3.1971 \(\int \frac{(1-2 x)^{5/2} (2+3 x)}{3+5 x} \, dx\)

Optimal. Leaf size=82 \[ -\frac{3}{35} (1-2 x)^{7/2}+\frac{2}{125} (1-2 x)^{5/2}+\frac{22}{375} (1-2 x)^{3/2}+\frac{242}{625} \sqrt{1-2 x}-\frac{242}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(242*Sqrt[1 - 2*x])/625 + (22*(1 - 2*x)^(3/2))/375 + (2*(1 - 2*x)^(5/2))/125 - (3*(1 - 2*x)^(7/2))/35 - (242*S
qrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

________________________________________________________________________________________

Rubi [A]  time = 0.0225279, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {80, 50, 63, 206} \[ -\frac{3}{35} (1-2 x)^{7/2}+\frac{2}{125} (1-2 x)^{5/2}+\frac{22}{375} (1-2 x)^{3/2}+\frac{242}{625} \sqrt{1-2 x}-\frac{242}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(242*Sqrt[1 - 2*x])/625 + (22*(1 - 2*x)^(3/2))/375 + (2*(1 - 2*x)^(5/2))/125 - (3*(1 - 2*x)^(7/2))/35 - (242*S
qrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/625

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{(1-2 x)^{5/2} (2+3 x)}{3+5 x} \, dx &=-\frac{3}{35} (1-2 x)^{7/2}+\frac{1}{5} \int \frac{(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac{2}{125} (1-2 x)^{5/2}-\frac{3}{35} (1-2 x)^{7/2}+\frac{11}{25} \int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac{22}{375} (1-2 x)^{3/2}+\frac{2}{125} (1-2 x)^{5/2}-\frac{3}{35} (1-2 x)^{7/2}+\frac{121}{125} \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx\\ &=\frac{242}{625} \sqrt{1-2 x}+\frac{22}{375} (1-2 x)^{3/2}+\frac{2}{125} (1-2 x)^{5/2}-\frac{3}{35} (1-2 x)^{7/2}+\frac{1331}{625} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{242}{625} \sqrt{1-2 x}+\frac{22}{375} (1-2 x)^{3/2}+\frac{2}{125} (1-2 x)^{5/2}-\frac{3}{35} (1-2 x)^{7/2}-\frac{1331}{625} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{242}{625} \sqrt{1-2 x}+\frac{22}{375} (1-2 x)^{3/2}+\frac{2}{125} (1-2 x)^{5/2}-\frac{3}{35} (1-2 x)^{7/2}-\frac{242}{625} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0220235, size = 56, normalized size = 0.68 \[ \frac{5 \sqrt{1-2 x} \left (9000 x^3-12660 x^2+4370 x+4937\right )-5082 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{65625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[1 - 2*x]*(4937 + 4370*x - 12660*x^2 + 9000*x^3) - 5082*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/656
25

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 56, normalized size = 0.7 \begin{align*}{\frac{22}{375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{125} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{3}{35} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{242\,\sqrt{55}}{3125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{242}{625}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

22/375*(1-2*x)^(3/2)+2/125*(1-2*x)^(5/2)-3/35*(1-2*x)^(7/2)-242/3125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(
1/2)+242/625*(1-2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.95172, size = 99, normalized size = 1.21 \begin{align*} -\frac{3}{35} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{125} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{22}{375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{3125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{242}{625} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/35*(-2*x + 1)^(7/2) + 2/125*(-2*x + 1)^(5/2) + 22/375*(-2*x + 1)^(3/2) + 121/3125*sqrt(55)*log(-(sqrt(55) -
5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/625*sqrt(-2*x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.35514, size = 203, normalized size = 2.48 \begin{align*} \frac{121}{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}{13125} \,{\left (9000 \, x^{3} - 12660 \, x^{2} + 4370 \, x + 4937\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/3125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 1/13125*(9000*x^3 - 126
60*x^2 + 4370*x + 4937)*sqrt(-2*x + 1)

________________________________________________________________________________________

Sympy [A]  time = 24.6752, size = 114, normalized size = 1.39 \begin{align*} - \frac{3 \left (1 - 2 x\right )^{\frac{7}{2}}}{35} + \frac{2 \left (1 - 2 x\right )^{\frac{5}{2}}}{125} + \frac{22 \left (1 - 2 x\right )^{\frac{3}{2}}}{375} + \frac{242 \sqrt{1 - 2 x}}{625} + \frac{2662 \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((1-2*x)**(5/2)*(2+3*x)/(3+5*x),x)

[Out]

-3*(1 - 2*x)**(7/2)/35 + 2*(1 - 2*x)**(5/2)/125 + 22*(1 - 2*x)**(3/2)/375 + 242*sqrt(1 - 2*x)/625 + 2662*Piece
wise((-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 = 1.85751, size = 122, normalized size = 1.49 \begin{align*} \frac{3}{35} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{125} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{22}{375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{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{242}{625} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/35*(2*x - 1)^3*sqrt(-2*x + 1) + 2/125*(2*x - 1)^2*sqrt(-2*x + 1) + 22/375*(-2*x + 1)^(3/2) + 121/3125*sqrt(5
5)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/625*sqrt(-2*x + 1)