3.2121 \(\int \frac{1}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{6}{121 \sqrt{1-2 x}}-\frac{1}{11 \sqrt{1-2 x} (5 x+3)}-\frac{6}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

6/(121*Sqrt[1 - 2*x]) - 1/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (6*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Rubi [A]  time = 0.014371, antiderivative size = 70, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {51, 63, 206} \[ -\frac{15 \sqrt{1-2 x}}{121 (5 x+3)}+\frac{2}{11 \sqrt{1-2 x} (5 x+3)}-\frac{6}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (15*Sqrt[1 - 2*x])/(121*(3 + 5*x)) - (6*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]])/121

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, 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)^{3/2} (3+5 x)^2} \, dx &=\frac{2}{11 \sqrt{1-2 x} (3+5 x)}+\frac{15}{11} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=\frac{2}{11 \sqrt{1-2 x} (3+5 x)}-\frac{15 \sqrt{1-2 x}}{121 (3+5 x)}+\frac{15}{121} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{2}{11 \sqrt{1-2 x} (3+5 x)}-\frac{15 \sqrt{1-2 x}}{121 (3+5 x)}-\frac{15}{121} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{2}{11 \sqrt{1-2 x} (3+5 x)}-\frac{15 \sqrt{1-2 x}}{121 (3+5 x)}-\frac{6}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0054955, size = 30, normalized size = 0.48 \[ \frac{4 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{121 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Hypergeometric2F1[-1/2, 2, 1/2, (5*(1 - 2*x))/11])/(121*Sqrt[1 - 2*x])

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Maple [A]  time = 0.009, size = 45, normalized size = 0.7 \begin{align*}{\frac{4}{121}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{121}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{6\,\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)^(3/2)/(3+5*x)^2,x)

[Out]

4/121/(1-2*x)^(1/2)+2/121*(1-2*x)^(1/2)/(-2*x-6/5)-6/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.4921, size = 88, normalized size = 1.4 \begin{align*} \frac{3}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (30 \, x + 7\right )}}{121 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2/121*(30*x + 7)/(5*(-2*x
+ 1)^(3/2) - 11*sqrt(-2*x + 1))

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Fricas [A]  time = 1.6156, size = 207, normalized size = 3.29 \begin{align*} \frac{3 \, \sqrt{11} \sqrt{5}{\left (10 \, x^{2} + x - 3\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \,{\left (30 \, x + 7\right )} \sqrt{-2 \, x + 1}}{1331 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1331*(3*sqrt(11)*sqrt(5)*(10*x^2 + x - 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 11*(3
0*x + 7)*sqrt(-2*x + 1))/(10*x^2 + x - 3)

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Sympy [A]  time = 2.45518, size = 175, normalized size = 2.78 \begin{align*} \begin{cases} - \frac{6 \sqrt{55} \operatorname{acosh}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{1331} + \frac{3 \sqrt{2}}{121 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} - \frac{\sqrt{2}}{110 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\\frac{6 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{1331} - \frac{3 \sqrt{2} i}{121 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \sqrt{x + \frac{3}{5}}} + \frac{\sqrt{2} i}{110 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-6*sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/1331 + 3*sqrt(2)/(121*sqrt(-1 + 11/(10*(x + 3/5)))*
sqrt(x + 3/5)) - sqrt(2)/(110*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)), 11/(10*Abs(x + 3/5)) > 1), (6*sq
rt(55)*I*asin(sqrt(110)/(10*sqrt(x + 3/5)))/1331 - 3*sqrt(2)*I/(121*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/5))
+ sqrt(2)*I/(110*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)), True))

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Giac [A]  time = 2.51358, size = 92, normalized size = 1.46 \begin{align*} \frac{3}{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{2 \,{\left (30 \, x + 7\right )}}{121 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 2/121*(30*x + 7)
/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))