3.21.46 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^5 (3+5 x)} \, dx\) [2046]

Optimal. Leaf size=137 \[ \frac {3 \sqrt {1-2 x}}{28 (2+3 x)^4}+\frac {3 \sqrt {1-2 x}}{4 (2+3 x)^3}+\frac {45 \sqrt {1-2 x}}{8 (2+3 x)^2}+\frac {3135 \sqrt {1-2 x}}{56 (2+3 x)}+\frac {36045}{28} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-1250 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

36045/196*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1250/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+3
/28*(1-2*x)^(1/2)/(2+3*x)^4+3/4*(1-2*x)^(1/2)/(2+3*x)^3+45/8*(1-2*x)^(1/2)/(2+3*x)^2+3135/56*(1-2*x)^(1/2)/(2+
3*x)

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Rubi [A]
time = 0.04, antiderivative size = 137, 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 = {105, 156, 162, 65, 212} \begin {gather*} \frac {3135 \sqrt {1-2 x}}{56 (3 x+2)}+\frac {45 \sqrt {1-2 x}}{8 (3 x+2)^2}+\frac {3 \sqrt {1-2 x}}{4 (3 x+2)^3}+\frac {3 \sqrt {1-2 x}}{28 (3 x+2)^4}+\frac {36045}{28} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-1250 \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)^5*(3 + 5*x)),x]

[Out]

(3*Sqrt[1 - 2*x])/(28*(2 + 3*x)^4) + (3*Sqrt[1 - 2*x])/(4*(2 + 3*x)^3) + (45*Sqrt[1 - 2*x])/(8*(2 + 3*x)^2) +
(3135*Sqrt[1 - 2*x])/(56*(2 + 3*x)) + (36045*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/28 - 1250*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)^5 (3+5 x)} \, dx &=\frac {3 \sqrt {1-2 x}}{28 (2+3 x)^4}+\frac {1}{28} \int \frac {77-105 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac {3 \sqrt {1-2 x}}{28 (2+3 x)^4}+\frac {3 \sqrt {1-2 x}}{4 (2+3 x)^3}+\frac {1}{588} \int \frac {8085-11025 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac {3 \sqrt {1-2 x}}{28 (2+3 x)^4}+\frac {3 \sqrt {1-2 x}}{4 (2+3 x)^3}+\frac {45 \sqrt {1-2 x}}{8 (2+3 x)^2}+\frac {\int \frac {612255-694575 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{8232}\\ &=\frac {3 \sqrt {1-2 x}}{28 (2+3 x)^4}+\frac {3 \sqrt {1-2 x}}{4 (2+3 x)^3}+\frac {45 \sqrt {1-2 x}}{8 (2+3 x)^2}+\frac {3135 \sqrt {1-2 x}}{56 (2+3 x)}+\frac {\int \frac {26337255-16129575 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{57624}\\ &=\frac {3 \sqrt {1-2 x}}{28 (2+3 x)^4}+\frac {3 \sqrt {1-2 x}}{4 (2+3 x)^3}+\frac {45 \sqrt {1-2 x}}{8 (2+3 x)^2}+\frac {3135 \sqrt {1-2 x}}{56 (2+3 x)}-\frac {108135}{56} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+3125 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {3 \sqrt {1-2 x}}{28 (2+3 x)^4}+\frac {3 \sqrt {1-2 x}}{4 (2+3 x)^3}+\frac {45 \sqrt {1-2 x}}{8 (2+3 x)^2}+\frac {3135 \sqrt {1-2 x}}{56 (2+3 x)}+\frac {108135}{56} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-3125 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {3 \sqrt {1-2 x}}{28 (2+3 x)^4}+\frac {3 \sqrt {1-2 x}}{4 (2+3 x)^3}+\frac {45 \sqrt {1-2 x}}{8 (2+3 x)^2}+\frac {3135 \sqrt {1-2 x}}{56 (2+3 x)}+\frac {36045}{28} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-1250 \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 = 0.35, size = 92, normalized size = 0.67 \begin {gather*} \frac {3 \sqrt {1-2 x} \left (8810+38922 x+57375 x^2+28215 x^3\right )}{56 (2+3 x)^4}+\frac {36045}{28} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-1250 \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)^5*(3 + 5*x)),x]

