3.22.24 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx\) [2124]

Optimal. Leaf size=146 \[ \frac {245865}{41503 \sqrt {1-2 x}}-\frac {36175}{1078 \sqrt {1-2 x} (3+5 x)}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}+\frac {165}{49 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {70065}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {24000}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-70065/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+24000/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1
/2)+245865/41503/(1-2*x)^(1/2)-36175/1078/(3+5*x)/(1-2*x)^(1/2)+3/14/(2+3*x)^2/(3+5*x)/(1-2*x)^(1/2)+165/49/(2
+3*x)/(3+5*x)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 146, 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 {245865}{41503 \sqrt {1-2 x}}-\frac {36175}{1078 \sqrt {1-2 x} (5 x+3)}+\frac {165}{49 \sqrt {1-2 x} (3 x+2) (5 x+3)}+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 (5 x+3)}-\frac {70065}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {24000}{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/((1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^2),x]

[Out]

245865/(41503*Sqrt[1 - 2*x]) - 36175/(1078*Sqrt[1 - 2*x]*(3 + 5*x)) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x
)) + 165/(49*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)) - (70065*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (24
000*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[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 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)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx &=\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}+\frac {1}{14} \int \frac {40-105 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}+\frac {165}{49 \sqrt {1-2 x} (2+3 x) (3+5 x)}+\frac {1}{98} \int \frac {2285-8250 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {36175}{1078 \sqrt {1-2 x} (3+5 x)}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}+\frac {165}{49 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {\int \frac {39855-325575 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{1078}\\ &=\frac {245865}{41503 \sqrt {1-2 x}}-\frac {36175}{1078 \sqrt {1-2 x} (3+5 x)}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}+\frac {165}{49 \sqrt {1-2 x} (2+3 x) (3+5 x)}+\frac {\int \frac {-\frac {6019215}{2}+\frac {3687975 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{41503}\\ &=\frac {245865}{41503 \sqrt {1-2 x}}-\frac {36175}{1078 \sqrt {1-2 x} (3+5 x)}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}+\frac {165}{49 \sqrt {1-2 x} (2+3 x) (3+5 x)}+\frac {210195}{686} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {60000}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {245865}{41503 \sqrt {1-2 x}}-\frac {36175}{1078 \sqrt {1-2 x} (3+5 x)}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}+\frac {165}{49 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {210195}{686} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {60000}{121} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {245865}{41503 \sqrt {1-2 x}}-\frac {36175}{1078 \sqrt {1-2 x} (3+5 x)}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}+\frac {165}{49 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {70065}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {24000}{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 = 0.32, size = 101, normalized size = 0.69 \begin {gather*} \frac {-4664333-5050290 x+17711235 x^2+22127850 x^3}{83006 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)}-\frac {70065}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {24000}{121} \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/((1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^2),x]

[Out]

(-4664333 - 5050290*x + 17711235*x^2 + 22127850*x^3)/(83006*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)) - (70065*Sqrt
[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (24000*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]
time = 0.18, size = 91, normalized size = 0.62

method	result	size
risch	\(\frac {22127850 x^{3}[phone] x^{2}-5050290 x -4664333}{83006 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )^{2}}+\frac {24000 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {70065 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(76\)
derivativedivides	\(\frac {250 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {24000 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {32}{41503 \sqrt {1-2 x}}+\frac {\frac {243 \left (1-2 x \right )^{\frac {3}{2}}}{7}-\frac {4023 \sqrt {1-2 x}}{49}}{\left (-4-6 x \right )^{2}}-\frac {70065 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(91\)
default	\(\frac {250 \sqrt {1-2 x}}{121 \left (-\frac {6}{5}-2 x \right )}+\frac {24000 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {32}{41503 \sqrt {1-2 x}}+\frac {\frac {243 \left (1-2 x \right )^{\frac {3}{2}}}{7}-\frac {4023 \sqrt {1-2 x}}{49}}{\left (-4-6 x \right )^{2}}-\frac {70065 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(91\)
trager	\(-\frac {\left (22127850 x^{3}[phone] x^{2}-5050290 x -4664333\right ) \sqrt {1-2 x}}{83006 \left (2+3 x \right )^{2} \left (10 x^{2}+x -3\right )}+\frac {405 \RootOf \left (\textit {\_Z}^{2}-628509\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-628509\right ) x +3633 \sqrt {1-2 x}-5 \RootOf \left (\textit {\_Z}^{2}-628509\right )}{2+3 x}\right )}{4802}+\frac {12000 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{1331}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

