3.22.89 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx\) [2189]

Optimal. Leaf size=159 \[ \frac {15185}{2541 (1-2 x)^{3/2}}+\frac {172105}{65219 \sqrt {1-2 x}}-\frac {745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac {24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {4455}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {117500 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \]

[Out]

15185/2541/(1-2*x)^(3/2)-745/22/(1-2*x)^(3/2)/(3+5*x)+3/14/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)+24/7/(1-2*x)^(3/2)/
(2+3*x)/(3+5*x)-4455/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+117500/14641*arctanh(1/11*55^(1/2)*(1-2*
x)^(1/2))*55^(1/2)+172105/65219/(1-2*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 159, 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 {172105}{65219 \sqrt {1-2 x}}+\frac {24}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac {745}{22 (1-2 x)^{3/2} (5 x+3)}+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}+\frac {15185}{2541 (1-2 x)^{3/2}}-\frac {4455}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {117500 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

15185/(2541*(1 - 2*x)^(3/2)) + 172105/(65219*Sqrt[1 - 2*x]) - 745/(22*(1 - 2*x)^(3/2)*(3 + 5*x)) + 3/(14*(1 -
2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)) + 24/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)) - (4455*Sqrt[3/7]*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/49 + (117500*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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)^3 (3+5 x)^2} \, dx &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac {1}{14} \int \frac {22-135 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac {24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac {1}{98} \int \frac {245-11760 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac {24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {\int \frac {-98245-547575 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx}{1078}\\ &=\frac {15185}{2541 (1-2 x)^{3/2}}-\frac {745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac {24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac {\int \frac {-\frac {4091745}{2}+\frac {33482925 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{124509}\\ &=\frac {15185}{2541 (1-2 x)^{3/2}}+\frac {172105}{65219 \sqrt {1-2 x}}-\frac {745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac {24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {2 \int \frac {\frac {618657585}{4}-\frac {379491525 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{9587193}\\ &=\frac {15185}{2541 (1-2 x)^{3/2}}+\frac {172105}{65219 \sqrt {1-2 x}}-\frac {745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac {24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac {13365}{98} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {293750 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac {15185}{2541 (1-2 x)^{3/2}}+\frac {172105}{65219 \sqrt {1-2 x}}-\frac {745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac {24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {13365}{98} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {293750 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1331}\\ &=\frac {15185}{2541 (1-2 x)^{3/2}}+\frac {172105}{65219 \sqrt {1-2 x}}-\frac {745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac {24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac {4455}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {117500 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 106, normalized size = 0.67 \begin {gather*} -\frac {9784671-9008764 x-58371045 x^2+27977220 x^3+92936700 x^4}{391314 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}-\frac {4455}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {117500 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/391314*(9784671 - 9008764*x - 58371045*x^2 + 27977220*x^3 + 92936700*x^4)/((1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 +
5*x)) - (4455*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (117500*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 -
2*x]])/1331

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Maple [A]
time = 0.17, size = 100, normalized size = 0.63

