3.22.98 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx\) [2198]

Optimal. Leaf size=152 \[ -\frac {667615}{195657 (1-2 x)^{3/2}}-\frac {7554245}{5021863 \sqrt {1-2 x}}-\frac {505}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {32765}{1694 (1-2 x)^{3/2} (3+5 x)}+\frac {17820}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {738625 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]

[Out]

-667615/195657/(1-2*x)^(3/2)-505/154/(1-2*x)^(3/2)/(3+5*x)^2+3/7/(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^2+32765/1694/(1
-2*x)^(3/2)/(3+5*x)+17820/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-738625/161051*arctanh(1/11*55^(1/2
)*(1-2*x)^(1/2))*55^(1/2)-7554245/5021863/(1-2*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {7554245}{5021863 \sqrt {1-2 x}}+\frac {32765}{1694 (1-2 x)^{3/2} (5 x+3)}-\frac {667615}{195657 (1-2 x)^{3/2}}+\frac {3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)^2}-\frac {505}{154 (1-2 x)^{3/2} (5 x+3)^2}+\frac {17820}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {738625 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-667615/(195657*(1 - 2*x)^(3/2)) - 7554245/(5021863*Sqrt[1 - 2*x]) - 505/(154*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + 3
/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^2) + 32765/(1694*(1 - 2*x)^(3/2)*(3 + 5*x)) + (17820*Sqrt[3/7]*ArcTanh
[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (738625*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

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)^{5/2} (2+3 x)^2 (3+5 x)^3} \, dx &=\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {1}{7} \int \frac {20-135 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {505}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}-\frac {1}{154} \int \frac {190-10605 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {505}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {32765}{1694 (1-2 x)^{3/2} (3+5 x)}+\frac {\int \frac {-88070-491475 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx}{1694}\\ &=-\frac {667615}{195657 (1-2 x)^{3/2}}-\frac {505}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {32765}{1694 (1-2 x)^{3/2} (3+5 x)}-\frac {\int \frac {-1844985+\frac {30042675 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{195657}\\ &=-\frac {667615}{195657 (1-2 x)^{3/2}}-\frac {7554245}{5021863 \sqrt {1-2 x}}-\frac {505}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {32765}{1694 (1-2 x)^{3/2} (3+5 x)}+\frac {2 \int \frac {\frac {278040255}{2}-\frac {339941025 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{15065589}\\ &=-\frac {667615}{195657 (1-2 x)^{3/2}}-\frac {7554245}{5021863 \sqrt {1-2 x}}-\frac {505}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {32765}{1694 (1-2 x)^{3/2} (3+5 x)}-\frac {26730}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {3693125 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{29282}\\ &=-\frac {667615}{195657 (1-2 x)^{3/2}}-\frac {7554245}{5021863 \sqrt {1-2 x}}-\frac {505}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {32765}{1694 (1-2 x)^{3/2} (3+5 x)}+\frac {26730}{343} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {3693125 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{29282}\\ &=-\frac {667615}{195657 (1-2 x)^{3/2}}-\frac {7554245}{5021863 \sqrt {1-2 x}}-\frac {505}{154 (1-2 x)^{3/2} (3+5 x)^2}+\frac {3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {32765}{1694 (1-2 x)^{3/2} (3+5 x)}+\frac {17820}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {738625 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 106, normalized size = 0.70 \begin {gather*} \frac {645558882-479695050 x-4110847595 x^2+1580768100 x^3+6798820500 x^4}{30131178 (1-2 x)^{3/2} (2+3 x) (3+5 x)^2}+\frac {17820}{343} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {738625 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(645558882 - 479695050*x - 4110847595*x^2 + 1580768100*x^3 + 6798820500*x^4)/(30131178*(1 - 2*x)^(3/2)*(2 + 3*
x)*(3 + 5*x)^2) + (17820*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (738625*Sqrt[5/11]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]])/14641

