3.1897 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx\)

Optimal. Leaf size=108 \[ \frac{81}{440} (1-2 x)^{11/2}-\frac{321}{200} (1-2 x)^{9/2}+\frac{34371 (1-2 x)^{7/2}}{7000}-\frac{136419 (1-2 x)^{5/2}}{25000}+\frac{2 (1-2 x)^{3/2}}{9375}+\frac{22 \sqrt{1-2 x}}{15625}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

[Out]

(22*Sqrt[1 - 2*x])/15625 + (2*(1 - 2*x)^(3/2))/9375 - (136419*(1 - 2*x)^(5/2))/25000 + (34371*(1 - 2*x)^(7/2))
/7000 - (321*(1 - 2*x)^(9/2))/200 + (81*(1 - 2*x)^(11/2))/440 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/15625

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Rubi [A]  time = 0.0338326, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ \frac{81}{440} (1-2 x)^{11/2}-\frac{321}{200} (1-2 x)^{9/2}+\frac{34371 (1-2 x)^{7/2}}{7000}-\frac{136419 (1-2 x)^{5/2}}{25000}+\frac{2 (1-2 x)^{3/2}}{9375}+\frac{22 \sqrt{1-2 x}}{15625}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x),x]

[Out]

(22*Sqrt[1 - 2*x])/15625 + (2*(1 - 2*x)^(3/2))/9375 - (136419*(1 - 2*x)^(5/2))/25000 + (34371*(1 - 2*x)^(7/2))
/7000 - (321*(1 - 2*x)^(9/2))/200 + (81*(1 - 2*x)^(11/2))/440 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x
]])/15625

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{(1-2 x)^{3/2} (2+3 x)^4}{3+5 x} \, dx &=\int \left (\frac{136419 (1-2 x)^{3/2}}{5000}-\frac{34371 (1-2 x)^{5/2}}{1000}+\frac{2889}{200} (1-2 x)^{7/2}-\frac{81}{40} (1-2 x)^{9/2}+\frac{(1-2 x)^{3/2}}{625 (3+5 x)}\right ) \, dx\\ &=-\frac{136419 (1-2 x)^{5/2}}{25000}+\frac{34371 (1-2 x)^{7/2}}{7000}-\frac{321}{200} (1-2 x)^{9/2}+\frac{81}{440} (1-2 x)^{11/2}+\frac{1}{625} \int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac{2 (1-2 x)^{3/2}}{9375}-\frac{136419 (1-2 x)^{5/2}}{25000}+\frac{34371 (1-2 x)^{7/2}}{7000}-\frac{321}{200} (1-2 x)^{9/2}+\frac{81}{440} (1-2 x)^{11/2}+\frac{11 \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx}{3125}\\ &=\frac{22 \sqrt{1-2 x}}{15625}+\frac{2 (1-2 x)^{3/2}}{9375}-\frac{136419 (1-2 x)^{5/2}}{25000}+\frac{34371 (1-2 x)^{7/2}}{7000}-\frac{321}{200} (1-2 x)^{9/2}+\frac{81}{440} (1-2 x)^{11/2}+\frac{121 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{15625}\\ &=\frac{22 \sqrt{1-2 x}}{15625}+\frac{2 (1-2 x)^{3/2}}{9375}-\frac{136419 (1-2 x)^{5/2}}{25000}+\frac{34371 (1-2 x)^{7/2}}{7000}-\frac{321}{200} (1-2 x)^{9/2}+\frac{81}{440} (1-2 x)^{11/2}-\frac{121 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{15625}\\ &=\frac{22 \sqrt{1-2 x}}{15625}+\frac{2 (1-2 x)^{3/2}}{9375}-\frac{136419 (1-2 x)^{5/2}}{25000}+\frac{34371 (1-2 x)^{7/2}}{7000}-\frac{321}{200} (1-2 x)^{9/2}+\frac{81}{440} (1-2 x)^{11/2}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625}\\ \end{align*}

