3.2186 \(\int \frac{1}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=76 \[ \frac{50}{1331 \sqrt{1-2 x}}-\frac{1}{11 (1-2 x)^{3/2} (5 x+3)}+\frac{10}{363 (1-2 x)^{3/2}}-\frac{50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

10/(363*(1 - 2*x)^(3/2)) + 50/(1331*Sqrt[1 - 2*x]) - 1/(11*(1 - 2*x)^(3/2)*(3 + 5*x)) - (50*Sqrt[5/11]*ArcTanh
[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.0195709, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {51, 63, 206} \[ \frac{50}{1331 \sqrt{1-2 x}}-\frac{1}{11 (1-2 x)^{3/2} (5 x+3)}+\frac{10}{363 (1-2 x)^{3/2}}-\frac{50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

10/(363*(1 - 2*x)^(3/2)) + 50/(1331*Sqrt[1 - 2*x]) - 1/(11*(1 - 2*x)^(3/2)*(3 + 5*x)) - (50*Sqrt[5/11]*ArcTanh
[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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)^{5/2} (3+5 x)^2} \, dx &=-\frac{1}{11 (1-2 x)^{3/2} (3+5 x)}+\frac{5}{11} \int \frac{1}{(1-2 x)^{5/2} (3+5 x)} \, dx\\ &=\frac{10}{363 (1-2 x)^{3/2}}-\frac{1}{11 (1-2 x)^{3/2} (3+5 x)}+\frac{25}{121} \int \frac{1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac{10}{363 (1-2 x)^{3/2}}+\frac{50}{1331 \sqrt{1-2 x}}-\frac{1}{11 (1-2 x)^{3/2} (3+5 x)}+\frac{125 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac{10}{363 (1-2 x)^{3/2}}+\frac{50}{1331 \sqrt{1-2 x}}-\frac{1}{11 (1-2 x)^{3/2} (3+5 x)}-\frac{125 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1331}\\ &=\frac{10}{363 (1-2 x)^{3/2}}+\frac{50}{1331 \sqrt{1-2 x}}-\frac{1}{11 (1-2 x)^{3/2} (3+5 x)}-\frac{50 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331}\\ \end{align*}

Mathematica [C]  time = 0.0055044, size = 30, normalized size = 0.39 \[ \frac{4 \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{363 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Hypergeometric2F1[-3/2, 2, -1/2, (5*(1 - 2*x))/11])/(363*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.012, size = 54, normalized size = 0.7 \begin{align*}{\frac{4}{363} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{40}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{10}{1331}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{50\,\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)/(3+5*x)^2,x)

[Out]

4/363/(1-2*x)^(3/2)+40/1331/(1-2*x)^(1/2)+10/1331*(1-2*x)^(1/2)/(-2*x-6/5)-50/14641*arctanh(1/11*55^(1/2)*(1-2
*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.66775, size = 100, normalized size = 1.32 \begin{align*} \frac{25}{14641} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (375 \,{\left (2 \, x - 1\right )}^{2} + 1100 \, x - 792\right )}}{3993 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\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)/(3+5*x)^2,x, algorithm="maxima")

[Out]