250/121*(1-2*x)^(1/2)/(-6/5-2*x)+24000/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+32/41503/(1-2*x)^(1/
2)+486/343*(49/2*(1-2*x)^(3/2)-1043/18*(1-2*x)^(1/2))/(-4-6*x)^2-70065/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2)
)*21^(1/2)

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Maxima [A]
time = 0.51, size = 137, normalized size = 0.94 \begin {gather*} -\frac {12000}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {70065}{4802} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {11063925 \, {\left (2 \, x - 1\right )}^{3} + 50903010 \, {\left (2 \, x - 1\right )}^{2} + 117027330 \, x - 58496417}{41503 \, {\left (45 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 309 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 707 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 539 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12000/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 70065/4802*sqrt(21)*l
og(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/41503*(11063925*(2*x - 1)^3 + 50903010*(2
*x - 1)^2 + 117027330*x - 58496417)/(45*(-2*x + 1)^(7/2) - 309*(-2*x + 1)^(5/2) + 707*(-2*x + 1)^(3/2) - 539*s
qrt(-2*x + 1))

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Fricas [A]
time = 1.24, size = 162, normalized size = 1.11 \begin {gather*} \frac {57624000 \, \sqrt {11} \sqrt {5} {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 93256515 \, \sqrt {7} \sqrt {3} {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (22127850 \, x^{3} + 17711235 \, x^{2} - 5050290 \, x - 4664333\right )} \sqrt {-2 \, x + 1}}{6391462 \, {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6391462*(57624000*sqrt(11)*sqrt(5)*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x
+ 1) - 5*x + 8)/(5*x + 3)) + 93256515*sqrt(7)*sqrt(3)*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*log((sqrt(7)*sqr
t(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(22127850*x^3 + 17711235*x^2 - 5050290*x - 4664333)*sqrt(-2*x +
1))/(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)

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Sympy [C] Result contains complex when optimal does not.
time = 13.49, size = 2222, normalized size = 15.22 \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)**(3/2)/(2+3*x)**3/(3+5*x)**2,x)

[Out]