method	result	size
risch	\(\frac {92936700 x^{4}[phone] x^{3}-58371045 x^{2}-9008764 x +9784671}{391314 \left (3+5 x \right ) \sqrt {1-2 x}\, \left (-1+2 x \right ) \left (2+3 x \right )^{2}}+\frac {117500 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}-\frac {4455 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(88\)
derivativedivides	\(\frac {1250 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {117500 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {32}{124509 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5408}{3195731 \sqrt {1-2 x}}+\frac {\frac {36693 \left (1-2 x \right )^{\frac {3}{2}}}{2401}-\frac {12393 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{2}}-\frac {4455 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(100\)
default	\(\frac {1250 \sqrt {1-2 x}}{1331 \left (-\frac {6}{5}-2 x \right )}+\frac {117500 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{14641}+\frac {32}{124509 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {5408}{3195731 \sqrt {1-2 x}}+\frac {\frac {36693 \left (1-2 x \right )^{\frac {3}{2}}}{2401}-\frac {12393 \sqrt {1-2 x}}{343}}{\left (-4-6 x \right )^{2}}-\frac {4455 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}\)	\(100\)
trager	\(-\frac {\left (92936700 x^{4}[phone] x^{3}-58371045 x^{2}-9008764 x +9784671\right ) \sqrt {1-2 x}}{391314 \left (6 x^{2}+x -2\right )^{2} \left (3+5 x \right )}-\frac {4455 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{686}+\frac {58750 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{14641}\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1250/1331*(1-2*x)^(1/2)/(-6/5-2*x)+117500/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+32/124509/(1-2*x
)^(3/2)+5408/3195731/(1-2*x)^(1/2)+4374/2401*(151/18*(1-2*x)^(3/2)-119/6*(1-2*x)^(1/2))/(-4-6*x)^2-4455/343*ar
ctanh(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 = 0.92 \begin {gather*} -\frac {58750}{14641} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4455}{686} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {23234175 \, {\left (2 \, x - 1\right )}^{4} + 106925310 \, {\left (2 \, x - 1\right )}^{3} + 122999835 \, {\left (2 \, x - 1\right )}^{2} + 285824 \, x - 170016}{195657 \, {\left (45 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 309 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 707 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 539 \, {\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)^3/(3+5*x)^2,x, algorithm="maxima")

[Out]

-58750/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 4455/686*sqrt(21)*lo
g(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/195657*(23234175*(2*x - 1)^4 + 106925310*(
2*x - 1)^3 + 122999835*(2*x - 1)^2 + 285824*x - 170016)/(45*(-2*x + 1)^(9/2) - 309*(-2*x + 1)^(7/2) + 707*(-2*
x + 1)^(5/2) - 539*(-2*x + 1)^(3/2))

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Fricas [A]
time = 1.02, size = 182, normalized size = 1.14 \begin {gather*} \frac {120907500 \, \sqrt {11} \sqrt {5} {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 195676965 \, \sqrt {7} \sqrt {3} {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (92936700 \, x^{4} + 27977220 \, x^{3} - 58371045 \, x^{2} - 9008764 \, x + 9784671\right )} \sqrt {-2 \, x + 1}}{30131178 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30131178*(120907500*sqrt(11)*sqrt(5)*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*log(-(sqrt(11)*sqrt(5)
*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 195676965*sqrt(7)*sqrt(3)*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x +
12)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(92936700*x^4 + 27977220*x^3 - 58371045*x^
2 - 9008764*x + 9784671)*sqrt(-2*x + 1))/(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)

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Sympy [C] Result contains complex when optimal does not.
time = 14.31, size = 2966, normalized size = 18.65 \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)**3/(3+5*x)**2,x)

[Out]