25/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/3993*(375*(2*x - 1)^2
+ 1100*x - 792)/(5*(-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.03315, size = 255, normalized size = 3.36 \begin{align*} \frac{75 \, \sqrt{11} \sqrt{5}{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \,{\left (1500 \, x^{2} - 400 \, x - 417\right )} \sqrt{-2 \, x + 1}}{43923 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/43923*(75*sqrt(11)*sqrt(5)*(20*x^3 - 8*x^2 - 7*x + 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x +
3)) - 11*(1500*x^2 - 400*x - 417)*sqrt(-2*x + 1))/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [C]  time = 3.90453, size = 2286, normalized size = 30.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((15000*sqrt(5)*I*(x + 3/5)**3*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(399300*sqrt(11)*(x + 3/5)**3 - 878
460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 7500*sqrt(5)*(x + 3/5)**3*log(110)/(399300*sqrt(11)*(
x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 7500*sqrt(5)*(x + 3/5)**3*log(11)/(3
99300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 15000*sqrt(5)*(x + 3
/5)**3*log(2)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 7500
*sqrt(5)*(x + 3/5)**3*log(10)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(
x + 3/5)) + 15000*sqrt(5)*(x + 3/5)**3*log(22)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 +
483153*sqrt(11)*(x + 3/5)) - 1500*sqrt(55)*I*(x + 3/5)**2*sqrt(10*x - 5)/(399300*sqrt(11)*(x + 3/5)**3 - 87846
0*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 33000*sqrt(5)*I*(x + 3/5)**2*asin(sqrt(110)/(10*sqrt(x
+ 3/5)))/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 33000*sqr
t(5)*(x + 3/5)**2*log(22)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x +
3/5)) - 16500*sqrt(5)*(x + 3/5)**2*log(10)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 4831
53*sqrt(11)*(x + 3/5)) + 33000*sqrt(5)*(x + 3/5)**2*log(2)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x
+ 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 16500*sqrt(5)*(x + 3/5)**2*log(11)/(399300*sqrt(11)*(x + 3/5)**3 - 87
8460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 16500*sqrt(5)*(x + 3/5)**2*log(110)/(399300*sqrt(11)
*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 2200*sqrt(55)*I*(x + 3/5)*sqrt(10*
x - 5)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 18150*sqrt(
5)*I*(x + 3/5)*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2
+ 483153*sqrt(11)*(x + 3/5)) - 9075*sqrt(5)*(x + 3/5)*log(110)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11
)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 9075*sqrt(5)*(x + 3/5)*log(11)/(399300*sqrt(11)*(x + 3/5)**3 - 8
78460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 18150*sqrt(5)*(x + 3/5)*log(2)/(399300*sqrt(11)*(x
+ 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 9075*sqrt(5)*(x + 3/5)*log(10)/(399300
*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 18150*sqrt(5)*(x + 3/5)*l
og(22)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 363*sqrt(55
)*I*sqrt(10*x - 5)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)),
10*Abs(x + 3/5)/11 > 1), (-1500*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)**2/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sq
rt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 2200*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)/(399300*sqrt(11)*(x
+ 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 363*sqrt(55)*sqrt(5 - 10*x)/(399300*sq
rt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 7500*sqrt(5)*(x + 3/5)**3*lo
g(x + 3/5)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 15000*s
qrt(5)*(x + 3/5)**3*log(sqrt(5/11 - 10*x/11) + 1)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2
+ 483153*sqrt(11)*(x + 3/5)) - 7500*sqrt(5)*(x + 3/5)**3*log(11)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(
11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 7500*sqrt(5)*(x + 3/5)**3*log(10)/(399300*sqrt(11)*(x + 3/5)**
3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 7500*sqrt(5)*I*pi*(x + 3/5)**3/(399300*sqrt(11
)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 16500*sqrt(5)*(x + 3/5)**2*log(x
+ 3/5)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 33000*sqrt(
5)*(x + 3/5)**2*log(sqrt(5/11 - 10*x/11) + 1)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 4
83153*sqrt(11)*(x + 3/5)) - 16500*sqrt(5)*(x + 3/5)**2*log(10)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)
*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 16500*sqrt(5)*(x + 3/5)**2*log(11)/(399300*sqrt(11)*(x + 3/5)**3
- 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 16500*sqrt(5)*I*pi*(x + 3/5)**2/(399300*sqrt(11)
*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 9075*sqrt(5)*(x + 3/5)*log(x + 3/5
)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) - 18150*sqrt(5)*(x
+ 3/5)*log(sqrt(5/11 - 10*x/11) + 1)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**2 + 483153*sq
rt(11)*(x + 3/5)) - 9075*sqrt(5)*(x + 3/5)*log(11)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11)*(x + 3/5)**
2 + 483153*sqrt(11)*(x + 3/5)) + 9075*sqrt(5)*(x + 3/5)*log(10)/(399300*sqrt(11)*(x + 3/5)**3 - 878460*sqrt(11
)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)) + 9075*sqrt(5)*I*pi*(x + 3/5)/(399300*sqrt(11)*(x + 3/5)**3 - 8784
60*sqrt(11)*(x + 3/5)**2 + 483153*sqrt(11)*(x + 3/5)), True))

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Giac [A]  time = 2.1625, size = 104, normalized size = 1.37 \begin{align*} \frac{25}{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{4 \,{\left (60 \, x - 41\right )}}{3993 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{25 \, \sqrt{-2 \, x + 1}}{1331 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 4/3993*(60*x -
41)/((2*x - 1)*sqrt(-2*x + 1)) - 25/1331*sqrt(-2*x + 1)/(5*x + 3)