3.2176 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\)

Optimal. Leaf size=85 \[ \frac{272}{5929 \sqrt{1-2 x}}+\frac{4}{231 (1-2 x)^{3/2}}+\frac{18}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{50}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

4/(231*(1 - 2*x)^(3/2)) + 272/(5929*Sqrt[1 - 2*x]) + (18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (50*
Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.0332659, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {85, 152, 156, 63, 206} \[ \frac{272}{5929 \sqrt{1-2 x}}+\frac{4}{231 (1-2 x)^{3/2}}+\frac{18}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{50}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(231*(1 - 2*x)^(3/2)) + 272/(5929*Sqrt[1 - 2*x]) + (18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (50*
Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

Rule 85

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[(f*(e + f*x)^(p +
1))/((p + 1)*(b*e - a*f)*(d*e - c*f)), x] + Dist[1/((b*e - a*f)*(d*e - c*f)), Int[((b*d*e - b*c*f - a*d*f - b
*d*f*x)*(e + f*x)^(p + 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -1]

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) (3+5 x)} \, dx &=\frac{4}{231 (1-2 x)^{3/2}}+\frac{1}{77} \int \frac{53+30 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=\frac{4}{231 (1-2 x)^{3/2}}+\frac{272}{5929 \sqrt{1-2 x}}-\frac{2 \int \frac{-\frac{2449}{2}-1020 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{5929}\\ &=\frac{4}{231 (1-2 x)^{3/2}}+\frac{272}{5929 \sqrt{1-2 x}}-\frac{27}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{125}{121} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{4}{231 (1-2 x)^{3/2}}+\frac{272}{5929 \sqrt{1-2 x}}+\frac{27}{49} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{125}{121} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{4}{231 (1-2 x)^{3/2}}+\frac{272}{5929 \sqrt{1-2 x}}+\frac{18}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{50}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0635437, size = 85, normalized size = 1. \[ \frac{272}{5929 \sqrt{1-2 x}}+\frac{4}{231 (1-2 x)^{3/2}}+\frac{18}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{50}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(231*(1 - 2*x)^(3/2)) + 272/(5929*Sqrt[1 - 2*x]) + (18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (50*
Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]  time = 0.011, size = 56, normalized size = 0.7 \begin{align*}{\frac{4}{231} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{18\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{50\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{272}{5929}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/231/(1-2*x)^(3/2)+18/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50/1331*arctanh(1/11*55^(1/2)*(1-2*x)^
(1/2))*55^(1/2)+272/5929/(1-2*x)^(1/2)

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Maxima [A]  time = 2.49401, size = 117, normalized size = 1.38 \begin{align*} \frac{25}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4 \,{\left (408 \, x - 281\right )}}{17787 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 9/343*sqrt(21)*log(-(sqrt
(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 4/17787*(408*x - 281)/(-2*x + 1)^(3/2)

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Fricas [B]  time = 1.24544, size = 360, normalized size = 4.24 \begin{align*} \frac{25725 \, \sqrt{11} \sqrt{5}{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 35937 \, \sqrt{7} \sqrt{3}{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) - 308 \,{\left (408 \, x - 281\right )} \sqrt{-2 \, x + 1}}{1369599 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1369599*(25725*sqrt(11)*sqrt(5)*(4*x^2 - 4*x + 1)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3))
+ 35937*sqrt(7)*sqrt(3)*(4*x^2 - 4*x + 1)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) - 308*(4
08*x - 281)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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Sympy [C]  time = 7.05976, size = 105, normalized size = 1.24 \begin{align*} - \frac{50 \sqrt{55} i \operatorname{atan}{\left (\frac{\sqrt{110} \sqrt{x - \frac{1}{2}}}{11} \right )}}{1331} + \frac{18 \sqrt{21} i \operatorname{atan}{\left (\frac{\sqrt{42} \sqrt{x - \frac{1}{2}}}{7} \right )}}{343} - \frac{136 \sqrt{2} i}{5929 \sqrt{x - \frac{1}{2}}} + \frac{\sqrt{2} i}{231 \left (x - \frac{1}{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{2} i}{20 \left (x - \frac{1}{2}\right )^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-50*sqrt(55)*I*atan(sqrt(110)*sqrt(x - 1/2)/11)/1331 + 18*sqrt(21)*I*atan(sqrt(42)*sqrt(x - 1/2)/7)/343 - 136*
sqrt(2)*I/(5929*sqrt(x - 1/2)) + sqrt(2)*I/(231*(x - 1/2)**(3/2)) + sqrt(2)*I/(20*(x - 1/2)**(5/2))

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Giac [A]  time = 2.60733, size = 135, normalized size = 1.59 \begin{align*} \frac{25}{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{9}{343} \, \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 (408 \, x - 281\right )}}{17787 \,{\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)/(3+5*x),x, algorithm="giac")

[Out]

25/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 9/343*sqrt(21)*
log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/17787*(408*x - 281)/((2*x - 1)*
sqrt(-2*x + 1))