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Maple [A]
time = 0.19, size = 100, normalized size = 0.66

method	result	size
risch	\(-\frac {6798820500 x^{4}[phone] x^{3}-4110847595 x^{2}-479695050 x [phone]}{30131178 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (-1+2 x \right ) \left (2+3 x \right )}-\frac {738625 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {17820 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(88\)
derivativedivides	\(\frac {-\frac {3125 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {74375 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {738625 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {32}{195657 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5472}{5021863 \sqrt {1-2 x}}-\frac {162 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}+\frac {17820 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(100\)
default	\(\frac {-\frac {3125 \left (1-2 x \right )^{\frac {3}{2}}}{121}+\frac {74375 \sqrt {1-2 x}}{1331}}{\left (-6-10 x \right )^{2}}-\frac {738625 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{161051}+\frac {32}{195657 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5472}{5021863 \sqrt {1-2 x}}-\frac {162 \sqrt {1-2 x}}{343 \left (-\frac {4}{3}-2 x \right )}+\frac {17820 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2401}\)	\(100\)
trager	\(\frac {\left (6798820500 x^{4}[phone] x^{3}-4110847595 x^{2}-479695050 x [phone]\right ) \sqrt {1-2 x}}{30131178 \left (10 x^{2}+x -3\right )^{2} \left (2+3 x \right )}+\frac {2375 \RootOf \left (\textit {\_Z}^{2}-5319655\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-5319655\right ) x +17105 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-5319655\right )}{3+5 x}\right )}{322102}-\frac {8910 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{2401}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

156250/14641*(-121/50*(1-2*x)^(3/2)+1309/250*(1-2*x)^(1/2))/(-6-10*x)^2-738625/161051*arctanh(1/11*55^(1/2)*(1
-2*x)^(1/2))*55^(1/2)+32/195657/(1-2*x)^(3/2)+5472/5021863/(1-2*x)^(1/2)-162/343*(1-2*x)^(1/2)/(-4/3-2*x)+1782
0/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.50, size = 146, normalized size = 0.96 \begin {gather*} \frac {738625}{322102} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {8910}{2401} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1699705125 \, {\left (2 \, x - 1\right )}^{4} + 7589204550 \, {\left (2 \, x - 1\right )}^{3} + 8458535305 \, {\left (2 \, x - 1\right )}^{2} - 22225280 \, x + 13199648}{15065589 \, {\left (75 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 505 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 1133 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 847 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

738625/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 8910/2401*sqrt(21)*
log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/15065589*(1699705125*(2*x - 1)^4 + 75892
04550*(2*x - 1)^3 + 8458535305*(2*x - 1)^2 - 22225280*x + 13199648)/(75*(-2*x + 1)^(9/2) - 505*(-2*x + 1)^(7/2
) + 1133*(-2*x + 1)^(5/2) - 847*(-2*x + 1)^(3/2))

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Fricas [A]
time = 0.98, size = 182, normalized size = 1.20 \begin {gather*} \frac {5320315875 \, \sqrt {11} \sqrt {5} {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 8609786460 \, \sqrt {7} \sqrt {3} {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (6798820500 \, x^{4} + 1580768100 \, x^{3} - 4110847595 \, x^{2} - 479695050 \, x + 645558882\right )} \sqrt {-2 \, x + 1}}{2320100706 \, {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2320100706*(5320315875*sqrt(11)*sqrt(5)*(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*log((sqrt(11)*sq
rt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 8609786460*sqrt(7)*sqrt(3)*(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2
+ 15*x + 18)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(6798820500*x^4 + 1580768100*x^3
- 4110847595*x^2 - 479695050*x + 645558882)*sqrt(-2*x + 1))/(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 1
8)

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Sympy [C] Result contains complex when optimal does not.
time = 14.47, size = 2966, normalized size = 19.51 \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)**(5/2)/(2+3*x)**2/(3+5*x)**3,x)

[Out]

