3.1961 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=121 \[ -\frac{(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}+\frac{55}{81} (1-2 x)^{5/2} (5 x+3)^2+\frac{220}{729} (1-2 x)^{3/2}-\frac{22}{567} (1-2 x)^{5/2} (100 x+69)+\frac{1540}{729} \sqrt{1-2 x}-\frac{1540}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(1540*Sqrt[1 - 2*x])/729 + (220*(1 - 2*x)^(3/2))/729 + (55*(1 - 2*x)^(5/2)*(3 + 5*x)^2)/81 - ((1 - 2*x)^(5/2)*
(3 + 5*x)^3)/(3*(2 + 3*x)) - (22*(1 - 2*x)^(5/2)*(69 + 100*x))/567 - (1540*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/729

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Rubi [A]  time = 0.0425597, antiderivative size = 121, 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, 153, 147, 50, 63, 206} \[ -\frac{(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}+\frac{55}{81} (1-2 x)^{5/2} (5 x+3)^2+\frac{220}{729} (1-2 x)^{3/2}-\frac{22}{567} (1-2 x)^{5/2} (100 x+69)+\frac{1540}{729} \sqrt{1-2 x}-\frac{1540}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1540*Sqrt[1 - 2*x])/729 + (220*(1 - 2*x)^(3/2))/729 + (55*(1 - 2*x)^(5/2)*(3 + 5*x)^2)/81 - ((1 - 2*x)^(5/2)*
(3 + 5*x)^3)/(3*(2 + 3*x)) - (22*(1 - 2*x)^(5/2)*(69 + 100*x))/567 - (1540*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/729

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 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 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} (3+5 x)^3}{(2+3 x)^2} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}+\frac{1}{3} \int -\frac{55 (1-2 x)^{3/2} x (3+5 x)^2}{2+3 x} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac{55}{3} \int \frac{(1-2 x)^{3/2} x (3+5 x)^2}{2+3 x} \, dx\\ &=\frac{55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}+\frac{55}{81} \int \frac{(1-2 x)^{3/2} (3+5 x) (10+24 x)}{2+3 x} \, dx\\ &=\frac{55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac{22}{567} (1-2 x)^{5/2} (69+100 x)+\frac{110}{81} \int \frac{(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac{220}{729} (1-2 x)^{3/2}+\frac{55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac{22}{567} (1-2 x)^{5/2} (69+100 x)+\frac{770}{243} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=\frac{1540}{729} \sqrt{1-2 x}+\frac{220}{729} (1-2 x)^{3/2}+\frac{55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac{22}{567} (1-2 x)^{5/2} (69+100 x)+\frac{5390}{729} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{1540}{729} \sqrt{1-2 x}+\frac{220}{729} (1-2 x)^{3/2}+\frac{55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac{22}{567} (1-2 x)^{5/2} (69+100 x)-\frac{5390}{729} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{1540}{729} \sqrt{1-2 x}+\frac{220}{729} (1-2 x)^{3/2}+\frac{55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac{(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac{22}{567} (1-2 x)^{5/2} (69+100 x)-\frac{1540}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0426771, size = 73, normalized size = 0.6 \[ \frac{\frac{3 \sqrt{1-2 x} \left (189000 x^5-17100 x^4-159714 x^3+25275 x^2+65558 x+13759\right )}{3 x+2}-10780 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{15309} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*Sqrt[1 - 2*x]*(13759 + 65558*x + 25275*x^2 - 159714*x^3 - 17100*x^4 + 189000*x^5))/(2 + 3*x) - 10780*Sqrt[
21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/15309

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Maple [A]  time = 0.01, size = 81, normalized size = 0.7 \begin{align*}{\frac{125}{162} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{725}{378} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{2}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{214}{729} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1526}{729}\sqrt{1-2\,x}}-{\frac{98}{2187}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{1540\,\sqrt{21}}{2187}{\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)^2,x)

[Out]

125/162*(1-2*x)^(9/2)-725/378*(1-2*x)^(7/2)+2/27*(1-2*x)^(5/2)+214/729*(1-2*x)^(3/2)+1526/729*(1-2*x)^(1/2)-98
/2187*(1-2*x)^(1/2)/(-2*x-4/3)-1540/2187*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 2.02131, size = 132, normalized size = 1.09 \begin{align*} \frac{125}{162} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{725}{378} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{27} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{214}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{770}{2187} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1526}{729} \, \sqrt{-2 \, x + 1} + \frac{49 \, \sqrt{-2 \, x + 1}}{729 \,{\left (3 \, x + 2\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)^2,x, algorithm="maxima")

[Out]

125/162*(-2*x + 1)^(9/2) - 725/378*(-2*x + 1)^(7/2) + 2/27*(-2*x + 1)^(5/2) + 214/729*(-2*x + 1)^(3/2) + 770/2
187*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1526/729*sqrt(-2*x + 1) + 49/
729*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 1.34584, size = 266, normalized size = 2.2 \begin{align*} \frac{5390 \, \sqrt{7} \sqrt{3}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 3 \,{\left (189000 \, x^{5} - 17100 \, x^{4} - 159714 \, x^{3} + 25275 \, x^{2} + 65558 \, x + 13759\right )} \sqrt{-2 \, x + 1}}{15309 \,{\left (3 \, x + 2\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)^2,x, algorithm="fricas")

[Out]

1/15309*(5390*sqrt(7)*sqrt(3)*(3*x + 2)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) + 3*(189000*
x^5 - 17100*x^4 - 159714*x^3 + 25275*x^2 + 65558*x + 13759)*sqrt(-2*x + 1))/(3*x + 2)

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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)**2,x)

[Out]

Timed out

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Giac [A]  time = 2.6945, size = 165, normalized size = 1.36 \begin{align*} \frac{125}{162} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{725}{378} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{27} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{214}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{770}{2187} \, \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{1526}{729} \, \sqrt{-2 \, x + 1} + \frac{49 \, \sqrt{-2 \, x + 1}}{729 \,{\left (3 \, x + 2\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)^2,x, algorithm="giac")

[Out]

125/162*(2*x - 1)^4*sqrt(-2*x + 1) + 725/378*(2*x - 1)^3*sqrt(-2*x + 1) + 2/27*(2*x - 1)^2*sqrt(-2*x + 1) + 21
4/729*(-2*x + 1)^(3/2) + 770/2187*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 1526/729*sqrt(-2*x + 1) + 49/729*sqrt(-2*x + 1)/(3*x + 2)