3.21.67 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx\) [2067]

Optimal. Leaf size=153 \[ -\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}+\frac {134217}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {321825}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

134217/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-321825/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1
/2)-35495/1078*(1-2*x)^(1/2)/(3+5*x)^2+3/14*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^2+429/98*(1-2*x)^(1/2)/(2+3*x)/(3+
5*x)^2+1177080/5929*(1-2*x)^(1/2)/(3+5*x)

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Rubi [A]
time = 0.04, antiderivative size = 153, 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 {1177080 \sqrt {1-2 x}}{5929 (5 x+3)}-\frac {35495 \sqrt {1-2 x}}{1078 (5 x+3)^2}+\frac {429 \sqrt {1-2 x}}{98 (3 x+2) (5 x+3)^2}+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^2}+\frac {134217}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {321825}{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)^3*(3 + 5*x)^3),x]

[Out]

(-35495*Sqrt[1 - 2*x])/(1078*(3 + 5*x)^2) + (3*Sqrt[1 - 2*x])/(14*(2 + 3*x)^2*(3 + 5*x)^2) + (429*Sqrt[1 - 2*x
])/(98*(2 + 3*x)*(3 + 5*x)^2) + (1177080*Sqrt[1 - 2*x])/(5929*(3 + 5*x)) + (134217*Sqrt[3/7]*ArcTanh[Sqrt[3/7]
*Sqrt[1 - 2*x]])/49 - (321825*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 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)^3 (3+5 x)^3} \, dx &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {1}{14} \int \frac {73-105 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1}{98} \int \frac {7763-10725 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}-\frac {\int \frac {558318-638910 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{2156}\\ &=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}+\frac {\int \frac {23063874-14124960 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{23716}\\ &=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}-\frac {402651}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {1609125}{242} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}+\frac {402651}{98} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {1609125}{242} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {35495 \sqrt {1-2 x}}{1078 (3+5 x)^2}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^2}+\frac {429 \sqrt {1-2 x}}{98 (2+3 x) (3+5 x)^2}+\frac {1177080 \sqrt {1-2 x}}{5929 (3+5 x)}+\frac {134217}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {321825}{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 = 0.29, size = 99, normalized size = 0.65 \begin {gather*} \frac {\sqrt {1-2 x} \left (26794499+127303347 x+201297915 x^2+105937200 x^3\right )}{11858 \left (6+19 x+15 x^2\right )^2}+\frac {134217}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {321825}{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/(Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^3),x]

[Out]

(Sqrt[1 - 2*x]*(26794499 + 127303347*x + 201297915*x^2 + 105937200*x^3))/(11858*(6 + 19*x + 15*x^2)^2) + (1342
17*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (321825*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]
time = 0.30, size = 94, normalized size = 0.61

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \left (105937200 x^{3}[phone] x^{2}[phone] x [phone]\right )}{11858 \left (15 x^{2}+19 x +6\right )^{2} \sqrt {1-2 x}}-\frac {321825 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {134217 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(79\)
derivativedivides	\(\frac {-\frac {121875 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {24125 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {321825 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {972 \left (\frac {71 \left (1-2 x \right )^{\frac {3}{2}}}{196}-\frac {215 \sqrt {1-2 x}}{252}\right )}{\left (-4-6 x \right )^{2}}+\frac {134217 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(94\)
default	\(\frac {-\frac {121875 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {24125 \sqrt {1-2 x}}{11}}{\left (-6-10 x \right )^{2}}-\frac {321825 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}-\frac {972 \left (\frac {71 \left (1-2 x \right )^{\frac {3}{2}}}{196}-\frac {215 \sqrt {1-2 x}}{252}\right )}{\left (-4-6 x \right )^{2}}+\frac {134217 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(94\)
trager	\(\frac {\left (105937200 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {1-2 x}}{11858 \left (15 x^{2}+19 x +6\right )^{2}}-\frac {525 \RootOf \left (\textit {\_Z}^{2}-20667295\right ) \ln \left (\frac {-5 \RootOf \left (\textit {\_Z}^{2}-20667295\right ) x +33715 \sqrt {1-2 x}+8 \RootOf \left (\textit {\_Z}^{2}-20667295\right )}{3+5 x}\right )}{2662}-\frac {81 \RootOf \left (\textit {\_Z}^{2}-57658629\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-57658629\right ) x +34797 \sqrt {1-2 x}-5 \RootOf \left (\textit {\_Z}^{2}-57658629\right )}{2+3 x}\right )}{686}\)	\(126\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

