3.2110 \(\int \frac {1}{(1-2 x)^{3/2} (3+5 x)} \, dx\)

Optimal. Leaf size=43 \[ \frac {2}{11 \sqrt {1-2 x}}-\frac {2}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

-2/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2/11/(1-2*x)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {51, 63, 206} \[ \frac {2}{11 \sqrt {1-2 x}}-\frac {2}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^(3/2)*(3 + 5*x)),x]

[Out]

2/(11*Sqrt[1 - 2*x]) - (2*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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)} \, dx &=\frac {2}{11 \sqrt {1-2 x}}+\frac {5}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {2}{11 \sqrt {1-2 x}}-\frac {5}{11} \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}}-\frac {2}{11} \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 = 0.70 \[ \frac {2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )}{11 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.65, size = 59, normalized size = 1.37 \[ \frac {\sqrt {11} \sqrt {5} {\left (2 \, x - 1\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 22 \, \sqrt {-2 \, x + 1}}{121 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/121*(sqrt(11)*sqrt(5)*(2*x - 1)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 22*sqrt(-2*x +
1))/(2*x - 1)

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giac [A]  time = 1.24, size = 49, normalized size = 1.14 \[ \frac {1}{121} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2}{11 \, \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/11/sqrt(-2*x +
1)

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maple [A]  time = 0.01, size = 29, normalized size = 0.67 \[ -\frac {2 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{121}+\frac {2}{11 \sqrt {-2 x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/121*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+2/11/(-2*x+1)^(1/2)

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maxima [A]  time = 1.22, size = 46, normalized size = 1.07 \[ \frac {1}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2}{11 \, \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/11/sqrt(-2*x + 1)

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mupad [B]  time = 0.06, size = 28, normalized size = 0.65 \[ \frac {2}{11\,\sqrt {1-2\,x}}-\frac {2\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - 2*x)^(3/2)*(5*x + 3)),x)

[Out]

2/(11*(1 - 2*x)^(1/2)) - (2*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121

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sympy [C]  time = 1.86, size = 830, normalized size = 19.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((20*sqrt(5)*I*(x + 3/5)*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) -
10*sqrt(5)*(x + 3/5)*log(110)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 10*sqrt(5)*(x + 3/5)*log(11)/(110*sqr
t(11)*(x + 3/5) - 121*sqrt(11)) - 20*sqrt(5)*(x + 3/5)*log(2)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 10*sqr
t(5)*(x + 3/5)*log(10)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 20*sqrt(5)*(x + 3/5)*log(22)/(110*sqrt(11)*(x
+ 3/5) - 121*sqrt(11)) - 2*sqrt(55)*I*sqrt(10*x - 5)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 22*sqrt(5)*I*a
sin(sqrt(110)/(10*sqrt(x + 3/5)))/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 22*sqrt(5)*log(22)/(110*sqrt(11)*(
x + 3/5) - 121*sqrt(11)) - 11*sqrt(5)*log(10)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 22*sqrt(5)*log(2)/(110
*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 11*sqrt(5)*log(11)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 11*sqrt(5)*
log(110)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)), 10*Abs(x + 3/5)/11 > 1), (-2*sqrt(55)*sqrt(5 - 10*x)/(110*sq
rt(11)*(x + 3/5) - 121*sqrt(11)) + 10*sqrt(5)*(x + 3/5)*log(x + 3/5)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) -
20*sqrt(5)*(x + 3/5)*log(sqrt(5/11 - 10*x/11) + 1)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 10*sqrt(5)*(x +
3/5)*log(11)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 10*sqrt(5)*(x + 3/5)*log(10)/(110*sqrt(11)*(x + 3/5) -
121*sqrt(11)) + 10*sqrt(5)*I*pi*(x + 3/5)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) - 11*sqrt(5)*log(x + 3/5)/(1
10*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 22*sqrt(5)*log(sqrt(5/11 - 10*x/11) + 1)/(110*sqrt(11)*(x + 3/5) - 121
*sqrt(11)) - 11*sqrt(5)*log(10)/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)) + 11*sqrt(5)*log(11)/(110*sqrt(11)*(x
+ 3/5) - 121*sqrt(11)) - 11*sqrt(5)*I*pi/(110*sqrt(11)*(x + 3/5) - 121*sqrt(11)), True))

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