3.1963 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=147 \[ -\frac{275 \sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^2}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac{1441}{27} \sqrt{1-2 x} (5 x+3)^2-\frac{22}{243} \sqrt{1-2 x} (1885 x+578)-\frac{41360 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{243 \sqrt{21}} \]

[Out]

(1441*Sqrt[1 - 2*x]*(3 + 5*x)^2)/27 - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(9*(2 + 3*x)^3) + (55*(1 - 2*x)^(3/2)*(3 +
5*x)^3)/(27*(2 + 3*x)^2) - (275*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(9*(2 + 3*x)) - (22*Sqrt[1 - 2*x]*(578 + 1885*x))/
243 - (41360*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(243*Sqrt[21])

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Rubi [A]  time = 0.0585161, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {97, 12, 149, 153, 147, 63, 206} \[ -\frac{275 \sqrt{1-2 x} (5 x+3)^3}{9 (3 x+2)}+\frac{55 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^2}-\frac{(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac{1441}{27} \sqrt{1-2 x} (5 x+3)^2-\frac{22}{243} \sqrt{1-2 x} (1885 x+578)-\frac{41360 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{243 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

(1441*Sqrt[1 - 2*x]*(3 + 5*x)^2)/27 - ((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(9*(2 + 3*x)^3) + (55*(1 - 2*x)^(3/2)*(3 +
5*x)^3)/(27*(2 + 3*x)^2) - (275*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(9*(2 + 3*x)) - (22*Sqrt[1 - 2*x]*(578 + 1885*x))/
243 - (41360*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(243*Sqrt[21])

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{1}{9} \int -\frac{55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}-\frac{55}{9} \int \frac{(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}+\frac{55}{54} \int \frac{\sqrt{1-2 x} (3+5 x)^2 (6+54 x)}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac{275 \sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)}-\frac{55}{162} \int \frac{(3+5 x)^2 (-684+2358 x)}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{1441}{27} \sqrt{1-2 x} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac{275 \sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)}+\frac{11}{486} \int \frac{(3+5 x) (-2232+13572 x)}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{1441}{27} \sqrt{1-2 x} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac{275 \sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)}-\frac{22}{243} \sqrt{1-2 x} (578+1885 x)+\frac{20680}{243} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{1441}{27} \sqrt{1-2 x} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac{275 \sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)}-\frac{22}{243} \sqrt{1-2 x} (578+1885 x)-\frac{20680}{243} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{1441}{27} \sqrt{1-2 x} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac{55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac{275 \sqrt{1-2 x} (3+5 x)^3}{9 (2+3 x)}-\frac{22}{243} \sqrt{1-2 x} (578+1885 x)-\frac{41360 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{243 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0190972, size = 59, normalized size = 0.4 \[ \frac{(1-2 x)^{7/2} \left (49632 (3 x+2)^3 \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{3}{7}-\frac{6 x}{7}\right )-343 \left (11025 x^2+14858 x+5003\right )\right )}{453789 (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

((1 - 2*x)^(7/2)*(-343*(5003 + 14858*x + 11025*x^2) + 49632*(2 + 3*x)^3*Hypergeometric2F1[2, 7/2, 9/2, 3/7 - (
6*x)/7]))/(453789*(2 + 3*x)^3)

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Maple [A]  time = 0.01, size = 84, normalized size = 0.6 \begin{align*}{\frac{50}{81} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{2050}{729} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{16570}{729}\sqrt{1-2\,x}}+{\frac{2}{27\, \left ( -6\,x-4 \right ) ^{3}} \left ( -{\frac{4153}{3} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{172130}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{198205}{27}\sqrt{1-2\,x}} \right ) }-{\frac{41360\,\sqrt{21}}{5103}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^4,x)

[Out]

50/81*(1-2*x)^(5/2)+2050/729*(1-2*x)^(3/2)+16570/729*(1-2*x)^(1/2)+2/27*(-4153/3*(1-2*x)^(5/2)+172130/27*(1-2*
x)^(3/2)-198205/27*(1-2*x)^(1/2))/(-6*x-4)^3-41360/5103*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 3.22556, size = 161, normalized size = 1.1 \begin{align*} \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2050}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{20680}{5103} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{16570}{729} \, \sqrt{-2 \, x + 1} + \frac{2 \,{\left (37377 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 172130 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 198205 \, \sqrt{-2 \, x + 1}\right )}}{729 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^4,x, algorithm="maxima")

[Out]

50/81*(-2*x + 1)^(5/2) + 2050/729*(-2*x + 1)^(3/2) + 20680/5103*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) + 16570/729*sqrt(-2*x + 1) + 2/729*(37377*(-2*x + 1)^(5/2) - 172130*(-2*x + 1)^(3
/2) + 198205*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 1.58521, size = 300, normalized size = 2.04 \begin{align*} \frac{20680 \, \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (16200 \, x^{5} - 20700 \, x^{4} + 87030 \, x^{3} + 289719 \, x^{2} + 229336 \, x + 56141\right )} \sqrt{-2 \, x + 1}}{5103 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/5103*(20680*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(1
6200*x^5 - 20700*x^4 + 87030*x^3 + 289719*x^2 + 229336*x + 56141)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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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)**(5/2)*(3+5*x)**3/(2+3*x)**4,x)

[Out]

Timed out

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Giac [A]  time = 1.92786, size = 159, normalized size = 1.08 \begin{align*} \frac{50}{81} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2050}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{20680}{5103} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16570}{729} \, \sqrt{-2 \, x + 1} + \frac{37377 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 172130 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 198205 \, \sqrt{-2 \, x + 1}}{2916 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^4,x, algorithm="giac")

[Out]

50/81*(2*x - 1)^2*sqrt(-2*x + 1) + 2050/729*(-2*x + 1)^(3/2) + 20680/5103*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6
*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16570/729*sqrt(-2*x + 1) + 1/2916*(37377*(2*x - 1)^2*sqrt(-2
*x + 1) - 172130*(-2*x + 1)^(3/2) + 198205*sqrt(-2*x + 1))/(3*x + 2)^3