62500*(-39/2420*(1-2*x)^(3/2)+193/5500*(1-2*x)^(1/2))/(-6-10*x)^2-321825/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1
/2))*55^(1/2)-972*(71/196*(1-2*x)^(3/2)-215/252*(1-2*x)^(1/2))/(-4-6*x)^2+134217/343*arctanh(1/7*21^(1/2)*(1-2
*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.51, size = 146, normalized size = 0.95 \begin {gather*} \frac {321825}{2662} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {134217}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (52968600 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 360203715 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 816108324 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 616051205 \, \sqrt {-2 \, x + 1}\right )}}{5929 \, {\left (225 \, {\left (2 \, x - 1\right )}^{4} + 2040 \, {\left (2 \, x - 1\right )}^{3} + 6934 \, {\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

321825/2662*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 134217/686*sqrt(21)*l
og(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/5929*(52968600*(-2*x + 1)^(7/2) - 3602037
15*(-2*x + 1)^(5/2) + 816108324*(-2*x + 1)^(3/2) - 616051205*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2040*(2*x - 1)
^3 + 6934*(2*x - 1)^2 + 20944*x - 4543)

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Fricas [A]
time = 0.69, size = 162, normalized size = 1.06 \begin {gather*} \frac {110385975 \, \sqrt {11} \sqrt {5} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 178642827 \, \sqrt {7} \sqrt {3} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (105937200 \, x^{3} + 201297915 \, x^{2} + 127303347 \, x + 26794499\right )} \sqrt {-2 \, x + 1}}{913066 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/913066*(110385975*sqrt(11)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((sqrt(11)*sqrt(5)*sqrt(-2*
x + 1) + 5*x - 8)/(5*x + 3)) + 178642827*sqrt(7)*sqrt(3)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(-(sqrt
(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(105937200*x^3 + 201297915*x^2 + 127303347*x + 26794499)
*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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Sympy [C] Result contains complex when optimal does not.
time = 16.21, size = 6346, normalized size = 41.48 \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)**3/(3+5*x)**3/(1-2*x)**(1/2),x)

[Out]

