3.2188 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=132 \[ \frac{159800}{456533 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac{340}{77 (1-2 x)^{3/2} (5 x+3)}+\frac{13900}{17787 (1-2 x)^{3/2}}-\frac{4050}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{15250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

13900/(17787*(1 - 2*x)^(3/2)) + 159800/(456533*Sqrt[1 - 2*x]) - 340/(77*(1 - 2*x)^(3/2)*(3 + 5*x)) + 3/(7*(1 -
2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)) - (4050*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (15250*Sqrt[5/11]*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.056628, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{159800}{456533 \sqrt{1-2 x}}+\frac{3}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac{340}{77 (1-2 x)^{3/2} (5 x+3)}+\frac{13900}{17787 (1-2 x)^{3/2}}-\frac{4050}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{15250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

13900/(17787*(1 - 2*x)^(3/2)) + 159800/(456533*Sqrt[1 - 2*x]) - 340/(77*(1 - 2*x)^(3/2)*(3 + 5*x)) + 3/(7*(1 -
2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)) - (4050*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (15250*Sqrt[5/11]*A
rcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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)^2 (3+5 x)^2} \, dx &=\frac{3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac{1}{7} \int \frac{5-105 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac{1}{77} \int \frac{-925-5100 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=\frac{13900}{17787 (1-2 x)^{3/2}}-\frac{340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac{2 \int \frac{-\frac{36525}{2}+156375 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{17787}\\ &=\frac{13900}{17787 (1-2 x)^{3/2}}+\frac{159800}{456533 \sqrt{1-2 x}}-\frac{340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac{4 \int \frac{\frac{5688825}{4}-898875 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{1369599}\\ &=\frac{13900}{17787 (1-2 x)^{3/2}}+\frac{159800}{456533 \sqrt{1-2 x}}-\frac{340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac{6075}{343} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-\frac{38125 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac{13900}{17787 (1-2 x)^{3/2}}+\frac{159800}{456533 \sqrt{1-2 x}}-\frac{340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac{6075}{343} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+\frac{38125 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1331}\\ &=\frac{13900}{17787 (1-2 x)^{3/2}}+\frac{159800}{456533 \sqrt{1-2 x}}-\frac{340}{77 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac{4050}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{15250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331}\\ \end{align*}

Mathematica [C]  time = 0.0326662, size = 93, normalized size = 0.7 \[ -\frac{-163350 \left (15 x^2+19 x+6\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+149450 \left (15 x^2+19 x+6\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+231 (1020 x+647)}{17787 (1-2 x)^{3/2} (3 x+2) (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(231*(647 + 1020*x) - 163350*(6 + 19*x + 15*x^2)*Hypergeometric2F1[-3/2, 1, -1/2, 3/7 - (6*x)/7] + 149450*(6
+ 19*x + 15*x^2)*Hypergeometric2F1[-3/2, 1, -1/2, (-5*(-1 + 2*x))/11])/(17787*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5
*x))

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Maple [A]  time = 0.016, size = 88, normalized size = 0.7 \begin{align*}{\frac{54}{343}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{4050\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{16}{17787} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2176}{456533}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{250}{1331}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{15250\,\sqrt{55}}{14641}{\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)^2/(3+5*x)^2,x)

[Out]

54/343*(1-2*x)^(1/2)/(-2*x-4/3)-4050/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+16/17787/(1-2*x)^(3/2)+
2176/456533/(1-2*x)^(1/2)+250/1331*(1-2*x)^(1/2)/(-2*x-6/5)+15250/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*5
5^(1/2)

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Maxima [A]  time = 2.93426, size = 173, normalized size = 1.31 \begin{align*} -\frac{7625}{14641} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2025}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4 \,{\left (1797750 \,{\left (2 \, x - 1\right )}^{3} + 4136175 \,{\left (2 \, x - 1\right )}^{2} + 209440 \, x - 128436\right )}}{1369599 \,{\left (15 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 68 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 77 \,{\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)^2/(3+5*x)^2,x, algorithm="maxima")

[Out]

-7625/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2025/2401*sqrt(21)*lo
g(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 4/1369599*(1797750*(2*x - 1)^3 + 4136175*(2*
x - 1)^2 + 209440*x - 128436)/(15*(-2*x + 1)^(7/2) - 68*(-2*x + 1)^(5/2) + 77*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.12483, size = 497, normalized size = 3.77 \begin{align*} \frac{54922875 \, \sqrt{11} \sqrt{5}{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 88944075 \, \sqrt{7} \sqrt{3}{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \,{\left (14382000 \, x^{3} - 5028300 \, x^{2} - 5548760 \, x + 2209989\right )} \sqrt{-2 \, x + 1}}{105459123 \,{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )}} \end{align*}

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)^2,x, algorithm="fricas")

[Out]

1/105459123*(54922875*sqrt(11)*sqrt(5)*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x +
1) - 5*x + 8)/(5*x + 3)) + 88944075*sqrt(7)*sqrt(3)*(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*log((sqrt(7)*sqrt(3)
*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(14382000*x^3 - 5028300*x^2 - 5548760*x + 2209989)*sqrt(-2*x + 1))/
(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)

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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)**2/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.521, size = 185, normalized size = 1.4 \begin{align*} -\frac{7625}{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{2025}{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{4 \,{\left (591090 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1343273 \, \sqrt{-2 \, x + 1}\right )}}{456533 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} + \frac{16 \,{\left (816 \, x - 485\right )}}{1369599 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \end{align*}

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)^2,x, algorithm="giac")

[Out]

-7625/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2025/2401*s
qrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/456533*(591090*(-2*x +
1)^(3/2) - 1343273*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9) + 16/1369599*(816*x - 485)/((2*x - 1)*sqrt(-2*
x + 1))