3.1797 \(\int \frac{\sqrt{1-2 x} (3+5 x)}{2+3 x} \, dx\)

Optimal. Leaf size=56 \[ -\frac{5}{9} (1-2 x)^{3/2}-\frac{2}{9} \sqrt{1-2 x}+\frac{2}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-2*Sqrt[1 - 2*x])/9 - (5*(1 - 2*x)^(3/2))/9 + (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9

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Rubi [A]  time = 0.0153621, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {80, 50, 63, 206} \[ -\frac{5}{9} (1-2 x)^{3/2}-\frac{2}{9} \sqrt{1-2 x}+\frac{2}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x))/(2 + 3*x),x]

[Out]

(-2*Sqrt[1 - 2*x])/9 - (5*(1 - 2*x)^(3/2))/9 + (2*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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{\sqrt{1-2 x} (3+5 x)}{2+3 x} \, dx &=-\frac{5}{9} (1-2 x)^{3/2}-\frac{1}{3} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=-\frac{2}{9} \sqrt{1-2 x}-\frac{5}{9} (1-2 x)^{3/2}-\frac{7}{9} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{2}{9} \sqrt{1-2 x}-\frac{5}{9} (1-2 x)^{3/2}+\frac{7}{9} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{2}{9} \sqrt{1-2 x}-\frac{5}{9} (1-2 x)^{3/2}+\frac{2}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0207419, size = 46, normalized size = 0.82 \[ \frac{1}{27} \left (3 \sqrt{1-2 x} (10 x-7)+2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[1 - 2*x]*(-7 + 10*x) + 2*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/27

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Maple [A]  time = 0.005, size = 38, normalized size = 0.7 \begin{align*} -{\frac{5}{9} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2\,\sqrt{21}}{27}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{2}{9}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/9*(1-2*x)^(3/2)+2/27*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2/9*(1-2*x)^(1/2)

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Maxima [A]  time = 1.52855, size = 74, normalized size = 1.32 \begin{align*} -\frac{5}{9} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{27} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2}{9} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/9*(-2*x + 1)^(3/2) - 1/27*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/9*
sqrt(-2*x + 1)

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Fricas [A]  time = 1.2222, size = 153, normalized size = 2.73 \begin{align*} \frac{1}{27} \, \sqrt{7} \sqrt{3} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + \frac{1}{9} \,{\left (10 \, x - 7\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*sqrt(7)*sqrt(3)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 1/9*(10*x - 7)*sqrt(-2*x + 1
)

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Sympy [A]  time = 3.96889, size = 92, normalized size = 1.64 \begin{align*} - \frac{5 \left (1 - 2 x\right )^{\frac{3}{2}}}{9} - \frac{2 \sqrt{1 - 2 x}}{9} - \frac{14 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 < - \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 > - \frac{7}{3} \end{cases}\right )}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*(1 - 2*x)**(3/2)/9 - 2*sqrt(1 - 2*x)/9 - 14*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x -
1 < -7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 > -7/3))/9

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Giac [A]  time = 2.06857, size = 78, normalized size = 1.39 \begin{align*} -\frac{5}{9} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{27} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{2}{9} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/9*(-2*x + 1)^(3/2) - 1/27*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 2/9*sqrt(-2*x + 1)