3.1901 \(\int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx\)

Optimal. Leaf size=56 \[ \frac{2}{15} (1-2 x)^{3/2}+\frac{22}{25} \sqrt{1-2 x}-\frac{22}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0124812, 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 = {50, 63, 206} \[ \frac{2}{15} (1-2 x)^{3/2}+\frac{22}{25} \sqrt{1-2 x}-\frac{22}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)/(3 + 5*x),x]

[Out]

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

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)^{3/2}}{3+5 x} \, dx &=\frac{2}{15} (1-2 x)^{3/2}+\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{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{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{22}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0205387, size = 46, normalized size = 0.82 \[ -\frac{2}{375} \left (10 \sqrt{1-2 x} (5 x-19)+33 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(10*Sqrt[1 - 2*x]*(-19 + 5*x) + 33*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/375

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Maple [A]  time = 0.004, size = 38, normalized size = 0.7 \begin{align*}{\frac{2}{15} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{22\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{25}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(1-2*x)^(3/2)-22/125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+22/25*(1-2*x)^(1/2)

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Maxima [A]  time = 1.92934, size = 74, normalized size = 1.32 \begin{align*} \frac{2}{15} \,{\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{22}{25} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(-2*x + 1)^(3/2) + 11/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22
/25*sqrt(-2*x + 1)

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Fricas [A]  time = 1.31478, size = 158, normalized size = 2.82 \begin{align*} \frac{11}{125} \, \sqrt{11} \sqrt{5} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac{4}{75} \,{\left (5 \, x - 19\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 4/75*(5*x - 19)*sqrt(-2*x
+ 1)

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Sympy [A]  time = 1.64456, size = 155, normalized size = 2.77 \begin{align*} \begin{cases} - \frac{4 \sqrt{5} i \left (x + \frac{3}{5}\right ) \sqrt{10 x - 5}}{75} + \frac{88 \sqrt{5} i \sqrt{10 x - 5}}{375} + \frac{22 \sqrt{55} i \operatorname{asin}{\left (\frac{\sqrt{110}}{10 \sqrt{x + \frac{3}{5}}} \right )}}{125} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{4 \sqrt{5} \sqrt{5 - 10 x} \left (x + \frac{3}{5}\right )}{75} + \frac{88 \sqrt{5} \sqrt{5 - 10 x}}{375} + \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)**(3/2)/(3+5*x),x)

[Out]

Piecewise((-4*sqrt(5)*I*(x + 3/5)*sqrt(10*x - 5)/75 + 88*sqrt(5)*I*sqrt(10*x - 5)/375 + 22*sqrt(55)*I*asin(sqr
t(110)/(10*sqrt(x + 3/5)))/125, 10*Abs(x + 3/5)/11 > 1), (-4*sqrt(5)*sqrt(5 - 10*x)*(x + 3/5)/75 + 88*sqrt(5)*
sqrt(5 - 10*x)/375 + 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 = 2.23719, size = 78, normalized size = 1.39 \begin{align*} \frac{2}{15} \,{\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{22}{25} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(-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))) + 22/25*sqrt(-2*x + 1)