3.2404 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=157 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}-\frac{8}{27} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{247}{270} (5 x+3)^{3/2} \sqrt{1-2 x}+\frac{24251 \sqrt{5 x+3} \sqrt{1-2 x}}{3240}+\frac{326717 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{9720 \sqrt{10}}+\frac{805}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(24251*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3240 - (247*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/270 - (8*(1 - 2*x)^(3/2)*(3 + 5
*x)^(3/2))/27 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(3*(2 + 3*x)) + (326717*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(9
720*Sqrt[10]) + (805*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

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Rubi [A]  time = 0.0642384, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {97, 154, 157, 54, 216, 93, 204} \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}-\frac{8}{27} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{247}{270} (5 x+3)^{3/2} \sqrt{1-2 x}+\frac{24251 \sqrt{5 x+3} \sqrt{1-2 x}}{3240}+\frac{326717 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{9720 \sqrt{10}}+\frac{805}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24251*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3240 - (247*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/270 - (8*(1 - 2*x)^(3/2)*(3 + 5
*x)^(3/2))/27 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(3*(2 + 3*x)) + (326717*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(9
720*Sqrt[10]) + (805*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

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 154

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] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{3 (2+3 x)}+\frac{1}{3} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{2+3 x} \, dx\\ &=-\frac{8}{27} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{3 (2+3 x)}+\frac{1}{135} \int \frac{\left (-\frac{915}{2}-3705 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{2+3 x} \, dx\\ &=-\frac{247}{270} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{8}{27} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{3 (2+3 x)}+\frac{\int \frac{\left (19620-\frac{363765 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx}{4050}\\ &=\frac{24251 \sqrt{1-2 x} \sqrt{3+5 x}}{3240}-\frac{247}{270} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{8}{27} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{3 (2+3 x)}-\frac{\int \frac{-\frac{1070085}{2}-\frac{4900755 x}{4}}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{24300}\\ &=\frac{24251 \sqrt{1-2 x} \sqrt{3+5 x}}{3240}-\frac{247}{270} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{8}{27} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{3 (2+3 x)}-\frac{5635}{486} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx+\frac{326717 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{19440}\\ &=\frac{24251 \sqrt{1-2 x} \sqrt{3+5 x}}{3240}-\frac{247}{270} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{8}{27} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{3 (2+3 x)}-\frac{5635}{243} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{326717 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{9720 \sqrt{5}}\\ &=\frac{24251 \sqrt{1-2 x} \sqrt{3+5 x}}{3240}-\frac{247}{270} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{8}{27} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{3 (2+3 x)}+\frac{326717 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{9720 \sqrt{10}}+\frac{805}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.1395, size = 127, normalized size = 0.81 \[ \frac{-30 \sqrt{5 x+3} \left (14400 x^4-34680 x^3+48294 x^2+26159 x-21718\right )-326717 \sqrt{10-20 x} (3 x+2) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+322000 \sqrt{7-14 x} (3 x+2) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{97200 \sqrt{1-2 x} (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*Sqrt[3 + 5*x]*(-21718 + 26159*x + 48294*x^2 - 34680*x^3 + 14400*x^4) - 326717*Sqrt[10 - 20*x]*(2 + 3*x)*A
rcSin[Sqrt[5/11]*Sqrt[1 - 2*x]] + 322000*Sqrt[7 - 14*x]*(2 + 3*x)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])
])/(97200*Sqrt[1 - 2*x]*(2 + 3*x))

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Maple [A]  time = 0.011, size = 180, normalized size = 1.2 \begin{align*}{\frac{1}{388800+583200\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 432000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+980151\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-966000\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-824400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+653434\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -644000\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1036620\,x\sqrt{-10\,{x}^{2}-x+3}+1303080\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/194400*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(432000*x^3*(-10*x^2-x+3)^(1/2)+980151*10^(1/2)*arcsin(20/11*x+1/11)*x-96
6000*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-824400*x^2*(-10*x^2-x+3)^(1/2)+653434*10^(1/
2)*arcsin(20/11*x+1/11)-644000*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1036620*x*(-10*x^2-x
+3)^(1/2)+1303080*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 2.5672, size = 140, normalized size = 0.89 \begin{align*} -\frac{2}{27} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{247}{54} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{326717}{194400} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{805}{486} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{15359}{3240} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{9 \,{\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)/(2+3*x)^2,x, algorithm="maxima")

[Out]

-2/27*(-10*x^2 - x + 3)^(3/2) - 247/54*sqrt(-10*x^2 - x + 3)*x + 326717/194400*sqrt(10)*arcsin(20/11*x + 1/11)
- 805/486*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 15359/3240*sqrt(-10*x^2 - x + 3) - 7/9*
(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 1.55427, size = 416, normalized size = 2.65 \begin{align*} \frac{322000 \, \sqrt{7}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 326717 \, \sqrt{10}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 60 \,{\left (7200 \, x^{3} - 13740 \, x^{2} + 17277 \, x + 21718\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{194400 \,{\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)/(2+3*x)^2,x, algorithm="fricas")

[Out]

1/194400*(322000*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x -
3)) - 326717*sqrt(10)*(3*x + 2)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
+ 60*(7200*x^3 - 13740*x^2 + 17277*x + 21718)*sqrt(5*x + 3)*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)/(2+3*x)**2,x)

[Out]

Timed out

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Giac [B]  time = 3.03012, size = 412, normalized size = 2.62 \begin{align*} -\frac{161}{972} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{5400} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 151 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4817 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{326717}{194400} \, \sqrt{10}{\left (\pi - 2 \, \arctan \left (\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1078 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{81 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\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)/(2+3*x)^2,x, algorithm="giac")

[Out]

-161/972*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^
2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 1/5400*(4*(8*sqrt(5)*(5*x + 3) - 151*sqrt(5))*(5*x +
3) + 4817*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 326717/194400*sqrt(10)*(pi - 2*arctan(1/4*sqrt(5*x + 3)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 1078/81*sqrt(10)
*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(
((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2
+ 280)