3.1829 \(\int \frac{\sqrt{1-2 x} (2+3 x)^3}{3+5 x} \, dx\)

Optimal. Leaf size=82 \[ -\frac{27}{140} (1-2 x)^{7/2}+\frac{162}{125} (1-2 x)^{5/2}-\frac{1299}{500} (1-2 x)^{3/2}+\frac{2}{625} \sqrt{1-2 x}-\frac{2}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(2*Sqrt[1 - 2*x])/625 - (1299*(1 - 2*x)^(3/2))/500 + (162*(1 - 2*x)^(5/2))/125 - (27*(1 - 2*x)^(7/2))/140 - (2
*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

________________________________________________________________________________________

Rubi [A]  time = 0.0272052, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ -\frac{27}{140} (1-2 x)^{7/2}+\frac{162}{125} (1-2 x)^{5/2}-\frac{1299}{500} (1-2 x)^{3/2}+\frac{2}{625} \sqrt{1-2 x}-\frac{2}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 - 2*x])/625 - (1299*(1 - 2*x)^(3/2))/500 + (162*(1 - 2*x)^(5/2))/125 - (27*(1 - 2*x)^(7/2))/140 - (2
*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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} (2+3 x)^3}{3+5 x} \, dx &=\int \left (\frac{3897}{500} \sqrt{1-2 x}-\frac{162}{25} (1-2 x)^{3/2}+\frac{27}{20} (1-2 x)^{5/2}+\frac{\sqrt{1-2 x}}{125 (3+5 x)}\right ) \, dx\\ &=-\frac{1299}{500} (1-2 x)^{3/2}+\frac{162}{125} (1-2 x)^{5/2}-\frac{27}{140} (1-2 x)^{7/2}+\frac{1}{125} \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx\\ &=\frac{2}{625} \sqrt{1-2 x}-\frac{1299}{500} (1-2 x)^{3/2}+\frac{162}{125} (1-2 x)^{5/2}-\frac{27}{140} (1-2 x)^{7/2}+\frac{11}{625} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{2}{625} \sqrt{1-2 x}-\frac{1299}{500} (1-2 x)^{3/2}+\frac{162}{125} (1-2 x)^{5/2}-\frac{27}{140} (1-2 x)^{7/2}-\frac{11}{625} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{2}{625} \sqrt{1-2 x}-\frac{1299}{500} (1-2 x)^{3/2}+\frac{162}{125} (1-2 x)^{5/2}-\frac{27}{140} (1-2 x)^{7/2}-\frac{2}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0302673, size = 56, normalized size = 0.68 \[ \frac{5 \sqrt{1-2 x} \left (6750 x^3+12555 x^2+5115 x-6526\right )-14 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{21875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[1 - 2*x]*(-6526 + 5115*x + 12555*x^2 + 6750*x^3) - 14*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/2187
5

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 56, normalized size = 0.7 \begin{align*} -{\frac{1299}{500} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{162}{125} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{27}{140} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2\,\sqrt{55}}{3125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2}{625}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1299/500*(1-2*x)^(3/2)+162/125*(1-2*x)^(5/2)-27/140*(1-2*x)^(7/2)-2/3125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
*55^(1/2)+2/625*(1-2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.65772, size = 99, normalized size = 1.21 \begin{align*} -\frac{27}{140} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{162}{125} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1299}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{3125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2}{625} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/140*(-2*x + 1)^(7/2) + 162/125*(-2*x + 1)^(5/2) - 1299/500*(-2*x + 1)^(3/2) + 1/3125*sqrt(55)*log(-(sqrt(5
5) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/625*sqrt(-2*x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.6494, size = 198, normalized size = 2.41 \begin{align*} \frac{1}{3125} \, \sqrt{11} \sqrt{5} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + \frac{1}{4375} \,{\left (6750 \, x^{3} + 12555 \, x^{2} + 5115 \, x - 6526\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 1/4375*(6750*x^3 + 12555*
x^2 + 5115*x - 6526)*sqrt(-2*x + 1)

________________________________________________________________________________________

Sympy [A]  time = 5.6609, size = 114, normalized size = 1.39 \begin{align*} - \frac{27 \left (1 - 2 x\right )^{\frac{7}{2}}}{140} + \frac{162 \left (1 - 2 x\right )^{\frac{5}{2}}}{125} - \frac{1299 \left (1 - 2 x\right )^{\frac{3}{2}}}{500} + \frac{2 \sqrt{1 - 2 x}}{625} + \frac{22 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 < - \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 > - \frac{11}{5} \end{cases}\right )}{625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*(1 - 2*x)**(7/2)/140 + 162*(1 - 2*x)**(5/2)/125 - 1299*(1 - 2*x)**(3/2)/500 + 2*sqrt(1 - 2*x)/625 + 22*Pie
cewise((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/55, 2*x - 1 < -11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(1 - 2*
x)/11)/55, 2*x - 1 > -11/5))/625

________________________________________________________________________________________

Giac [A]  time = 1.54703, size = 122, normalized size = 1.49 \begin{align*} \frac{27}{140} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{162}{125} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{1299}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{3125} \, \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{2}{625} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/140*(2*x - 1)^3*sqrt(-2*x + 1) + 162/125*(2*x - 1)^2*sqrt(-2*x + 1) - 1299/500*(-2*x + 1)^(3/2) + 1/3125*sq
rt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2/625*sqrt(-2*x + 1)