[Out]

(3*Sqrt[1 - 2*x]*(8810 + 38922*x + 57375*x^2 + 28215*x^3))/(56*(2 + 3*x)^4) + (36045*Sqrt[3/7]*ArcTanh[Sqrt[3/
7]*Sqrt[1 - 2*x]])/28 - 1250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]
time = 0.15, size = 84, normalized size = 0.61

method	result	size
risch	\(-\frac {3 \left (56430 x^{4}+86535 x^{3}+20469 x^{2}-21302 x -8810\right )}{56 \left (2+3 x \right )^{4} \sqrt {1-2 x}}-\frac {1250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {36045 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{196}\)	\(74\)
derivativedivides	\(-\frac {1250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {486 \left (\frac {1045 \left (1-2 x \right )^{\frac {7}{2}}}{168}-\frac {1055 \left (1-2 x \right )^{\frac {5}{2}}}{24}+\frac {22373 \left (1-2 x \right )^{\frac {3}{2}}}{216}-\frac {369133 \sqrt {1-2 x}}{4536}\right )}{\left (-4-6 x \right )^{4}}+\frac {36045 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{196}\)	\(84\)
default	\(-\frac {1250 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {486 \left (\frac {1045 \left (1-2 x \right )^{\frac {7}{2}}}{168}-\frac {1055 \left (1-2 x \right )^{\frac {5}{2}}}{24}+\frac {22373 \left (1-2 x \right )^{\frac {3}{2}}}{216}-\frac {369133 \sqrt {1-2 x}}{4536}\right )}{\left (-4-6 x \right )^{4}}+\frac {36045 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{196}\)	\(84\)
trager	\(\frac {3 \left (28215 x^{3}+57375 x^{2}+38922 x +8810\right ) \sqrt {1-2 x}}{56 \left (2+3 x \right )^{4}}-\frac {36045 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{392}-\frac {625 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{11}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1250/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-486*(1045/168*(1-2*x)^(7/2)-1055/24*(1-2*x)^(5/2)+22373
/216*(1-2*x)^(3/2)-369133/4536*(1-2*x)^(1/2))/(-4-6*x)^4+36045/196*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2
)

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Maxima [A]
time = 0.52, size = 146, normalized size = 1.07 \begin {gather*} \frac {625}{11} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {36045}{392} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {3 \, {\left (28215 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 199395 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 469833 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 369133 \, \sqrt {-2 \, x + 1}\right )}}{28 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 36045/392*sqrt(21)*log(-(s
qrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 3/28*(28215*(-2*x + 1)^(7/2) - 199395*(-2*x + 1)^
(5/2) + 469833*(-2*x + 1)^(3/2) - 369133*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2
+ 8232*x - 1715)

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Fricas [A]
time = 0.83, size = 162, normalized size = 1.18 \begin {gather*} \frac {245000 \, \sqrt {11} \sqrt {5} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 396495 \, \sqrt {7} \sqrt {3} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 231 \, {\left (28215 \, x^{3} + 57375 \, x^{2} + 38922 \, x + 8810\right )} \sqrt {-2 \, x + 1}}{4312 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4312*(245000*sqrt(11)*sqrt(5)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1)
+ 5*x - 8)/(5*x + 3)) + 396495*sqrt(7)*sqrt(3)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(-(sqrt(7)*sqrt(3)*
sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 231*(28215*x^3 + 57375*x^2 + 38922*x + 8810)*sqrt(-2*x + 1))/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [C] Result contains complex when optimal does not.
time = 33.28, size = 25400, normalized size = 185.40 \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)**5/(3+5*x)/(1-2*x)**(1/2),x)

[Out]