Mathematica [A]  time = 0.0571759, size = 66, normalized size = 0.61 \[ \frac{-5 \sqrt{1-2 x} \left (21262500 x^5+39532500 x^4+9559125 x^3-21433590 x^2-12144995 x+7095688\right )-5082 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{18046875} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x),x]

[Out]

(-5*Sqrt[1 - 2*x]*(7095688 - 12144995*x - 21433590*x^2 + 9559125*x^3 + 39532500*x^4 + 21262500*x^5) - 5082*Sqr
t[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/18046875

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Maple [A]  time = 0.007, size = 74, normalized size = 0.7 \begin{align*}{\frac{2}{9375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{136419}{25000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{34371}{7000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{321}{200} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{81}{440} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}-{\frac{22\,\sqrt{55}}{78125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{15625}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x)

[Out]

2/9375*(1-2*x)^(3/2)-136419/25000*(1-2*x)^(5/2)+34371/7000*(1-2*x)^(7/2)-321/200*(1-2*x)^(9/2)+81/440*(1-2*x)^
(11/2)-22/78125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+22/15625*(1-2*x)^(1/2)

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Maxima [A]  time = 2.98812, size = 123, normalized size = 1.14 \begin{align*} \frac{81}{440} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{321}{200} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{34371}{7000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{136419}{25000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{78125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22}{15625} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x, algorithm="maxima")

[Out]

81/440*(-2*x + 1)^(11/2) - 321/200*(-2*x + 1)^(9/2) + 34371/7000*(-2*x + 1)^(7/2) - 136419/25000*(-2*x + 1)^(5
/2) + 2/9375*(-2*x + 1)^(3/2) + 11/78125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 22/15625*sqrt(-2*x + 1)

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Fricas [A]  time = 1.38534, size = 263, normalized size = 2.44 \begin{align*} \frac{11}{78125} \, \sqrt{11} \sqrt{5} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac{1}{3609375} \,{\left (21262500 \, x^{5} + 39532500 \, x^{4} + 9559125 \, x^{3} - 21433590 \, x^{2} - 12144995 \, x + 7095688\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x, algorithm="fricas")

[Out]

11/78125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 1/3609375*(21262500*x^5
+ 39532500*x^4 + 9559125*x^3 - 21433590*x^2 - 12144995*x + 7095688)*sqrt(-2*x + 1)

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Sympy [A]  time = 57.0217, size = 138, normalized size = 1.28 \begin{align*} \frac{81 \left (1 - 2 x\right )^{\frac{11}{2}}}{440} - \frac{321 \left (1 - 2 x\right )^{\frac{9}{2}}}{200} + \frac{34371 \left (1 - 2 x\right )^{\frac{7}{2}}}{7000} - \frac{136419 \left (1 - 2 x\right )^{\frac{5}{2}}}{25000} + \frac{2 \left (1 - 2 x\right )^{\frac{3}{2}}}{9375} + \frac{22 \sqrt{1 - 2 x}}{15625} + \frac{242 \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 )}{15625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(3/2)*(2+3*x)**4/(3+5*x),x)

[Out]

81*(1 - 2*x)**(11/2)/440 - 321*(1 - 2*x)**(9/2)/200 + 34371*(1 - 2*x)**(7/2)/7000 - 136419*(1 - 2*x)**(5/2)/25
000 + 2*(1 - 2*x)**(3/2)/9375 + 22*sqrt(1 - 2*x)/15625 + 242*Piecewise((-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))/15625

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Giac [A]  time = 2.36578, size = 165, normalized size = 1.53 \begin{align*} -\frac{81}{440} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{321}{200} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{34371}{7000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{136419}{25000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{78125} \, \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}{15625} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x),x, algorithm="giac")

[Out]

-81/440*(2*x - 1)^5*sqrt(-2*x + 1) - 321/200*(2*x - 1)^4*sqrt(-2*x + 1) - 34371/7000*(2*x - 1)^3*sqrt(-2*x + 1
) - 136419/25000*(2*x - 1)^2*sqrt(-2*x + 1) + 2/9375*(-2*x + 1)^(3/2) + 11/78125*sqrt(55)*log(1/2*abs(-2*sqrt(
55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/15625*sqrt(-2*x + 1)