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

Optimal. Leaf size=95 \[ -\frac {3}{20} (1-2 x)^{9/2}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {2 (1-2 x)^{3/2}}{1875}+\frac {22 \sqrt {1-2 x}}{3125}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {3}{20} (1-2 x)^{9/2}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {2 (1-2 x)^{3/2}}{1875}+\frac {22 \sqrt {1-2 x}}{3125}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(22*Sqrt[1 - 2*x])/3125 + (2*(1 - 2*x)^(3/2))/1875 - (3897*(1 - 2*x)^(5/2))/2500 + (162*(1 - 2*x)^(7/2))/175 -
(3*(1 - 2*x)^(9/2))/20 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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 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 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)^3}{3+5 x} \, dx &=\int \left (\frac {3897}{500} (1-2 x)^{3/2}-\frac {162}{25} (1-2 x)^{5/2}+\frac {27}{20} (1-2 x)^{7/2}+\frac {(1-2 x)^{3/2}}{125 (3+5 x)}\right ) \, dx\\ &=-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}+\frac {1}{125} \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{1875}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}+\frac {11}{625} \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx\\ &=\frac {22 \sqrt {1-2 x}}{3125}+\frac {2 (1-2 x)^{3/2}}{1875}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}+\frac {121 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac {22 \sqrt {1-2 x}}{3125}+\frac {2 (1-2 x)^{3/2}}{1875}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}-\frac {121 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{3125}\\ &=\frac {22 \sqrt {1-2 x}}{3125}+\frac {2 (1-2 x)^{3/2}}{1875}-\frac {3897 (1-2 x)^{5/2}}{2500}+\frac {162}{175} (1-2 x)^{7/2}-\frac {3}{20} (1-2 x)^{9/2}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 61, normalized size = 0.64 \begin {gather*} \frac {-5 \sqrt {1-2 x} \left (157500 x^4+171000 x^3-83565 x^2-123295 x+50858\right )-462 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{328125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[1 - 2*x]*(50858 - 123295*x - 83565*x^2 + 171000*x^3 + 157500*x^4) - 462*Sqrt[55]*ArcTanh[Sqrt[5/11]*S
qrt[1 - 2*x]])/328125

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.07, size = 79, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {1-2 x} \left (39375 (1-2 x)^4-243000 (1-2 x)^3+409185 (1-2 x)^2-280 (1-2 x)-1848\right )}{262500}-\frac {22 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/262500*((-1848 - 280*(1 - 2*x) + 409185*(1 - 2*x)^2 - 243000*(1 - 2*x)^3 + 39375*(1 - 2*x)^4)*Sqrt[1 - 2*x]
) - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

________________________________________________________________________________________

fricas [A]  time = 1.20, size = 66, normalized size = 0.69 \begin {gather*} \frac {11}{15625} \, \sqrt {11} \sqrt {5} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - \frac {1}{65625} \, {\left (157500 \, x^{4} + 171000 \, x^{3} - 83565 \, x^{2} - 123295 \, x + 50858\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/15625*sqrt(11)*sqrt(5)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 1/65625*(157500*x^4 + 1
71000*x^3 - 83565*x^2 - 123295*x + 50858)*sqrt(-2*x + 1)

________________________________________________________________________________________

giac [A]  time = 1.03, size = 106, normalized size = 1.12 \begin {gather*} -\frac {3}{20} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {162}{175} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {3897}{2500} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{15625} \, \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}{3125} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/20*(2*x - 1)^4*sqrt(-2*x + 1) - 162/175*(2*x - 1)^3*sqrt(-2*x + 1) - 3897/2500*(2*x - 1)^2*sqrt(-2*x + 1) +
2/1875*(-2*x + 1)^(3/2) + 11/15625*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) + 22/3125*sqrt(-2*x + 1)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 65, normalized size = 0.68 \begin {gather*} -\frac {22 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{15625}+\frac {2 \left (-2 x +1\right )^{\frac {3}{2}}}{1875}-\frac {3897 \left (-2 x +1\right )^{\frac {5}{2}}}{2500}+\frac {162 \left (-2 x +1\right )^{\frac {7}{2}}}{175}-\frac {3 \left (-2 x +1\right )^{\frac {9}{2}}}{20}+\frac {22 \sqrt {-2 x +1}}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1875*(-2*x+1)^(3/2)-3897/2500*(-2*x+1)^(5/2)+162/175*(-2*x+1)^(7/2)-3/20*(-2*x+1)^(9/2)-22/15625*arctanh(1/1
1*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+22/3125*(-2*x+1)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.24, size = 82, normalized size = 0.86 \begin {gather*} -\frac {3}{20} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {162}{175} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {3897}{2500} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2}{1875} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {11}{15625} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {22}{3125} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/20*(-2*x + 1)^(9/2) + 162/175*(-2*x + 1)^(7/2) - 3897/2500*(-2*x + 1)^(5/2) + 2/1875*(-2*x + 1)^(3/2) + 11/
15625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/3125*sqrt(-2*x + 1)

________________________________________________________________________________________

mupad [B]  time = 1.17, size = 66, normalized size = 0.69 \begin {gather*} \frac {22\,\sqrt {1-2\,x}}{3125}+\frac {2\,{\left (1-2\,x\right )}^{3/2}}{1875}-\frac {3897\,{\left (1-2\,x\right )}^{5/2}}{2500}+\frac {162\,{\left (1-2\,x\right )}^{7/2}}{175}-\frac {3\,{\left (1-2\,x\right )}^{9/2}}{20}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,22{}\mathrm {i}}{15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*22i)/15625 + (22*(1 - 2*x)^(1/2))/3125 + (2*(1 - 2*x)^(3/2))/
1875 - (3897*(1 - 2*x)^(5/2))/2500 + (162*(1 - 2*x)^(7/2))/175 - (3*(1 - 2*x)^(9/2))/20

________________________________________________________________________________________

sympy [A]  time = 55.70, size = 126, normalized size = 1.33 \begin {gather*} - \frac {3 \left (1 - 2 x\right )^{\frac {9}{2}}}{20} + \frac {162 \left (1 - 2 x\right )^{\frac {7}{2}}}{175} - \frac {3897 \left (1 - 2 x\right )^{\frac {5}{2}}}{2500} + \frac {2 \left (1 - 2 x\right )^{\frac {3}{2}}}{1875} + \frac {22 \sqrt {1 - 2 x}}{3125} + \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 )}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(1 - 2*x)**(9/2)/20 + 162*(1 - 2*x)**(7/2)/175 - 3897*(1 - 2*x)**(5/2)/2500 + 2*(1 - 2*x)**(3/2)/1875 + 22*
sqrt(1 - 2*x)/3125 + 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))/3125

________________________________________________________________________________________