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

Optimal. Leaf size=109 \[ -\frac {129 (1-2 x)^{7/2}}{6050 (5 x+3)}-\frac {(1-2 x)^{7/2}}{550 (5 x+3)^2}+\frac {1533 (1-2 x)^{5/2}}{75625}+\frac {511 (1-2 x)^{3/2}}{6875}+\frac {1533 \sqrt {1-2 x}}{3125}-\frac {1533 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {89, 78, 50, 63, 206} \begin {gather*} -\frac {129 (1-2 x)^{7/2}}{6050 (5 x+3)}-\frac {(1-2 x)^{7/2}}{550 (5 x+3)^2}+\frac {1533 (1-2 x)^{5/2}}{75625}+\frac {511 (1-2 x)^{3/2}}{6875}+\frac {1533 \sqrt {1-2 x}}{3125}-\frac {1533 \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)^(5/2)*(2 + 3*x)^2)/(3 + 5*x)^3,x]

[Out]

(1533*Sqrt[1 - 2*x])/3125 + (511*(1 - 2*x)^(3/2))/6875 + (1533*(1 - 2*x)^(5/2))/75625 - (1 - 2*x)^(7/2)/(550*(
3 + 5*x)^2) - (129*(1 - 2*x)^(7/2))/(6050*(3 + 5*x)) - (1533*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/312
5

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 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)^{5/2} (2+3 x)^2}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{7/2}}{550 (3+5 x)^2}+\frac {1}{550} \int \frac {(1-2 x)^{5/2} (723+990 x)}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{550 (3+5 x)^2}-\frac {129 (1-2 x)^{7/2}}{6050 (3+5 x)}+\frac {1533 \int \frac {(1-2 x)^{5/2}}{3+5 x} \, dx}{6050}\\ &=\frac {1533 (1-2 x)^{5/2}}{75625}-\frac {(1-2 x)^{7/2}}{550 (3+5 x)^2}-\frac {129 (1-2 x)^{7/2}}{6050 (3+5 x)}+\frac {1533 \int \frac {(1-2 x)^{3/2}}{3+5 x} \, dx}{2750}\\ &=\frac {511 (1-2 x)^{3/2}}{6875}+\frac {1533 (1-2 x)^{5/2}}{75625}-\frac {(1-2 x)^{7/2}}{550 (3+5 x)^2}-\frac {129 (1-2 x)^{7/2}}{6050 (3+5 x)}+\frac {1533 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{1250}\\ &=\frac {1533 \sqrt {1-2 x}}{3125}+\frac {511 (1-2 x)^{3/2}}{6875}+\frac {1533 (1-2 x)^{5/2}}{75625}-\frac {(1-2 x)^{7/2}}{550 (3+5 x)^2}-\frac {129 (1-2 x)^{7/2}}{6050 (3+5 x)}+\frac {16863 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{6250}\\ &=\frac {1533 \sqrt {1-2 x}}{3125}+\frac {511 (1-2 x)^{3/2}}{6875}+\frac {1533 (1-2 x)^{5/2}}{75625}-\frac {(1-2 x)^{7/2}}{550 (3+5 x)^2}-\frac {129 (1-2 x)^{7/2}}{6050 (3+5 x)}-\frac {16863 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{6250}\\ &=\frac {1533 \sqrt {1-2 x}}{3125}+\frac {511 (1-2 x)^{3/2}}{6875}+\frac {1533 (1-2 x)^{5/2}}{75625}-\frac {(1-2 x)^{7/2}}{550 (3+5 x)^2}-\frac {129 (1-2 x)^{7/2}}{6050 (3+5 x)}-\frac {1533 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{3125}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 68, normalized size = 0.62 \begin {gather*} \frac {\frac {5 \sqrt {1-2 x} \left (18000 x^4-25400 x^3+51980 x^2+98595 x+32504\right )}{(5 x+3)^2}-3066 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{31250} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((5*Sqrt[1 - 2*x]*(32504 + 98595*x + 51980*x^2 - 25400*x^3 + 18000*x^4))/(3 + 5*x)^2 - 3066*Sqrt[55]*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]])/31250

