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

Optimal. Leaf size=56 \[ \frac {10}{121 \sqrt {1-2 x}}+\frac {2}{33 (1-2 x)^{3/2}}-\frac {10}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {10}{121 \sqrt {1-2 x}}+\frac {2}{33 (1-2 x)^{3/2}}-\frac {10}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^(5/2)*(3 + 5*x)),x]

[Out]

2/(33*(1 - 2*x)^(3/2)) + 10/(121*Sqrt[1 - 2*x]) - (10*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)^{5/2} (3+5 x)} \, dx &=\frac {2}{33 (1-2 x)^{3/2}}+\frac {5}{11} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2}}+\frac {10}{121 \sqrt {1-2 x}}+\frac {25}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2}}+\frac {10}{121 \sqrt {1-2 x}}-\frac {25}{121} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {2}{33 (1-2 x)^{3/2}}+\frac {10}{121 \sqrt {1-2 x}}-\frac {10}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 30, normalized size = 0.54 \begin {gather*} \frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )}{33 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Hypergeometric2F1[-3/2, 1, -1/2, (-5*(-1 + 2*x))/11])/(33*(1 - 2*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.06, size = 52, normalized size = 0.93 \begin {gather*} \frac {2 (15 (1-2 x)+11)}{363 (1-2 x)^{3/2}}-\frac {10}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(11 + 15*(1 - 2*x)))/(363*(1 - 2*x)^(3/2)) - (10*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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fricas [B]  time = 1.18, size = 75, normalized size = 1.34 \begin {gather*} \frac {15 \, \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 ) - 44 \, {\left (15 \, x - 13\right )} \sqrt {-2 \, x + 1}}{3993 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3993*(15*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)) - 44*
(15*x - 13)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.28, size = 61, normalized size = 1.09 \begin {gather*} \frac {5}{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 {4 \, {\left (15 \, x - 13\right )}}{363 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 4/363*(15*x - 13
)/((2*x - 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.01, size = 38, normalized size = 0.68 \begin {gather*} -\frac {10 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1331}+\frac {2}{33 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {10}{121 \sqrt {-2 x +1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/33/(-2*x+1)^(3/2)-10/1331*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+10/121/(-2*x+1)^(1/2)

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maxima [A]  time = 1.09, size = 51, normalized size = 0.91 \begin {gather*} \frac {5}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4 \, {\left (15 \, x - 13\right )}}{363 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 4/363*(15*x - 13)/(-2*x +
1)^(3/2)

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mupad [B]  time = 0.07, size = 33, normalized size = 0.59 \begin {gather*} -\frac {\frac {20\,x}{121}-\frac {52}{363}}{{\left (1-2\,x\right )}^{3/2}}-\frac {10\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - 2*x)^(5/2)*(5*x + 3)),x)

[Out]

- ((20*x)/121 - 52/363)/(1 - 2*x)^(3/2) - (10*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1331

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sympy [C]  time = 2.96, size = 1836, normalized size = 32.79

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),x)

[Out]

Piecewise((-3000*sqrt(5)*I*(x + 3/5)**2*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(-36300*sqrt(11)*(x + 3/5)**2 + 798
60*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 3000*sqrt(5)*(x + 3/5)**2*log(22)/(-36300*sqrt(11)*(x + 3/5)**2 + 79
860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 1500*sqrt(5)*(x + 3/5)**2*log(10)/(-36300*sqrt(11)*(x + 3/5)**2 + 7
9860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 3000*sqrt(5)*(x + 3/5)**2*log(2)/(-36300*sqrt(11)*(x + 3/5)**2 + 7
9860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 1500*sqrt(5)*(x + 3/5)**2*log(11)/(-36300*sqrt(11)*(x + 3/5)**2 +
79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 1500*sqrt(5)*(x + 3/5)**2*log(110)/(-36300*sqrt(11)*(x + 3/5)**2
+ 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 300*sqrt(55)*I*(x + 3/5)*sqrt(10*x - 5)/(-36300*sqrt(11)*(x + 3
/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 6600*sqrt(5)*I*(x + 3/5)*asin(sqrt(110)/(10*sqrt(x + 3/5
)))/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 3300*sqrt(5)*(x + 3/5)*log(11
0)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 3300*sqrt(5)*(x + 3/5)*log(11)
/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 6600*sqrt(5)*(x + 3/5)*log(2)/(-
36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 3300*sqrt(5)*(x + 3/5)*log(10)/(-36
300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 6600*sqrt(5)*(x + 3/5)*log(22)/(-3630
0*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 440*sqrt(55)*I*sqrt(10*x - 5)/(-36300*s
qrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 3630*sqrt(5)*I*asin(sqrt(110)/(10*sqrt(x +
3/5)))/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 3630*sqrt(5)*log(22)/(-36
300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 1815*sqrt(5)*log(10)/(-36300*sqrt(11)
*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 3630*sqrt(5)*log(2)/(-36300*sqrt(11)*(x + 3/5)**2
+ 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 1815*sqrt(5)*log(11)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqr
t(11)*(x + 3/5) - 43923*sqrt(11)) + 1815*sqrt(5)*log(110)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x +
3/5) - 43923*sqrt(11)), 10*Abs(x + 3/5)/11 > 1), (300*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)/(-36300*sqrt(11)*(x +
3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 440*sqrt(55)*sqrt(5 - 10*x)/(-36300*sqrt(11)*(x + 3/5)*
*2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 1500*sqrt(5)*(x + 3/5)**2*log(x + 3/5)/(-36300*sqrt(11)*(x +
3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 3000*sqrt(5)*(x + 3/5)**2*log(sqrt(5/11 - 10*x/11) + 1
)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 1500*sqrt(5)*(x + 3/5)**2*log(1
0)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 1500*sqrt(5)*(x + 3/5)**2*log(
11)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 1500*sqrt(5)*I*pi*(x + 3/5)**
2/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 3300*sqrt(5)*(x + 3/5)*log(x +
3/5)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 6600*sqrt(5)*(x + 3/5)*log(s
qrt(5/11 - 10*x/11) + 1)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 3300*sqr
t(5)*(x + 3/5)*log(11)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 3300*sqrt(
5)*(x + 3/5)*log(10)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 3300*sqrt(5)
*I*pi*(x + 3/5)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 1815*sqrt(5)*log(
x + 3/5)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 3630*sqrt(5)*log(sqrt(5/
11 - 10*x/11) + 1)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 1815*sqrt(5)*l
og(10)/(-36300*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) + 1815*sqrt(5)*log(11)/(-363
00*sqrt(11)*(x + 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)) - 1815*sqrt(5)*I*pi/(-36300*sqrt(11)*(x
+ 3/5)**2 + 79860*sqrt(11)*(x + 3/5) - 43923*sqrt(11)), True))

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