50454063292862952898560*sqrt(2)*I*(x - 1/2)**(83/2)/(2703758415694091255808*(x - 1/2)**42 + 630876963661954626
35520*(x - 1/2)**41 + 699221968058666377543680*(x - 1/2)**40 + 4894553776410664642805760*(x - 1/2)**39 + 24268
829141369545520578560*(x - 1/2)**38 + 90603628794446303276826624*(x - 1/2)**37 + 264260583983801717890744320*(
x - 1/2)**36 + 616608029295537341745070080*(x - 1/2)**35 + 1168986055539456210391695360*(x - 1/2)**34 + 181842
2753061376327275970560*(x - 1/2)**33 + 2333642533095432953337495552*(x - 1/2)**32 + 24750754138890955565700710
40*(x - 1/2)**31 + 2165690987152958611998812160*(x - 1/2)**30 + 1554855067699560029127352320*(x - 1/2)**29 + 9
06998789491410016990955520*(x - 1/2)**28 + 423266101762658007929112576*(x - 1/2)**27 + 15431576626763573205748
8960*(x - 1/2)**26 + 42361190740135298996173440*(x - 1/2)**25 + 8236898199470752582589280*(x - 1/2)**24 + 1011
548901689390668037280*(x - 1/2)**23 + 59007019265214455635508*(x - 1/2)**22) + 1120088252001604262952960*sqrt(
2)*I*(x - 1/2)**(81/2)/(2703758415694091255808*(x - 1/2)**42 + 63087696366195462635520*(x - 1/2)**41 + 6992219
68058666377543680*(x - 1/2)**40 + 4894553776410664642805760*(x - 1/2)**39 + 24268829141369545520578560*(x - 1/
2)**38 + 90603628794446303276826624*(x - 1/2)**37 + 264260583983801717890744320*(x - 1/2)**36 + 61660802929553
7341745070080*(x - 1/2)**35 + 1168986055539456210391695360*(x - 1/2)**34 + 1818422753061376327275970560*(x - 1
/2)**33 + 2333642533095432953337495552*(x - 1/2)**32 + 2475075413889095556570071040*(x - 1/2)**31 + 2165690987
152958611998812160*(x - 1/2)**30 + 1554855067699560029127352320*(x - 1/2)**29 + 906998789491410016990955520*(x
- 1/2)**28 + 423266101762658007929112576*(x - 1/2)**27 + 154315766267635732057488960*(x - 1/2)**26 + 42361190
740135298996173440*(x - 1/2)**25 + 8236898199470752582589280*(x - 1/2)**24 + 1011548901689390668037280*(x - 1/
2)**23 + 59007019265214455635508*(x - 1/2)**22) + 11778745165020273181851648*sqrt(2)*I*(x - 1/2)**(79/2)/(2703
758415694091255808*(x - 1/2)**42 + 63087696366195462635520*(x - 1/2)**41 + 699221968058666377543680*(x - 1/2)*
*40 + 4894553776410664642805760*(x - 1/2)**39 + 24268829141369545520578560*(x - 1/2)**38 + 9060362879444630327
6826624*(x - 1/2)**37 + 264260583983801717890744320*(x - 1/2)**36 + 616608029295537341745070080*(x - 1/2)**35
+ 1168986055539456210391695360*(x - 1/2)**34 + 1818422753061376327275970560*(x - 1/2)**33 + 233364253309543295
3337495552*(x - 1/2)**32 + 2475075413889095556570071040*(x - 1/2)**31 + 2165690987152958611998812160*(x - 1/2)
**30 + 1554855067699560029127352320*(x - 1/2)**29 + 906998789491410016990955520*(x - 1/2)**28 + 42326610176265
8007929112576*(x - 1/2)**27 + 154315766267635732057488960*(x - 1/2)**26 + 42361190740135298996173440*(x - 1/2)
**25 + 8236898199470752582589280*(x - 1/2)**24 + 1011548901689390668037280*(x - 1/2)**23 + 5900701926521445563
5508*(x - 1/2)**22) + 77988893344315152337797120*sqrt(2)*I*(x - 1/2)**(77/2)/(2703758415694091255808*(x - 1/2)
**42 + 63087696366195462635520*(x - 1/2)**41 + 699221968058666377543680*(x - 1/2)**40 + 4894553776410664642805
760*(x - 1/2)**39 + 24268829141369545520578560*(x - 1/2)**38 + 90603628794446303276826624*(x - 1/2)**37 + 2642
60583983801717890744320*(x - 1/2)**36 + 616608029295537341745070080*(x - 1/2)**35 + 11689860555394562103916953
60*(x - 1/2)**34 + 1818422753061376327275970560*(x - 1/2)**33 + 2333642533095432953337495552*(x - 1/2)**32 + 2
475075413889095556570071040*(x - 1/2)**31 + 2165690987152958611998812160*(x - 1/2)**30 + 155485506769956002912
7352320*(x - 1/2)**29 + 906998789491410016990955520*(x - 1/2)**28 + 423266101762658007929112576*(x - 1/2)**27
+ 154315766267635732057488960*(x - 1/2)**26 + 42361190740135298996173440*(x - 1/2)**25 + 823689819947075258258
9280*(x - 1/2)**24 + 1011548901689390668037280*(x - 1/2)**23 + 59007019265214455635508*(x - 1/2)**22) + 364502
587537922192734814208*sqrt(2)*I*(x - 1/2)**(75/2)/(2703758415694091255808*(x - 1/2)**42 + 63087696366195462635
520*(x - 1/2)**41 + 699221968058666377543680*(x - 1/2)**40 + 4894553776410664642805760*(x - 1/2)**39 + 2426882
9141369545520578560*(x - 1/2)**38 + 90603628794446303276826624*(x - 1/2)**37 + 264260583983801717890744320*(x
- 1/2)**36 + 616608029295537341745070080*(x - 1/2)**35 + 1168986055539456210391695360*(x - 1/2)**34 + 18184227
53061376327275970560*(x - 1/2)**33 + 2333642533095432953337495552*(x - 1/2)**32 + 2475075413889095556570071040
*(x - 1/2)**31 + 2165690987152958611998812160*(x - 1/2)**30 + 1554855067699560029127352320*(x - 1/2)**29 + 906
998789491410016990955520*(x - 1/2)**28 + 423266101762658007929112576*(x - 1/2)**27 + 1543157662676357320574889
60*(x - 1/2)**26 + 42361190740135298996173440*(x - 1/2)**25 + 8236898199470752582589280*(x - 1/2)**24 + 101154
8901689390668037280*(x - 1/2)**23 + 59007019265214455635508*(x - 1/2)**22) + 1277707806603000396814417920*sqrt
(2)*I*(x - 1/2)**(73/2)/(2703758415694091255808*(x - 1/2)**42 + 63087696366195462635520*(x - 1/2)**41 + 699221
968058666377543680*(x - 1/2)**40 + 489455377641...

