3.1809 \(\int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=74 \[ -\frac{(1-2 x)^{3/2}}{63 (3 x+2)}-\frac{25}{27} (1-2 x)^{3/2}-\frac{142}{189} \sqrt{1-2 x}+\frac{142 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

[Out]

(-142*Sqrt[1 - 2*x])/189 - (25*(1 - 2*x)^(3/2))/27 - (1 - 2*x)^(3/2)/(63*(2 + 3*x)) + (142*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/(27*Sqrt[21])

________________________________________________________________________________________

Rubi [A]  time = 0.0200904, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 80, 50, 63, 206} \[ -\frac{(1-2 x)^{3/2}}{63 (3 x+2)}-\frac{25}{27} (1-2 x)^{3/2}-\frac{142}{189} \sqrt{1-2 x}+\frac{142 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-142*Sqrt[1 - 2*x])/189 - (25*(1 - 2*x)^(3/2))/27 - (1 - 2*x)^(3/2)/(63*(2 + 3*x)) + (142*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/(27*Sqrt[21])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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}{(2+3 x)^2} \, dx &=-\frac{(1-2 x)^{3/2}}{63 (2+3 x)}+\frac{1}{63} \int \frac{\sqrt{1-2 x} (279+525 x)}{2+3 x} \, dx\\ &=-\frac{25}{27} (1-2 x)^{3/2}-\frac{(1-2 x)^{3/2}}{63 (2+3 x)}-\frac{71}{63} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=-\frac{142}{189} \sqrt{1-2 x}-\frac{25}{27} (1-2 x)^{3/2}-\frac{(1-2 x)^{3/2}}{63 (2+3 x)}-\frac{71}{27} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{142}{189} \sqrt{1-2 x}-\frac{25}{27} (1-2 x)^{3/2}-\frac{(1-2 x)^{3/2}}{63 (2+3 x)}+\frac{71}{27} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{142}{189} \sqrt{1-2 x}-\frac{25}{27} (1-2 x)^{3/2}-\frac{(1-2 x)^{3/2}}{63 (2+3 x)}+\frac{142 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0360349, size = 55, normalized size = 0.74 \[ \frac{\sqrt{1-2 x} \left (150 x^2-35 x-91\right )}{81 x+54}+\frac{142 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(-91 - 35*x + 150*x^2))/(54 + 81*x) + (142*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 54, normalized size = 0.7 \begin{align*} -{\frac{25}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{20}{27}\sqrt{1-2\,x}}+{\frac{2}{81}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}+{\frac{142\,\sqrt{21}}{567}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/27*(1-2*x)^(3/2)-20/27*(1-2*x)^(1/2)+2/81*(1-2*x)^(1/2)/(-2*x-4/3)+142/567*arctanh(1/7*21^(1/2)*(1-2*x)^(1
/2))*21^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 2.22573, size = 96, normalized size = 1.3 \begin{align*} -\frac{25}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{71}{567} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20}{27} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{27 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/27*(-2*x + 1)^(3/2) - 71/567*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) -
20/27*sqrt(-2*x + 1) - 1/27*sqrt(-2*x + 1)/(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.6058, size = 181, normalized size = 2.45 \begin{align*} \frac{71 \, \sqrt{21}{\left (3 \, x + 2\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (150 \, x^{2} - 35 \, x - 91\right )} \sqrt{-2 \, x + 1}}{567 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/567*(71*sqrt(21)*(3*x + 2)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(150*x^2 - 35*x - 91)*sqr
t(-2*x + 1))/(3*x + 2)

________________________________________________________________________________________

Sympy [A]  time = 66.3321, size = 192, normalized size = 2.59 \begin{align*} - \frac{25 \left (1 - 2 x\right )^{\frac{3}{2}}}{27} - \frac{20 \sqrt{1 - 2 x}}{27} - \frac{28 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{27} - \frac{16 \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 )}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25*(1 - 2*x)**(3/2)/27 - 20*sqrt(1 - 2*x)/27 - 28*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 +
log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 -
1)))/147, (x <= 1/2) & (x > -2/3)))/27 - 16*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))/3

________________________________________________________________________________________

Giac [A]  time = 1.60841, size = 100, normalized size = 1.35 \begin{align*} -\frac{25}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{71}{567} \, \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{20}{27} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{27 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/27*(-2*x + 1)^(3/2) - 71/567*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 20/27*sqrt(-2*x + 1) - 1/27*sqrt(-2*x + 1)/(3*x + 2)