3.1969 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx\)

Optimal. Leaf size=108 \[ -\frac{27}{220} (1-2 x)^{11/2}+\frac{18}{25} (1-2 x)^{9/2}-\frac{3897 (1-2 x)^{7/2}}{3500}+\frac{2 (1-2 x)^{5/2}}{3125}+\frac{22 (1-2 x)^{3/2}}{9375}+\frac{242 \sqrt{1-2 x}}{15625}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

[Out]

(242*Sqrt[1 - 2*x])/15625 + (22*(1 - 2*x)^(3/2))/9375 + (2*(1 - 2*x)^(5/2))/3125 - (3897*(1 - 2*x)^(7/2))/3500
+ (18*(1 - 2*x)^(9/2))/25 - (27*(1 - 2*x)^(11/2))/220 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/15
625

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Rubi [A]  time = 0.037433, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 206} \[ -\frac{27}{220} (1-2 x)^{11/2}+\frac{18}{25} (1-2 x)^{9/2}-\frac{3897 (1-2 x)^{7/2}}{3500}+\frac{2 (1-2 x)^{5/2}}{3125}+\frac{22 (1-2 x)^{3/2}}{9375}+\frac{242 \sqrt{1-2 x}}{15625}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(242*Sqrt[1 - 2*x])/15625 + (22*(1 - 2*x)^(3/2))/9375 + (2*(1 - 2*x)^(5/2))/3125 - (3897*(1 - 2*x)^(7/2))/3500
+ (18*(1 - 2*x)^(9/2))/25 - (27*(1 - 2*x)^(11/2))/220 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/15
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{(1-2 x)^{5/2} (2+3 x)^3}{3+5 x} \, dx &=\int \left (\frac{3897}{500} (1-2 x)^{5/2}-\frac{162}{25} (1-2 x)^{7/2}+\frac{27}{20} (1-2 x)^{9/2}+\frac{(1-2 x)^{5/2}}{125 (3+5 x)}\right ) \, dx\\ &=-\frac{3897 (1-2 x)^{7/2}}{3500}+\frac{18}{25} (1-2 x)^{9/2}-\frac{27}{220} (1-2 x)^{11/2}+\frac{1}{125} \int \frac{(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac{2 (1-2 x)^{5/2}}{3125}-\frac{3897 (1-2 x)^{7/2}}{3500}+\frac{18}{25} (1-2 x)^{9/2}-\frac{27}{220} (1-2 x)^{11/2}+\frac{11}{625} \int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac{22 (1-2 x)^{3/2}}{9375}+\frac{2 (1-2 x)^{5/2}}{3125}-\frac{3897 (1-2 x)^{7/2}}{3500}+\frac{18}{25} (1-2 x)^{9/2}-\frac{27}{220} (1-2 x)^{11/2}+\frac{121 \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx}{3125}\\ &=\frac{242 \sqrt{1-2 x}}{15625}+\frac{22 (1-2 x)^{3/2}}{9375}+\frac{2 (1-2 x)^{5/2}}{3125}-\frac{3897 (1-2 x)^{7/2}}{3500}+\frac{18}{25} (1-2 x)^{9/2}-\frac{27}{220} (1-2 x)^{11/2}+\frac{1331 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{15625}\\ &=\frac{242 \sqrt{1-2 x}}{15625}+\frac{22 (1-2 x)^{3/2}}{9375}+\frac{2 (1-2 x)^{5/2}}{3125}-\frac{3897 (1-2 x)^{7/2}}{3500}+\frac{18}{25} (1-2 x)^{9/2}-\frac{27}{220} (1-2 x)^{11/2}-\frac{1331 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{15625}\\ &=\frac{242 \sqrt{1-2 x}}{15625}+\frac{22 (1-2 x)^{3/2}}{9375}+\frac{2 (1-2 x)^{5/2}}{3125}-\frac{3897 (1-2 x)^{7/2}}{3500}+\frac{18}{25} (1-2 x)^{9/2}-\frac{27}{220} (1-2 x)^{11/2}-\frac{242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{15625}\\ \end{align*}

