3.1982 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^2}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=102 \[ -\frac{(1-2 x)^{7/2}}{275 (5 x+3)}-\frac{9}{175} (1-2 x)^{7/2}+\frac{122 (1-2 x)^{5/2}}{6875}+\frac{122 (1-2 x)^{3/2}}{1875}+\frac{1342 \sqrt{1-2 x}}{3125}-\frac{1342 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

[Out]

(1342*Sqrt[1 - 2*x])/3125 + (122*(1 - 2*x)^(3/2))/1875 + (122*(1 - 2*x)^(5/2))/6875 - (9*(1 - 2*x)^(7/2))/175
- (1 - 2*x)^(7/2)/(275*(3 + 5*x)) - (1342*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

________________________________________________________________________________________

Rubi [A]  time = 0.0294716, antiderivative size = 102, normalized size of antiderivative = 1., 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, 80, 50, 63, 206} \[ -\frac{(1-2 x)^{7/2}}{275 (5 x+3)}-\frac{9}{175} (1-2 x)^{7/2}+\frac{122 (1-2 x)^{5/2}}{6875}+\frac{122 (1-2 x)^{3/2}}{1875}+\frac{1342 \sqrt{1-2 x}}{3125}-\frac{1342 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1342*Sqrt[1 - 2*x])/3125 + (122*(1 - 2*x)^(3/2))/1875 + (122*(1 - 2*x)^(5/2))/6875 - (9*(1 - 2*x)^(7/2))/175
- (1 - 2*x)^(7/2)/(275*(3 + 5*x)) - (1342*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/3125

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 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)^2}{(3+5 x)^2} \, dx &=-\frac{(1-2 x)^{7/2}}{275 (3+5 x)}+\frac{1}{275} \int \frac{(1-2 x)^{5/2} (358+495 x)}{3+5 x} \, dx\\ &=-\frac{9}{175} (1-2 x)^{7/2}-\frac{(1-2 x)^{7/2}}{275 (3+5 x)}+\frac{61}{275} \int \frac{(1-2 x)^{5/2}}{3+5 x} \, dx\\ &=\frac{122 (1-2 x)^{5/2}}{6875}-\frac{9}{175} (1-2 x)^{7/2}-\frac{(1-2 x)^{7/2}}{275 (3+5 x)}+\frac{61}{125} \int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx\\ &=\frac{122 (1-2 x)^{3/2}}{1875}+\frac{122 (1-2 x)^{5/2}}{6875}-\frac{9}{175} (1-2 x)^{7/2}-\frac{(1-2 x)^{7/2}}{275 (3+5 x)}+\frac{671}{625} \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx\\ &=\frac{1342 \sqrt{1-2 x}}{3125}+\frac{122 (1-2 x)^{3/2}}{1875}+\frac{122 (1-2 x)^{5/2}}{6875}-\frac{9}{175} (1-2 x)^{7/2}-\frac{(1-2 x)^{7/2}}{275 (3+5 x)}+\frac{7381 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{3125}\\ &=\frac{1342 \sqrt{1-2 x}}{3125}+\frac{122 (1-2 x)^{3/2}}{1875}+\frac{122 (1-2 x)^{5/2}}{6875}-\frac{9}{175} (1-2 x)^{7/2}-\frac{(1-2 x)^{7/2}}{275 (3+5 x)}-\frac{7381 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{3125}\\ &=\frac{1342 \sqrt{1-2 x}}{3125}+\frac{122 (1-2 x)^{3/2}}{1875}+\frac{122 (1-2 x)^{5/2}}{6875}-\frac{9}{175} (1-2 x)^{7/2}-\frac{(1-2 x)^{7/2}}{275 (3+5 x)}-\frac{1342 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0395907, size = 68, normalized size = 0.67 \[ \frac{\frac{5 \sqrt{1-2 x} \left (135000 x^4-96300 x^3-75130 x^2+173795 x+90486\right )}{5 x+3}-28182 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{328125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((5*Sqrt[1 - 2*x]*(90486 + 173795*x - 75130*x^2 - 96300*x^3 + 135000*x^4))/(3 + 5*x) - 28182*Sqrt[55]*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]])/328125

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 72, normalized size = 0.7 \begin{align*} -{\frac{9}{175} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{12}{625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{128}{1875} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1364}{3125}\sqrt{1-2\,x}}+{\frac{242}{15625}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}-{\frac{1342\,\sqrt{55}}{15625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/175*(1-2*x)^(7/2)+12/625*(1-2*x)^(5/2)+128/1875*(1-2*x)^(3/2)+1364/3125*(1-2*x)^(1/2)+242/15625*(1-2*x)^(1/
2)/(-2*x-6/5)-1342/15625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 2.267, size = 120, normalized size = 1.18 \begin{align*} -\frac{9}{175} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{12}{625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{128}{1875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{671}{15625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1364}{3125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/175*(-2*x + 1)^(7/2) + 12/625*(-2*x + 1)^(5/2) + 128/1875*(-2*x + 1)^(3/2) + 671/15625*sqrt(55)*log(-(sqrt(
55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1364/3125*sqrt(-2*x + 1) - 121/3125*sqrt(-2*x + 1)/(5
*x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.37158, size = 255, normalized size = 2.5 \begin{align*} \frac{14091 \, \sqrt{11} \sqrt{5}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \,{\left (135000 \, x^{4} - 96300 \, x^{3} - 75130 \, x^{2} + 173795 \, x + 90486\right )} \sqrt{-2 \, x + 1}}{328125 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/328125*(14091*sqrt(11)*sqrt(5)*(5*x + 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 5*(135
000*x^4 - 96300*x^3 - 75130*x^2 + 173795*x + 90486)*sqrt(-2*x + 1))/(5*x + 3)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 2.46658, size = 143, normalized size = 1.4 \begin{align*} \frac{9}{175} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{12}{625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{128}{1875} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{671}{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{1364}{3125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{3125 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/175*(2*x - 1)^3*sqrt(-2*x + 1) + 12/625*(2*x - 1)^2*sqrt(-2*x + 1) + 128/1875*(-2*x + 1)^(3/2) + 671/15625*s
qrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1364/3125*sqrt(-2*x + 1)
- 121/3125*sqrt(-2*x + 1)/(5*x + 3)