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Giac [A]
time = 0.80, size = 139, normalized size = 1.01 \begin {gather*} \frac {625}{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 {36045}{392} \, \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 (28215 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 199395 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 469833 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 369133 \, \sqrt {-2 \, x + 1}\right )}}{448 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 36045/392*sqrt(2
1)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 3/448*(28215*(2*x - 1)^3*sqrt(
-2*x + 1) + 199395*(2*x - 1)^2*sqrt(-2*x + 1) - 469833*(-2*x + 1)^(3/2) + 369133*sqrt(-2*x + 1))/(3*x + 2)^4

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Mupad [B]
time = 1.28, size = 107, normalized size = 0.78 \begin {gather*} \frac {36045\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{196}-\frac {1250\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {369133\,\sqrt {1-2\,x}}{756}-\frac {22373\,{\left (1-2\,x\right )}^{3/2}}{36}+\frac {1055\,{\left (1-2\,x\right )}^{5/2}}{4}-\frac {1045\,{\left (1-2\,x\right )}^{7/2}}{28}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(36045*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/196 - (1250*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11)
)/11 + ((369133*(1 - 2*x)^(1/2))/756 - (22373*(1 - 2*x)^(3/2))/36 + (1055*(1 - 2*x)^(5/2))/4 - (1045*(1 - 2*x)
^(7/2))/28)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)

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