-8587351080000*sqrt(2)*I*(x - 1/2)**(17/2)/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 1253722
15916640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)
**4 + 13755877085874*(x - 1/2)**3) - 48670545924000*sqrt(2)*I*(x - 1/2)**(15/2)/(6508334448000*(x - 1/2)**9 +
44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x
- 1/2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) - 110321398202400*sqrt(2)*I*(x - 1/2)*
*(13/2)/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 18938578305
2928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3)
- 125018036238480*sqrt(2)*I*(x - 1/2)**(11/2)/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 1253
72215916640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1
/2)**4 + 13755877085874*(x - 1/2)**3) - 70838364022580*sqrt(2)*I*(x - 1/2)**(9/2)/(6508334448000*(x - 1/2)**9
+ 44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*
(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) - 16066680171234*sqrt(2)*I*(x - 1/2)
**(7/2)/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 18938578305
2928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3)
- 6955997664*sqrt(2)*I*(x - 1/2)**(5/2)/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 1253722159
16640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4
+ 13755877085874*(x - 1/2)**3) + 883847888*sqrt(2)*I*(x - 1/2)**(3/2)/(6508334448000*(x - 1/2)**9 + 442566742
46400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**
5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) + 52232040000000*sqrt(55)*I*(x - 1/2)**9*atan(s
qrt(110)*sqrt(x - 1/2)/11)/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/
2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 1375587708
5874*(x - 1/2)**3) - 84532448880000*sqrt(21)*I*(x - 1/2)**9*atan(sqrt(42)*sqrt(x - 1/2)/7)/(6508334448000*(x -
1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 1608943
43759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) - 26116020000000*sqrt(55)*I
*pi*(x - 1/2)**9/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 18
9385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x -
1/2)**3) + 42266224440000*sqrt(21)*I*pi*(x - 1/2)**9/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8
+ 125372215916640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696
*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) + 355177872000000*sqrt(55)*I*(x - 1/2)**8*atan(sqrt(110)*sqrt(x -
1/2)/11)/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783
052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3
) - 574820652384000*sqrt(21)*I*(x - 1/2)**8*atan(sqrt(42)*sqrt(x - 1/2)/7)/(6508334448000*(x - 1/2)**9 + 44256
674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/
2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) - 177588936000000*sqrt(55)*I*pi*(x - 1/2)**
8/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*(
x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) + 2874
10326192000*sqrt(21)*I*pi*(x - 1/2)**8/(6508334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 12537221591
6640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4
+ 13755877085874*(x - 1/2)**3) + 1006163197200000*sqrt(55)*I*(x - 1/2)**7*atan(sqrt(110)*sqrt(x - 1/2)/11)/(65
08334448000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*(x - 1
/2)**6 + 160894343759688*(x - 1/2)**5 + 72888283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) - 162837674
0258400*sqrt(21)*I*(x - 1/2)**7*atan(sqrt(42)*sqrt(x - 1/2)/7)/(6508334448000*(x - 1/2)**9 + 44256674246400*(x
- 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6 + 160894343759688*(x - 1/2)**5 + 7288
8283779696*(x - 1/2)**4 + 13755877085874*(x - 1/2)**3) - 503081598600000*sqrt(55)*I*pi*(x - 1/2)**7/(650833444
8000*(x - 1/2)**9 + 44256674246400*(x - 1/2)**8 + 125372215916640*(x - 1/2)**7 + 189385783052928*(x - 1/2)**6
+ 160894343759688*(x - 1/2)**5 + 72888283779696...

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Giac [A]
time = 1.15, size = 144, normalized size = 0.91 \begin {gather*} -\frac {58750}{14641} \, \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 {4455}{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 {64 \, {\left (507 \, x - 292\right )}}{9587193 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {3125 \, \sqrt {-2 \, x + 1}}{1331 \, {\left (5 \, x + 3\right )}} + \frac {243 \, {\left (151 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 357 \, \sqrt {-2 \, x + 1}\right )}}{9604 \, {\left (3 \, x + 2\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)^3/(3+5*x)^2,x, algorithm="giac")

[Out]

-58750/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 4455/686*s
qrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/9587193*(507*x - 292)/
((2*x - 1)*sqrt(-2*x + 1)) - 3125/1331*sqrt(-2*x + 1)/(5*x + 3) + 243/9604*(151*(-2*x + 1)^(3/2) - 357*sqrt(-2
*x + 1))/(3*x + 2)^2

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Mupad [B]
time = 1.26, size = 110, normalized size = 0.69 \begin {gather*} \frac {117500\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641}-\frac {4455\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {3712\,x}{114345}+\frac {1171427\,{\left (2\,x-1\right )}^2}{83853}+\frac {2376118\,{\left (2\,x-1\right )}^3}{195657}+\frac {172105\,{\left (2\,x-1\right )}^4}{65219}-\frac {736}{38115}}{\frac {539\,{\left (1-2\,x\right )}^{3/2}}{45}-\frac {707\,{\left (1-2\,x\right )}^{5/2}}{45}+\frac {103\,{\left (1-2\,x\right )}^{7/2}}{15}-{\left (1-2\,x\right )}^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(117500*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641 - (4455*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))
/7))/343 - ((3712*x)/114345 + (1171427*(2*x - 1)^2)/83853 + (2376118*(2*x - 1)^3)/195657 + (172105*(2*x - 1)^4
)/65219 - 736/38115)/((539*(1 - 2*x)^(3/2))/45 - (707*(1 - 2*x)^(5/2))/45 + (103*(1 - 2*x)^(7/2))/15 - (1 - 2*
x)^(9/2))

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