376926608520000*sqrt(2)*I*(x - 1/2)**(17/2)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 96
53660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 56123978510365
92*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 2135605689756000*sqrt(2)*I*(x - 1/2)**(15/2)/(5011417524960
00*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)*
*6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 4838544
255837600*sqrt(2)*I*(x - 1/2)**(13/2)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 96536606
25581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x
- 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 5479255948116720*sqrt(2)*I*(x - 1/2)**(11/2)/(501141752496000*(x
- 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 1
2388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 3100767153386
980*sqrt(2)*I*(x - 1/2)**(9/2)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 965366062558128
0*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)*
*4 + 1059202535612298*(x - 1/2)**3) + 700998571871598*sqrt(2)*I*(x - 1/2)**(7/2)/(501141752496000*(x - 1/2)**9
+ 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 1238886446
9495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) - 347593269408*sqrt(2)*I
*(x - 1/2)**(5/2)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**
7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 10592025
35612298*(x - 1/2)**3) + 43308546512*sqrt(2)*I*(x - 1/2)**(3/2)/(501141752496000*(x - 1/2)**9 + 34077639169728
00*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)
**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) - 2298376458000000*sqrt(55)*I*(x - 1/2)**
9*atan(sqrt(110)*sqrt(x - 1/2)/11)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 96536606255
81280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1
/2)**4 + 1059202535612298*(x - 1/2)**3) + 3719427750720000*sqrt(21)*I*(x - 1/2)**9*atan(sqrt(42)*sqrt(x - 1/2)
/7)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 1458270529
5075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612298*(x -
1/2)**3) - 1859713875360000*sqrt(21)*I*pi*(x - 1/2)**9/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1
/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 561
2397851036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 1149188229000000*sqrt(55)*I*pi*(x - 1/2)**9/(501
141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*
(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3)
- 15628959914400000*sqrt(55)*I*(x - 1/2)**8*atan(sqrt(110)*sqrt(x - 1/2)/11)/(501141752496000*(x - 1/2)**9 +
3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 1238886446949
5976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 25292108704896000*sqrt(21
)*I*(x - 1/2)**8*atan(sqrt(42)*sqrt(x - 1/2)/7)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8
+ 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851
036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) - 12646054352448000*sqrt(21)*I*pi*(x - 1/2)**8/(501141752
496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1
/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) + 781
4479957200000*sqrt(55)*I*pi*(x - 1/2)**8/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 96536
60625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*
(x - 1/2)**4 + 1059202535612298*(x - 1/2)**3) - 44274391835940000*sqrt(55)*I*(x - 1/2)**7*atan(sqrt(110)*sqrt(
x - 1/2)/11)/(501141752496000*(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 1
4582705295075456*(x - 1/2)**6 + 12388864469495976*(x - 1/2)**5 + 5612397851036592*(x - 1/2)**4 + 1059202535612
298*(x - 1/2)**3) + 71648576571369600*sqrt(21)*I*(x - 1/2)**7*atan(sqrt(42)*sqrt(x - 1/2)/7)/(501141752496000*
(x - 1/2)**9 + 3407763916972800*(x - 1/2)**8 + 9653660625581280*(x - 1/2)**7 + 14582705295075456*(x - 1/2)**6
+ 12388864469495976*(x - 1/2)**5 + 561239785103...

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Giac [A]
time = 1.62, size = 144, normalized size = 0.95 \begin {gather*} \frac {738625}{322102} \, \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 {8910}{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 {64 \, {\left (513 \, x - 295\right )}}{15065589 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {243 \, \sqrt {-2 \, x + 1}}{343 \, {\left (3 \, x + 2\right )}} - \frac {625 \, {\left (55 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 119 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

738625/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 8910/2401
*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/15065589*(513*x - 29
5)/((2*x - 1)*sqrt(-2*x + 1)) + 243/343*sqrt(-2*x + 1)/(3*x + 2) - 625/5324*(55*(-2*x + 1)^(3/2) - 119*sqrt(-2
*x + 1))/(5*x + 3)^2

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Mupad [B]
time = 1.24, size = 109, normalized size = 0.72 \begin {gather*} \frac {\frac {153791551\,{\left (2\,x-1\right )}^2}{20543985}-\frac {5248\,x}{266805}+\frac {33729798\,{\left (2\,x-1\right )}^3}{5021863}+\frac {7554245\,{\left (2\,x-1\right )}^4}{5021863}+\frac {15584}{1334025}}{\frac {847\,{\left (1-2\,x\right )}^{3/2}}{75}-\frac {1133\,{\left (1-2\,x\right )}^{5/2}}{75}+\frac {101\,{\left (1-2\,x\right )}^{7/2}}{15}-{\left (1-2\,x\right )}^{9/2}}+\frac {17820\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{2401}-\frac {738625\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{161051} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((153791551*(2*x - 1)^2)/20543985 - (5248*x)/266805 + (33729798*(2*x - 1)^3)/5021863 + (7554245*(2*x - 1)^4)/5
021863 + 15584/1334025)/((847*(1 - 2*x)^(3/2))/75 - (1133*(1 - 2*x)^(5/2))/75 + (101*(1 - 2*x)^(7/2))/15 - (1
- 2*x)^(9/2)) + (17820*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/2401 - (738625*55^(1/2)*atanh((55^(1/2)*(
1 - 2*x)^(1/2))/11))/161051

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