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.18, size = 90, normalized size = 0.83 \begin {gather*} \frac {\left (2250 (1-2 x)^4-2650 (1-2 x)^3+20440 (1-2 x)^2-140525 (1-2 x)+185493\right ) \sqrt {1-2 x}}{3125 (5 (1-2 x)-11)^2}-\frac {1533 \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)^(5/2)*(2 + 3*x)^2)/(3 + 5*x)^3,x]

[Out]

((185493 - 140525*(1 - 2*x) + 20440*(1 - 2*x)^2 - 2650*(1 - 2*x)^3 + 2250*(1 - 2*x)^4)*Sqrt[1 - 2*x])/(3125*(-
11 + 5*(1 - 2*x))^2) - (1533*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

________________________________________________________________________________________

fricas [A]  time = 1.35, size = 90, normalized size = 0.83 \begin {gather*} \frac {1533 \, \sqrt {11} \sqrt {5} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (18000 \, x^{4} - 25400 \, x^{3} + 51980 \, x^{2} + 98595 \, x + 32504\right )} \sqrt {-2 \, x + 1}}{31250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/31250*(1533*sqrt(11)*sqrt(5)*(25*x^2 + 30*x + 9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3))
+ 5*(18000*x^4 - 25400*x^3 + 51980*x^2 + 98595*x + 32504)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

________________________________________________________________________________________

giac [A]  time = 0.92, size = 102, normalized size = 0.94 \begin {gather*} \frac {18}{625} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {58}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1533}{31250} \, \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 {1658}{3125} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (123 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 275 \, \sqrt {-2 \, x + 1}\right )}}{2500 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18/625*(2*x - 1)^2*sqrt(-2*x + 1) + 58/625*(-2*x + 1)^(3/2) + 1533/31250*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10
*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1658/3125*sqrt(-2*x + 1) + 11/2500*(123*(-2*x + 1)^(3/2) - 2
75*sqrt(-2*x + 1))/(5*x + 3)^2

________________________________________________________________________________________

maple [A]  time = 0.01, size = 75, normalized size = 0.69 \begin {gather*} -\frac {1533 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{15625}+\frac {18 \left (-2 x +1\right )^{\frac {5}{2}}}{625}+\frac {58 \left (-2 x +1\right )^{\frac {3}{2}}}{625}+\frac {1658 \sqrt {-2 x +1}}{3125}+\frac {\frac {1353 \left (-2 x +1\right )^{\frac {3}{2}}}{625}-\frac {121 \sqrt {-2 x +1}}{25}}{\left (-10 x -6\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18/625*(-2*x+1)^(5/2)+58/625*(-2*x+1)^(3/2)+1658/3125*(-2*x+1)^(1/2)+22/125*(123/10*(-2*x+1)^(3/2)-55/2*(-2*x+
1)^(1/2))/(-10*x-6)^2-1533/15625*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.22, size = 101, normalized size = 0.93 \begin {gather*} \frac {18}{625} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {58}{625} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1533}{31250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1658}{3125} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (123 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 275 \, \sqrt {-2 \, x + 1}\right )}}{625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18/625*(-2*x + 1)^(5/2) + 58/625*(-2*x + 1)^(3/2) + 1533/31250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sq
rt(55) + 5*sqrt(-2*x + 1))) + 1658/3125*sqrt(-2*x + 1) + 11/625*(123*(-2*x + 1)^(3/2) - 275*sqrt(-2*x + 1))/(2
5*(2*x - 1)^2 + 220*x + 11)

________________________________________________________________________________________

mupad [B]  time = 1.17, size = 83, normalized size = 0.76 \begin {gather*} \frac {1658\,\sqrt {1-2\,x}}{3125}+\frac {58\,{\left (1-2\,x\right )}^{3/2}}{625}+\frac {18\,{\left (1-2\,x\right )}^{5/2}}{625}-\frac {\frac {121\,\sqrt {1-2\,x}}{625}-\frac {1353\,{\left (1-2\,x\right )}^{3/2}}{15625}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,1533{}\mathrm {i}}{15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*1533i)/15625 + (1658*(1 - 2*x)^(1/2))/3125 + (58*(1 - 2*x)^(3
/2))/625 + (18*(1 - 2*x)^(5/2))/625 - ((121*(1 - 2*x)^(1/2))/625 - (1353*(1 - 2*x)^(3/2))/15625)/((44*x)/5 + (
2*x - 1)^2 + 11/25)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________