3.1984 \(\int \frac{(1-2 x)^{5/2}}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=76 \[ -\frac{(1-2 x)^{5/2}}{5 (5 x+3)}-\frac{2}{15} (1-2 x)^{3/2}-\frac{22}{25} \sqrt{1-2 x}+\frac{22}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-22*Sqrt[1 - 2*x])/25 - (2*(1 - 2*x)^(3/2))/15 - (1 - 2*x)^(5/2)/(5*(3 + 5*x)) + (22*Sqrt[11/5]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/25

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Rubi [A]  time = 0.0206487, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {47, 50, 63, 206} \[ -\frac{(1-2 x)^{5/2}}{5 (5 x+3)}-\frac{2}{15} (1-2 x)^{3/2}-\frac{22}{25} \sqrt{1-2 x}+\frac{22}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/(3 + 5*x)^2,x]

[Out]

(-22*Sqrt[1 - 2*x])/25 - (2*(1 - 2*x)^(3/2))/15 - (1 - 2*x)^(5/2)/(5*(3 + 5*x)) + (22*Sqrt[11/5]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/25

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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{(1-2 x)^{5/2}}{(3+5 x)^2} \, dx &=-\frac{(1-2 x)^{5/2}}{5 (3+5 x)}-\int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=-\frac{2}{15} (1-2 x)^{3/2}-\frac{(1-2 x)^{5/2}}{5 (3+5 x)}-\frac{11}{5} \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx\\ &=-\frac{22}{25} \sqrt{1-2 x}-\frac{2}{15} (1-2 x)^{3/2}-\frac{(1-2 x)^{5/2}}{5 (3+5 x)}-\frac{121}{25} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{22}{25} \sqrt{1-2 x}-\frac{2}{15} (1-2 x)^{3/2}-\frac{(1-2 x)^{5/2}}{5 (3+5 x)}+\frac{121}{25} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{22}{25} \sqrt{1-2 x}-\frac{2}{15} (1-2 x)^{3/2}-\frac{(1-2 x)^{5/2}}{5 (3+5 x)}+\frac{22}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0060298, size = 30, normalized size = 0.39 \[ -\frac{4}{847} (1-2 x)^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{5}{11} (1-2 x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(1 - 2*x)^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, (5*(1 - 2*x))/11])/847

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Maple [A]  time = 0.009, size = 54, normalized size = 0.7 \begin{align*} -{\frac{4}{75} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{88}{125}\sqrt{1-2\,x}}+{\frac{242}{625}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{22\,\sqrt{55}}{125}{\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((1-2*x)^(5/2)/(3+5*x)^2,x)

[Out]

-4/75*(1-2*x)^(3/2)-88/125*(1-2*x)^(1/2)+242/625*(1-2*x)^(1/2)/(-2*x-6/5)+22/125*arctanh(1/11*55^(1/2)*(1-2*x)
^(1/2))*55^(1/2)

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Maxima [A]  time = 3.35702, size = 96, normalized size = 1.26 \begin{align*} -\frac{4}{75} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{11}{125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{88}{125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{125 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/75*(-2*x + 1)^(3/2) - 11/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 8
8/125*sqrt(-2*x + 1) - 121/125*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 1.43027, size = 204, normalized size = 2.68 \begin{align*} \frac{33 \, \sqrt{11} \sqrt{5}{\left (5 \, x + 3\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 5 \,{\left (40 \, x^{2} - 260 \, x - 243\right )} \sqrt{-2 \, x + 1}}{375 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/375*(33*sqrt(11)*sqrt(5)*(5*x + 3)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 5*(40*x^2 -
260*x - 243)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [A]  time = 2.8901, size = 197, normalized size = 2.59 \begin{align*} \begin{cases} \frac{8 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{375} - \frac{308 \sqrt{5} i \sqrt{10 x - 5}}{1875} - \frac{22 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{125} - \frac{121 \sqrt{5} i \sqrt{10 x - 5}}{3125 \left (x + \frac{3}{5}\right )} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{8 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{375} - \frac{308 \sqrt{5} \sqrt{5 - 10 x}}{1875} - \frac{121 \sqrt{5} \sqrt{5 - 10 x}}{3125 \left (x + \frac{3}{5}\right )} - \frac{11 \sqrt{55} \log{\left (x + \frac{3}{5} \right )}}{125} + \frac{22 \sqrt{55} \log{\left (\sqrt{\frac{5}{11} - \frac{10 x}{11}} + 1 \right )}}{125} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((8*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/375 - 308*sqrt(5)*I*sqrt(10*x - 5)/1875 - 22*sqrt(55)*I*asin(s
qrt(110)/(10*sqrt(x + 3/5)))/125 - 121*sqrt(5)*I*sqrt(10*x - 5)/(3125*(x + 3/5)), 10*Abs(x + 3/5)/11 > 1), (8*
sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)/375 - 308*sqrt(5)*sqrt(5 - 10*x)/1875 - 121*sqrt(5)*sqrt(5 - 10*x)/(3125*(x +
3/5)) - 11*sqrt(55)*log(x + 3/5)/125 + 22*sqrt(55)*log(sqrt(5/11 - 10*x/11) + 1)/125, True))

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Giac [A]  time = 1.94621, size = 100, normalized size = 1.32 \begin{align*} -\frac{4}{75} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{11}{125} \, \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{88}{125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{125 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/75*(-2*x + 1)^(3/2) - 11/125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 88/125*sqrt(-2*x + 1) - 121/125*sqrt(-2*x + 1)/(5*x + 3)