-245353600800*sqrt(2)*I*(x - 1/2)**(13/2)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 110168352201
6*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 1136904
310080*sqrt(2)*I*(x - 1/2)**(11/2)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x -
1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 1975315945680*
sqrt(2)*I*(x - 1/2)**(9/2)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5
+ 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 1525208808816*sqrt(2)*
I*(x - 1/2)**(7/2)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 127026
4723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 441655676154*sqrt(2)*I*(x - 1/
2)**(5/2)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(
x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 65076704*sqrt(2)*I*(x - 1/2)**(3/2)/(82
833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 +
732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) + 1493614080000*sqrt(55)*I*(x - 1/2)**7*atan(sqrt(110)
*sqrt(x - 1/2)/11)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 127026
4723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 2417208868800*sqrt(21)*I*(x -
1/2)**7*atan(sqrt(42)*sqrt(x - 1/2)/7)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(
x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 7468070400
00*sqrt(55)*I*pi*(x - 1/2)**7/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)*
*5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) + 1208604434400*sqrt(
21)*I*pi*(x - 1/2)**7/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 127
0264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) + 8613174528000*sqrt(55)*I*(x
- 1/2)**6*atan(sqrt(110)*sqrt(x - 1/2)/11)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522
016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 13939
237810080*sqrt(21)*I*(x - 1/2)**6*atan(sqrt(42)*sqrt(x - 1/2)/7)/(82833347520*(x - 1/2)**7 + 477672304032*(x -
1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*
(x - 1/2)**2) - 4306587264000*sqrt(55)*I*pi*(x - 1/2)**6/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6
+ 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2
)**2) + 6969618905040*sqrt(21)*I*pi*(x - 1/2)**6/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 11016
83522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) +
19865067264000*sqrt(55)*I*(x - 1/2)**5*atan(sqrt(110)*sqrt(x - 1/2)/11)/(82833347520*(x - 1/2)**7 + 4776723040
32*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804
902882*(x - 1/2)**2) - 32148877955040*sqrt(21)*I*(x - 1/2)**5*atan(sqrt(42)*sqrt(x - 1/2)/7)/(82833347520*(x -
1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*
(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 9932533632000*sqrt(55)*I*pi*(x - 1/2)**5/(82833347520*(x - 1/2)**7
+ 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2
)**3 + 168804902882*(x - 1/2)**2) + 16074438977520*sqrt(21)*I*pi*(x - 1/2)**5/(82833347520*(x - 1/2)**7 + 4776
72304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 +
168804902882*(x - 1/2)**2) + 22904848512000*sqrt(55)*I*(x - 1/2)**4*atan(sqrt(110)*sqrt(x - 1/2)/11)/(82833347
520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4 + 73221
8669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 37068345634320*sqrt(21)*I*(x - 1/2)**4*atan(sqrt(42)*sqrt(
x - 1/2)/7)/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728
*(x - 1/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) - 11452424256000*sqrt(55)*I*pi*(x - 1/2
)**4/(82833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1
/2)**4 + 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) + 18534172817160*sqrt(21)*I*pi*(x - 1/2)**4/(8
2833347520*(x - 1/2)**7 + 477672304032*(x - 1/2)**6 + 1101683522016*(x - 1/2)**5 + 1270264723728*(x - 1/2)**4
+ 732218669644*(x - 1/2)**3 + 168804902882*(x - 1/2)**2) + 13203041376000*sqrt(55)*I*(x - 1/2)**3*atan(sqrt(11
0)*sqrt(x - 1/2)/11)/(82833347520*(x - 1/2)**7 ...

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Giac [A]
time = 1.19, size = 135, normalized size = 0.92 \begin {gather*} -\frac {12000}{1331} \, \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 {70065}{4802} \, \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 {2 \, {\left (428910 \, x - 214279\right )}}{41503 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \sqrt {-2 \, x + 1}\right )}} + \frac {27 \, {\left (63 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 149 \, \sqrt {-2 \, x + 1}\right )}}{196 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12000/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 70065/4802*
sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/41503*(428910*x - 2142
79)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1)) + 27/196*(63*(-2*x + 1)^(3/2) - 149*sqrt(-2*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 0.11, size = 100, normalized size = 0.68 \begin {gather*} \frac {\frac {1114546\,x}{17787}+\frac {1131178\,{\left (2\,x-1\right )}^2}{41503}+\frac {245865\,{\left (2\,x-1\right )}^3}{41503}-\frac {8356631}{266805}}{\frac {539\,\sqrt {1-2\,x}}{45}-\frac {707\,{\left (1-2\,x\right )}^{3/2}}{45}+\frac {103\,{\left (1-2\,x\right )}^{5/2}}{15}-{\left (1-2\,x\right )}^{7/2}}-\frac {70065\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}+\frac {24000\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1114546*x)/17787 + (1131178*(2*x - 1)^2)/41503 + (245865*(2*x - 1)^3)/41503 - 8356631/266805)/((539*(1 - 2*x
)^(1/2))/45 - (707*(1 - 2*x)^(3/2))/45 + (103*(1 - 2*x)^(5/2))/15 - (1 - 2*x)^(7/2)) - (70065*21^(1/2)*atanh((
21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 + (24000*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1331

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