Mathematica [A]  time = 0.0598353, size = 66, normalized size = 0.61 \[ \frac{5 \sqrt{1-2 x} \left (14175000 x^5+6142500 x^4-15572250 x^3-3564885 x^2+7726195 x-1796318\right )-55902 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{18046875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[1 - 2*x]*(-1796318 + 7726195*x - 3564885*x^2 - 15572250*x^3 + 6142500*x^4 + 14175000*x^5) - 55902*Sqrt
[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/18046875

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Maple [A]  time = 0.005, size = 74, normalized size = 0.7 \begin{align*}{\frac{22}{9375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{3125} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{3897}{3500} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{18}{25} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{27}{220} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}-{\frac{242\,\sqrt{55}}{78125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{242}{15625}\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/(3+5*x),x)

[Out]

22/9375*(1-2*x)^(3/2)+2/3125*(1-2*x)^(5/2)-3897/3500*(1-2*x)^(7/2)+18/25*(1-2*x)^(9/2)-27/220*(1-2*x)^(11/2)-2
42/78125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+242/15625*(1-2*x)^(1/2)

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Maxima [A]  time = 2.0409, size = 123, normalized size = 1.14 \begin{align*} -\frac{27}{220} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{18}{25} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{3897}{3500} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{3125} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{22}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{78125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{242}{15625} \, \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/(3+5*x),x, algorithm="maxima")

[Out]

-27/220*(-2*x + 1)^(11/2) + 18/25*(-2*x + 1)^(9/2) - 3897/3500*(-2*x + 1)^(7/2) + 2/3125*(-2*x + 1)^(5/2) + 22
/9375*(-2*x + 1)^(3/2) + 121/78125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 242/15625*sqrt(-2*x + 1)

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Fricas [A]  time = 1.41586, size = 262, normalized size = 2.43 \begin{align*} \frac{121}{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 (14175000 \, x^{5} + 6142500 \, x^{4} - 15572250 \, x^{3} - 3564885 \, x^{2} + 7726195 \, x - 1796318\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/(3+5*x),x, algorithm="fricas")

[Out]

121/78125*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 1/3609375*(14175000*x^
5 + 6142500*x^4 - 15572250*x^3 - 3564885*x^2 + 7726195*x - 1796318)*sqrt(-2*x + 1)

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Sympy [A]  time = 55.6912, size = 138, normalized size = 1.28 \begin{align*} - \frac{27 \left (1 - 2 x\right )^{\frac{11}{2}}}{220} + \frac{18 \left (1 - 2 x\right )^{\frac{9}{2}}}{25} - \frac{3897 \left (1 - 2 x\right )^{\frac{7}{2}}}{3500} + \frac{2 \left (1 - 2 x\right )^{\frac{5}{2}}}{3125} + \frac{22 \left (1 - 2 x\right )^{\frac{3}{2}}}{9375} + \frac{242 \sqrt{1 - 2 x}}{15625} + \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 )}{15625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*(1 - 2*x)**(11/2)/220 + 18*(1 - 2*x)**(9/2)/25 - 3897*(1 - 2*x)**(7/2)/3500 + 2*(1 - 2*x)**(5/2)/3125 + 22
*(1 - 2*x)**(3/2)/9375 + 242*sqrt(1 - 2*x)/15625 + 2662*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 = 2.03055, size = 165, normalized size = 1.53 \begin{align*} \frac{27}{220} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{18}{25} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{3897}{3500} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{3125} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{22}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{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{242}{15625} \, \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/(3+5*x),x, algorithm="giac")

[Out]

27/220*(2*x - 1)^5*sqrt(-2*x + 1) + 18/25*(2*x - 1)^4*sqrt(-2*x + 1) + 3897/3500*(2*x - 1)^3*sqrt(-2*x + 1) +
2/3125*(2*x - 1)^2*sqrt(-2*x + 1) + 22/9375*(-2*x + 1)^(3/2) + 121/78125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10
*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 242/15625*sqrt(-2*x + 1)