3.1921 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=94 \[ -\frac{131 (1-2 x)^{5/2}}{6050 (5 x+3)}-\frac{(1-2 x)^{5/2}}{550 (5 x+3)^2}+\frac{119 (1-2 x)^{3/2}}{3025}+\frac{357 \sqrt{1-2 x}}{1375}-\frac{357 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}} \]

[Out]

(357*Sqrt[1 - 2*x])/1375 + (119*(1 - 2*x)^(3/2))/3025 - (1 - 2*x)^(5/2)/(550*(3 + 5*x)^2) - (131*(1 - 2*x)^(5/
2))/(6050*(3 + 5*x)) - (357*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(125*Sqrt[55])

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Rubi [A]  time = 0.0260342, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{131 (1-2 x)^{5/2}}{6050 (5 x+3)}-\frac{(1-2 x)^{5/2}}{550 (5 x+3)^2}+\frac{119 (1-2 x)^{3/2}}{3025}+\frac{357 \sqrt{1-2 x}}{1375}-\frac{357 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(357*Sqrt[1 - 2*x])/1375 + (119*(1 - 2*x)^(3/2))/3025 - (1 - 2*x)^(5/2)/(550*(3 + 5*x)^2) - (131*(1 - 2*x)^(5/
2))/(6050*(3 + 5*x)) - (357*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(125*Sqrt[55])

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 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 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)^2}{(3+5 x)^3} \, dx &=-\frac{(1-2 x)^{5/2}}{550 (3+5 x)^2}+\frac{1}{550} \int \frac{(1-2 x)^{3/2} (725+990 x)}{(3+5 x)^2} \, dx\\ &=-\frac{(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac{131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac{357 \int \frac{(1-2 x)^{3/2}}{3+5 x} \, dx}{1210}\\ &=\frac{119 (1-2 x)^{3/2}}{3025}-\frac{(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac{131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac{357}{550} \int \frac{\sqrt{1-2 x}}{3+5 x} \, dx\\ &=\frac{357 \sqrt{1-2 x}}{1375}+\frac{119 (1-2 x)^{3/2}}{3025}-\frac{(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac{131 (1-2 x)^{5/2}}{6050 (3+5 x)}+\frac{357}{250} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{357 \sqrt{1-2 x}}{1375}+\frac{119 (1-2 x)^{3/2}}{3025}-\frac{(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac{131 (1-2 x)^{5/2}}{6050 (3+5 x)}-\frac{357}{250} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{357 \sqrt{1-2 x}}{1375}+\frac{119 (1-2 x)^{3/2}}{3025}-\frac{(1-2 x)^{5/2}}{550 (3+5 x)^2}-\frac{131 (1-2 x)^{5/2}}{6050 (3+5 x)}-\frac{357 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}}\\ \end{align*}

Mathematica [A]  time = 0.0448974, size = 63, normalized size = 0.67 \[ \frac{\sqrt{1-2 x} \left (-600 x^3+1320 x^2+2105 x+656\right )}{250 (5 x+3)^2}-\frac{357 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{125 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(656 + 2105*x + 1320*x^2 - 600*x^3))/(250*(3 + 5*x)^2) - (357*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]
)/(125*Sqrt[55])

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Maple [A]  time = 0.01, size = 66, normalized size = 0.7 \begin{align*}{\frac{6}{125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{174}{625}\sqrt{1-2\,x}}+{\frac{2}{25\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{127}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1419}{50}\sqrt{1-2\,x}} \right ) }-{\frac{357\,\sqrt{55}}{6875}{\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)^(3/2)*(2+3*x)^2/(3+5*x)^3,x)

[Out]

6/125*(1-2*x)^(3/2)+174/625*(1-2*x)^(1/2)+2/25*(127/10*(1-2*x)^(3/2)-1419/50*(1-2*x)^(1/2))/(-10*x-6)^2-357/68
75*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 4.08003, size = 124, normalized size = 1.32 \begin{align*} \frac{6}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{357}{13750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{174}{625} \, \sqrt{-2 \, x + 1} + \frac{635 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1419 \, \sqrt{-2 \, x + 1}}{625 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/125*(-2*x + 1)^(3/2) + 357/13750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 174/625*sqrt(-2*x + 1) + 1/625*(635*(-2*x + 1)^(3/2) - 1419*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 1.52152, size = 231, normalized size = 2.46 \begin{align*} \frac{357 \, \sqrt{55}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (600 \, x^{3} - 1320 \, x^{2} - 2105 \, x - 656\right )} \sqrt{-2 \, x + 1}}{13750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13750*(357*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(600*x^3 - 1
320*x^2 - 2105*x - 656)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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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)**(3/2)*(2+3*x)**2/(3+5*x)**3,x)

[Out]

Timed out

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Giac [A]  time = 2.07146, size = 116, normalized size = 1.23 \begin{align*} \frac{6}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{357}{13750} \, \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{174}{625} \, \sqrt{-2 \, x + 1} + \frac{635 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1419 \, \sqrt{-2 \, x + 1}}{2500 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/125*(-2*x + 1)^(3/2) + 357/13750*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 174/625*sqrt(-2*x + 1) + 1/2500*(635*(-2*x + 1)^(3/2) - 1419*sqrt(-2*x + 1))/(5*x + 3)^2