3.1920 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=120 \[ -\frac{48 \sqrt{1-2 x} (3 x+2)^3}{25 (5 x+3)}-\frac{(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}+\frac{693}{625} \sqrt{1-2 x} (3 x+2)^2+\frac{63 \sqrt{1-2 x} (125 x+92)}{6250}-\frac{5943 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

[Out]

(693*Sqrt[1 - 2*x]*(2 + 3*x)^2)/625 - ((1 - 2*x)^(3/2)*(2 + 3*x)^3)/(10*(3 + 5*x)^2) - (48*Sqrt[1 - 2*x]*(2 +
3*x)^3)/(25*(3 + 5*x)) + (63*Sqrt[1 - 2*x]*(92 + 125*x))/6250 - (5943*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125
*Sqrt[55])

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Rubi [A]  time = 0.0365559, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 149, 153, 147, 63, 206} \[ -\frac{48 \sqrt{1-2 x} (3 x+2)^3}{25 (5 x+3)}-\frac{(1-2 x)^{3/2} (3 x+2)^3}{10 (5 x+3)^2}+\frac{693}{625} \sqrt{1-2 x} (3 x+2)^2+\frac{63 \sqrt{1-2 x} (125 x+92)}{6250}-\frac{5943 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(693*Sqrt[1 - 2*x]*(2 + 3*x)^2)/625 - ((1 - 2*x)^(3/2)*(2 + 3*x)^3)/(10*(3 + 5*x)^2) - (48*Sqrt[1 - 2*x]*(2 +
3*x)^3)/(25*(3 + 5*x)) + (63*Sqrt[1 - 2*x]*(92 + 125*x))/6250 - (5943*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125
*Sqrt[55])

Rule 97

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

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

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)^3}{(3+5 x)^3} \, dx &=-\frac{(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}+\frac{1}{10} \int \frac{(3-27 x) \sqrt{1-2 x} (2+3 x)^2}{(3+5 x)^2} \, dx\\ &=-\frac{(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac{48 \sqrt{1-2 x} (2+3 x)^3}{25 (3+5 x)}+\frac{1}{50} \int \frac{(357-1386 x) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{693}{625} \sqrt{1-2 x} (2+3 x)^2-\frac{(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac{48 \sqrt{1-2 x} (2+3 x)^3}{25 (3+5 x)}-\frac{\int \frac{(2+3 x) (-1218+7875 x)}{\sqrt{1-2 x} (3+5 x)} \, dx}{1250}\\ &=\frac{693}{625} \sqrt{1-2 x} (2+3 x)^2-\frac{(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac{48 \sqrt{1-2 x} (2+3 x)^3}{25 (3+5 x)}+\frac{63 \sqrt{1-2 x} (92+125 x)}{6250}+\frac{5943 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{6250}\\ &=\frac{693}{625} \sqrt{1-2 x} (2+3 x)^2-\frac{(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac{48 \sqrt{1-2 x} (2+3 x)^3}{25 (3+5 x)}+\frac{63 \sqrt{1-2 x} (92+125 x)}{6250}-\frac{5943 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{6250}\\ &=\frac{693}{625} \sqrt{1-2 x} (2+3 x)^2-\frac{(1-2 x)^{3/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac{48 \sqrt{1-2 x} (2+3 x)^3}{25 (3+5 x)}+\frac{63 \sqrt{1-2 x} (92+125 x)}{6250}-\frac{5943 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}}\\ \end{align*}

Mathematica [A]  time = 0.051221, size = 68, normalized size = 0.57 \[ \frac{\sqrt{1-2 x} \left (-27000 x^4-14400 x^3+37530 x^2+36295 x+8644\right )}{6250 (5 x+3)^2}-\frac{5943 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(8644 + 36295*x + 37530*x^2 - 14400*x^3 - 27000*x^4))/(6250*(3 + 5*x)^2) - (5943*ArcTanh[Sqrt[5
/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

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Maple [A]  time = 0.01, size = 75, normalized size = 0.6 \begin{align*} -{\frac{27}{625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{18}{625} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{558}{3125}\sqrt{1-2\,x}}+{\frac{2}{125\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{193}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{429}{10}\sqrt{1-2\,x}} \right ) }-{\frac{5943\,\sqrt{55}}{171875}{\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)^3/(3+5*x)^3,x)

[Out]

-27/625*(1-2*x)^(5/2)+18/625*(1-2*x)^(3/2)+558/3125*(1-2*x)^(1/2)+2/125*(193/10*(1-2*x)^(3/2)-429/10*(1-2*x)^(
1/2))/(-10*x-6)^2-5943/171875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.5059, size = 136, normalized size = 1.13 \begin{align*} -\frac{27}{625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{18}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{5943}{343750} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{558}{3125} \, \sqrt{-2 \, x + 1} + \frac{193 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 429 \, \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)^3/(3+5*x)^3,x, algorithm="maxima")

[Out]

-27/625*(-2*x + 1)^(5/2) + 18/625*(-2*x + 1)^(3/2) + 5943/343750*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(
sqrt(55) + 5*sqrt(-2*x + 1))) + 558/3125*sqrt(-2*x + 1) + 1/625*(193*(-2*x + 1)^(3/2) - 429*sqrt(-2*x + 1))/(2
5*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 1.65334, size = 257, normalized size = 2.14 \begin{align*} \frac{5943 \, \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 (27000 \, x^{4} + 14400 \, x^{3} - 37530 \, x^{2} - 36295 \, x - 8644\right )} \sqrt{-2 \, x + 1}}{343750 \,{\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)^3/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/343750*(5943*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(27000*x^4
+ 14400*x^3 - 37530*x^2 - 36295*x - 8644)*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)**3/(3+5*x)**3,x)

[Out]

Timed out

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Giac [A]  time = 2.73903, size = 138, normalized size = 1.15 \begin{align*} -\frac{27}{625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{18}{625} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{5943}{343750} \, \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{558}{3125} \, \sqrt{-2 \, x + 1} + \frac{193 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 429 \, \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)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

-27/625*(2*x - 1)^2*sqrt(-2*x + 1) + 18/625*(-2*x + 1)^(3/2) + 5943/343750*sqrt(55)*log(1/2*abs(-2*sqrt(55) +
10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 558/3125*sqrt(-2*x + 1) + 1/2500*(193*(-2*x + 1)^(3/2) - 4
29*sqrt(-2*x + 1))/(5*x + 3)^2