3.2179 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=145 \[ -\frac{446660}{290521 \sqrt{1-2 x}}+\frac{582}{49 (1-2 x)^{3/2} (3 x+2)}-\frac{39520}{11319 (1-2 x)^{3/2}}+\frac{57}{49 (1-2 x)^{3/2} (3 x+2)^2}+\frac{1}{7 (1-2 x)^{3/2} (3 x+2)^3}+\frac{127710 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401}-\frac{6250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

-39520/(11319*(1 - 2*x)^(3/2)) - 446660/(290521*Sqrt[1 - 2*x]) + 1/(7*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + 57/(49*(1
- 2*x)^(3/2)*(2 + 3*x)^2) + 582/(49*(1 - 2*x)^(3/2)*(2 + 3*x)) + (127710*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/2401 - (6250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.0694505, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {103, 151, 152, 156, 63, 206} \[ -\frac{446660}{290521 \sqrt{1-2 x}}+\frac{582}{49 (1-2 x)^{3/2} (3 x+2)}-\frac{39520}{11319 (1-2 x)^{3/2}}+\frac{57}{49 (1-2 x)^{3/2} (3 x+2)^2}+\frac{1}{7 (1-2 x)^{3/2} (3 x+2)^3}+\frac{127710 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401}-\frac{6250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^(5/2)*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

-39520/(11319*(1 - 2*x)^(3/2)) - 446660/(290521*Sqrt[1 - 2*x]) + 1/(7*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + 57/(49*(1
- 2*x)^(3/2)*(2 + 3*x)^2) + 582/(49*(1 - 2*x)^(3/2)*(2 + 3*x)) + (127710*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/2401 - (6250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

Rule 103

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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

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] && IntegerQ[m]

Rule 152

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 156

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 63

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 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)^4 (3+5 x)} \, dx &=\frac{1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{1}{21} \int \frac{24-135 x}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac{1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac{1}{294} \int \frac{168-11970 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac{582}{49 (1-2 x)^{3/2} (2+3 x)}+\frac{\int \frac{-109410-611100 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx}{2058}\\ &=-\frac{39520}{11319 (1-2 x)^{3/2}}+\frac{1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac{582}{49 (1-2 x)^{3/2} (2+3 x)}-\frac{\int \frac{-2301705+18673200 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{237699}\\ &=-\frac{39520}{11319 (1-2 x)^{3/2}}-\frac{446660}{290521 \sqrt{1-2 x}}+\frac{1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac{582}{49 (1-2 x)^{3/2} (2+3 x)}+\frac{2 \int \frac{\frac{346068765}{2}-105523425 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{18302823}\\ &=-\frac{39520}{11319 (1-2 x)^{3/2}}-\frac{446660}{290521 \sqrt{1-2 x}}+\frac{1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac{582}{49 (1-2 x)^{3/2} (2+3 x)}-\frac{191565 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2401}+\frac{15625}{121} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{39520}{11319 (1-2 x)^{3/2}}-\frac{446660}{290521 \sqrt{1-2 x}}+\frac{1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac{582}{49 (1-2 x)^{3/2} (2+3 x)}+\frac{191565 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2401}-\frac{15625}{121} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{39520}{11319 (1-2 x)^{3/2}}-\frac{446660}{290521 \sqrt{1-2 x}}+\frac{1}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{57}{49 (1-2 x)^{3/2} (2+3 x)^2}+\frac{582}{49 (1-2 x)^{3/2} (2+3 x)}+\frac{127710 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401}-\frac{6250}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0472234, size = 71, normalized size = 0.49 \[ \frac{-468270 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+428750 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+\frac{231 \left (5238 x^2+7155 x+2449\right )}{(3 x+2)^3}}{11319 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^(5/2)*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

((231*(2449 + 7155*x + 5238*x^2))/(2 + 3*x)^3 - 468270*Hypergeometric2F1[-3/2, 1, -1/2, 3/7 - (6*x)/7] + 42875
0*Hypergeometric2F1[-3/2, 1, -1/2, (-5*(-1 + 2*x))/11])/(11319*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.014, size = 93, normalized size = 0.6 \begin{align*} -{\frac{1458}{16807\, \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{1438}{3} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{61250}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{72520}{27}\sqrt{1-2\,x}} \right ) }+{\frac{127710\,\sqrt{21}}{16807}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{32}{79233} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{5344}{2033647}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{6250\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^(5/2)/(2+3*x)^4/(3+5*x),x)

[Out]

-1458/16807*(1438/3*(1-2*x)^(5/2)-61250/27*(1-2*x)^(3/2)+72520/27*(1-2*x)^(1/2))/(-6*x-4)^3+127710/16807*arcta
nh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+32/79233/(1-2*x)^(3/2)+5344/2033647/(1-2*x)^(1/2)-6250/1331*arctanh(1/
11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.83963, size = 197, normalized size = 1.36 \begin{align*} \frac{3125}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{63855}{16807} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4 \,{\left (9044865 \,{\left (2 \, x - 1\right )}^{4} + 42773535 \,{\left (2 \, x - 1\right )}^{3} + 50533308 \,{\left (2 \, x - 1\right )}^{2} - 315168 \, x + 187768\right )}}{871563 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 343 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(5/2)/(2+3*x)^4/(3+5*x),x, algorithm="maxima")

[Out]

3125/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 63855/16807*sqrt(21)*lo
g(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 4/871563*(9044865*(2*x - 1)^4 + 42773535*(2*
x - 1)^3 + 50533308*(2*x - 1)^2 - 315168*x + 187768)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2) + 441*(-2*x +
1)^(5/2) - 343*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.39361, size = 564, normalized size = 3.89 \begin{align*} \frac{157565625 \, \sqrt{11} \sqrt{5}{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 254973015 \, \sqrt{7} \sqrt{3}{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (72358920 \, x^{4} + 26376300 \, x^{3} - 47036214 \, x^{2} - 9083055 \, x + 8496203\right )} \sqrt{-2 \, x + 1}}{67110351 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(5/2)/(2+3*x)^4/(3+5*x),x, algorithm="fricas")

[Out]

1/67110351*(157565625*sqrt(11)*sqrt(5)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log((sqrt(11)*sqrt(5)*s
qrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 254973015*sqrt(7)*sqrt(3)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8
)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(72358920*x^4 + 26376300*x^3 - 47036214*x^2
- 9083055*x + 8496203)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)**(5/2)/(2+3*x)**4/(3+5*x),x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.45406, size = 181, normalized size = 1.25 \begin{align*} \frac{3125}{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{63855}{16807} \, \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{4 \,{\left (9044865 \,{\left (2 \, x - 1\right )}^{4} + 42773535 \,{\left (2 \, x - 1\right )}^{3} + 50533308 \,{\left (2 \, x - 1\right )}^{2} - 315168 \, x + 187768\right )}}{871563 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(5/2)/(2+3*x)^4/(3+5*x),x, algorithm="giac")

[Out]

3125/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 63855/16807*s
qrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 4/871563*(9044865*(2*x -
1)^4 + 42773535*(2*x - 1)^3 + 50533308*(2*x - 1)^2 - 315168*x + 187768)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1)
)^3