, \sqrt {7} \sqrt {3} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (1313415 \, x^{3} + 2583264 \, x^{2} + 1692159 \, x + 369116\right )} \sqrt {-2 \, x + 1}}{7546 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7546*(2280950*sqrt(11)*sqrt(5)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x +
1) - 5*x + 8)/(5*x + 3)) + 3691369*sqrt(7)*sqrt(3)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((sqrt(7)*sq
rt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(1313415*x^3 + 2583264*x^2 + 1692159*x + 369116)*sqrt(-2*x + 1
))/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)

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Sympy [A]
time = 195.97, size = 733, normalized size = 4.64 \begin {gather*} - 6060 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) + 1632 \left (\begin {cases} \frac {\sqrt {21} \cdot \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 336 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{32} + \frac {5 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{32} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{3}} - \frac {5}{32 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}} - \frac {1}{48 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{3}}\right )}{7203} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {21}}{3} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {21}}{3} \end {cases}\right ) - 5500 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right ) + 20100 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x < - \frac {2}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: x > - \frac {2}{3} \end {cases}\right ) - 33500 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6060*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(
sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (sqrt(1 - 2*x) > -sqrt(21)/3) & (s
qrt(1 - 2*x) < sqrt(21)/3))) + 1632*Piecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(2
1)*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2)
+ 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (sqrt(1 - 2*x) > -sq
rt(21)/3) & (sqrt(1 - 2*x) < sqrt(21)/3))) - 336*Piecewise((sqrt(21)*(-5*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/32
+ 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*
x)/7 + 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt
(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**3))/7203, (sqrt(1 - 2*x) > -sqrt(21)/3)
& (sqrt(1 - 2*x) < sqrt(21)/3))) - 5500*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(
55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/6
05, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5))) + 20100*Piecewise((-sqrt(21)*acoth(sqrt(21)
*sqrt(1 - 2*x)/7)/21, x < -2/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, x > -2/3)) - 33500*Piecewise((
-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/55, x < -3/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(1 - 2*x)/11)/55, x >
-3/5))

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Giac [A]
time = 1.38, size = 139, normalized size = 0.88 \begin {gather*} -\frac {3325}{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 {335579}{686} \, \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 {125 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {3 \, {\left (65511 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 308644 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 363629 \, \sqrt {-2 \, x + 1}\right )}}{392 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3325/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 335579/686*sqr
t(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125*sqrt(-2*x + 1)/(5*x + 3
) - 3/392*(65511*(2*x - 1)^2*sqrt(-2*x + 1) - 308644*(-2*x + 1)^(3/2) + 363629*sqrt(-2*x + 1))/(3*x + 2)^3

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Mupad [B]
time = 1.22, size = 108, normalized size = 0.68 \begin {gather*} \frac {6650\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}-\frac {335579\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {330643\,\sqrt {1-2\,x}}{135}-\frac {111333\,{\left (1-2\,x\right )}^{3/2}}{35}+\frac {3035591\,{\left (1-2\,x\right )}^{5/2}}{2205}-\frac {9729\,{\left (1-2\,x\right )}^{7/2}}{49}}{\frac {13132\,x}{135}+\frac {476\,{\left (2\,x-1\right )}^2}{15}+\frac {46\,{\left (2\,x-1\right )}^3}{5}+{\left (2\,x-1\right )}^4-\frac {931}{45}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6650*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/11 - (335579*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7)
)/343 - ((330643*(1 - 2*x)^(1/2))/135 - (111333*(1 - 2*x)^(3/2))/35 + (3035591*(1 - 2*x)^(5/2))/2205 - (9729*(
1 - 2*x)^(7/2))/49)/((13132*x)/135 + (476*(2*x - 1)^2)/15 + (46*(2*x - 1)^3)/5 + (2*x - 1)^4 - 931/45)

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