234926334720000*sqrt(2)*I*(x - 1/2)**(23/2)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 2127
63369772800*(x - 1/2)**10 + 482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x -
1/2)**7 + 350409449828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 186378
7142448000*sqrt(2)*I*(x - 1/2)**(21/2)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369
772800*(x - 1/2)**10 + 482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)
**7 + 350409449828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 63360489753
79200*sqrt(2)*I*(x - 1/2)**(19/2)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 21276336977280
0*(x - 1/2)**10 + 482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 +
350409449828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 1196472136205808
0*sqrt(2)*I*(x - 1/2)**(17/2)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x
- 1/2)**10 + 482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 350
409449828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 13554148250345472*sq
rt(2)*I*(x - 1/2)**(15/2)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1
/2)**10 + 482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 3504094
49828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 9211438082389928*sqrt(2)
*I*(x - 1/2)**(13/2)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**
10 + 482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 350409449828
592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 3477318297300848*sqrt(2)*I*(x
- 1/2)**(11/2)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**10 +
482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 350409449828592*(
x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 562495409544530*sqrt(2)*I*(x - 1/2
)**(9/2)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**10 + 4821444
28254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 350409449828592*(x - 1/2
)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 21448476000000*sqrt(55)*I*(x - 1/2)**12*a
tan(sqrt(110)/(10*sqrt(x - 1/2)))/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 21276336977280
0*(x - 1/2)**10 + 482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 +
350409449828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) - 1409153760000000
*sqrt(55)*I*(x - 1/2)**12*atan(sqrt(110)*sqrt(x - 1/2)/11)/(5916667680000*(x - 1/2)**12 + 53644453632000*(x -
1/2)**11 + 212763369772800*(x - 1/2)**10 + 482144428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 61875
2016260224*(x - 1/2)**7 + 350409449828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 + 16048523266853*(x - 1/
2)**4) + 33378285600000*sqrt(21)*I*(x - 1/2)**12*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(5916667680000*(x - 1/2)**12
+ 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**10 + 482144428254720*(x - 1/2)**9 + 6827846628236
48*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 350409449828592*(x - 1/2)**6 + 113381774768416*(x - 1/2)**5 +
16048523266853*(x - 1/2)**4) + 2348589323520000*sqrt(21)*I*(x - 1/2)**12*atan(sqrt(42)*sqrt(x - 1/2)/7)/(5916
667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**10 + 482144428254720*(x -
1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 350409449828592*(x - 1/2)**6 + 1133817
74768416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) - 1174294661760000*sqrt(21)*I*pi*(x - 1/2)**12/(591666768
0000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**10 + 482144428254720*(x - 1/2)*
*9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 350409449828592*(x - 1/2)**6 + 113381774768
416*(x - 1/2)**5 + 16048523266853*(x - 1/2)**4) + 704576880000000*sqrt(55)*I*pi*(x - 1/2)**12/(5916667680000*(
x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**10 + 482144428254720*(x - 1/2)**9 + 6
82784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 350409449828592*(x - 1/2)**6 + 113381774768416*(x
- 1/2)**5 + 16048523266853*(x - 1/2)**4) + 194466182400000*sqrt(55)*I*(x - 1/2)**11*atan(sqrt(110)/(10*sqrt(x
- 1/2)))/(5916667680000*(x - 1/2)**12 + 53644453632000*(x - 1/2)**11 + 212763369772800*(x - 1/2)**10 + 482144
428254720*(x - 1/2)**9 + 682784662823648*(x - 1/2)**8 + 618752016260224*(x - 1/2)**7 + 350409449828592*(x - 1/
2)**6 + 113381774768416*(x - 1/2)**5 + 16048523...

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Giac [A]
time = 1.17, size = 148, normalized size = 0.97 \begin {gather*} \frac {321825}{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 {134217}{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 {2 \, {\left (52968600 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 360203715 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 816108324 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 616051205 \, \sqrt {-2 \, x + 1}\right )}}{5929 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

321825/2662*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 134217/686*
sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/5929*(52968600*(2*x -
1)^3*sqrt(-2*x + 1) + 360203715*(2*x - 1)^2*sqrt(-2*x + 1) - 816108324*(-2*x + 1)^(3/2) + 616051205*sqrt(-2*x
+ 1))/(15*(2*x - 1)^2 + 136*x + 9)^2

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Mupad [B]
time = 0.11, size = 107, normalized size = 0.70 \begin {gather*} \frac {134217\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {321825\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}+\frac {\frac {3200266\,\sqrt {1-2\,x}}{3465}-\frac {544072216\,{\left (1-2\,x\right )}^{3/2}}{444675}+\frac {16009054\,{\left (1-2\,x\right )}^{5/2}}{29645}-\frac {470832\,{\left (1-2\,x\right )}^{7/2}}{5929}}{\frac {20944\,x}{225}+\frac {6934\,{\left (2\,x-1\right )}^2}{225}+\frac {136\,{\left (2\,x-1\right )}^3}{15}+{\left (2\,x-1\right )}^4-\frac {4543}{225}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(134217*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - (321825*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/
11))/1331 + ((3200266*(1 - 2*x)^(1/2))/3465 - (544072216*(1 - 2*x)^(3/2))/444675 + (16009054*(1 - 2*x)^(5/2))/
29645 - (470832*(1 - 2*x)^(7/2))/5929)/((20944*x)/225 + (6934*(2*x - 1)^2)/225 + (136*(2*x - 1)^3)/15 + (2*x -
1)^4 - 4543/225)

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