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)-1/11/(1-2*x)^(3/2)/(3+5*x)-50/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+50/1331
/(1-2*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, 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 = 0.01, size = 30, normalized size = 0.39 \[ \frac {4 \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )}{363 (1-2 x)^{3/2}} \]

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))

________________________________________________________________________________________

fricas [A]  time = 1.15, size = 90, normalized size = 1.18 \[ \frac {75 \, \sqrt {11} \sqrt {5} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (1500 \, x^{2} - 400 \, x - 417\right )} \sqrt {-2 \, x + 1}}{43923 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 1.21, size = 77, normalized size = 1.01 \[ \frac {25}{14641} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {4 \, {\left (60 \, x - 41\right )}}{3993 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {25 \, \sqrt {-2 \, x + 1}}{1331 \, {\left (5 \, x + 3\right )}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 54, normalized size = 0.71 \[ -\frac {50 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{14641}+\frac {4}{363 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {40}{1331 \sqrt {-2 x +1}}+\frac {10 \sqrt {-2 x +1}}{1331 \left (-2 x -\frac {6}{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 74, normalized size = 0.97 \[ \frac {25}{14641} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (375 \, {\left (2 \, x - 1\right )}^{2} + 1100 \, x - 792\right )}}{3993 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 11 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]

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))

________________________________________________________________________________________

mupad [B]  time = 1.20, size = 56, normalized size = 0.74 \[ -\frac {\frac {40\,x}{363}+\frac {50\,{\left (2\,x-1\right )}^2}{1331}-\frac {48}{605}}{\frac {11\,{\left (1-2\,x\right )}^{3/2}}{5}-{\left (1-2\,x\right )}^{5/2}}-\frac {50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{14641} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((40*x)/363 + (50*(2*x - 1)^2)/1331 - 48/605)/((11*(1 - 2*x)^(3/2))/5 - (1 - 2*x)^(5/2)) - (50*55^(1/2)*atan
h((55^(1/2)*(1 - 2*x)^(1/2))/11))/14641

________________________________________________________________________________________

sympy [C]  time = 4.59, size = 2286, normalized size = 30.08 \[ \text {result too large to display} \]

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))

________________________________________________________________________________________