3.2175 \(\int \frac{1}{(1-2 x)^{5/2} (3+5 x)} \, dx\)
Optimal. Leaf size=56 \[ \frac{10}{121 \sqrt{1-2 x}}+\frac{2}{33 (1-2 x)^{3/2}}-\frac{10}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
[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
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Rubi [A] time = 0.013328, antiderivative size = 56, normalized size of antiderivative = 1.,
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} \[ \frac{10}{121 \sqrt{1-2 x}}+\frac{2}{33 (1-2 x)^{3/2}}-\frac{10}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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*}
Mathematica [C] time = 0.0058196, size = 30, normalized size = 0.54 \[ \frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{5}{11} (2 x-1)\right )}{33 (1-2 x)^{3/2}} \]
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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Maple [A] time = 0.007, size = 38, normalized size = 0.7 \begin{align*}{\frac{2}{33} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{10\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{10}{121}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(1-2*x)^(5/2)/(3+5*x),x)
[Out]
2/33/(1-2*x)^(3/2)-10/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+10/121/(1-2*x)^(1/2)
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Maxima [A] time = 3.53009, size = 69, normalized size = 1.23 \begin{align*} \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{align*}
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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Fricas [B] time = 1.31831, size = 212, normalized size = 3.79 \begin{align*} \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{align*}
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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Sympy [C] time = 2.34526, size = 1836, normalized size = 32.79 \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),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 - 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 - 7986
0*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 1500*sqrt(5)*(x + 3/5)**2*log(11)/(36300*sqrt(11)*(x + 3/5)**2 - 7986
0*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 3000*sqrt(5)*(x + 3/5)**2*log(2)/(36300*sqrt(11)*(x + 3/5)**2 - 79860
*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 1500*sqrt(5)*(x + 3/5)**2*log(10)/(36300*sqrt(11)*(x + 3/5)**2 - 79860
*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 3000*sqrt(5)*(x + 3/5)**2*log(22)/(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)))/(363
00*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) - 6600*sqrt(5)*(x + 3/5)*log(22)/(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*sqr
t(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(11)/(36300*sqrt(11)*(
x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 3300*sqrt(5)*(x + 3/5)*log(110)/(36300*sqrt(11)*(x
+ 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 43923*sqrt(11)) + 440*sqrt(55)*I*sqrt(10*x - 5)/(36300*sqrt(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)) - 1815*sqrt(5)*log(110)/(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 - 7
9860*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(10)/(36300*sqrt(11)*(x + 3/5)**2 - 79860*sqrt(11)*(x + 3/5) + 4
3923*sqrt(11)) + 3630*sqrt(5)*log(22)/(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(1
1)*(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(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(10)/(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(sqrt(5/11 - 10*x/11) + 1)/(36
300*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)*(x + 3/5)*log(11)/(36300*sq
rt(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)*log(11)/(36300*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*sqr
t(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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Giac [A] time = 1.81046, size = 82, normalized size = 1.46 \begin{align*} \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{align*}
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))