3.2416 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=179 \[ -\frac{5}{18} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{247}{324} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac{1453}{288} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{155777 \sqrt{1-2 x} \sqrt{5 x+3}}{31104}-\frac{660959 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{93312 \sqrt{10}}-\frac{1295}{729} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-155777*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/31104 + (1453*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/288 - (247*Sqrt[1 - 2*x]*(3
+ 5*x)^(5/2))/324 - (5*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/18 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(3*(2 + 3*x))
- (660959*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(93312*Sqrt[10]) - (1295*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr
t[3 + 5*x])])/729

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Rubi [A]  time = 0.07942, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{5}{18} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac{247}{324} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac{1453}{288} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{155777 \sqrt{1-2 x} \sqrt{5 x+3}}{31104}-\frac{660959 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{93312 \sqrt{10}}-\frac{1295}{729} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-155777*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/31104 + (1453*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/288 - (247*Sqrt[1 - 2*x]*(3
+ 5*x)^(5/2))/324 - (5*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/18 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(3*(2 + 3*x))
- (660959*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(93312*Sqrt[10]) - (1295*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr
t[3 + 5*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 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)^{5/2}}{(2+3 x)^2} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac{1}{3} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac{5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac{1}{180} \int \frac{(200-6175 x) \sqrt{1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=-\frac{247}{324} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac{\int \frac{\left (126325-\frac{980775 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)} \, dx}{8100}\\ &=\frac{1453}{288} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{247}{324} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac{\int \frac{\left (-\frac{268425}{2}-\frac{11683275 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx}{97200}\\ &=-\frac{155777 \sqrt{1-2 x} \sqrt{3+5 x}}{31104}+\frac{1453}{288} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{247}{324} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac{\int \frac{-\frac{2019975}{4}-\frac{49571925 x}{8}}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{583200}\\ &=-\frac{155777 \sqrt{1-2 x} \sqrt{3+5 x}}{31104}+\frac{1453}{288} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{247}{324} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac{660959 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{186624}+\frac{9065 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{1458}\\ &=-\frac{155777 \sqrt{1-2 x} \sqrt{3+5 x}}{31104}+\frac{1453}{288} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{247}{324} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}+\frac{9065}{729} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )-\frac{660959 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{93312 \sqrt{5}}\\ &=-\frac{155777 \sqrt{1-2 x} \sqrt{3+5 x}}{31104}+\frac{1453}{288} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{247}{324} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{5}{18} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{3 (2+3 x)}-\frac{660959 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{93312 \sqrt{10}}-\frac{1295}{729} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.152681, size = 132, normalized size = 0.74 \[ \frac{-30 \sqrt{5 x+3} \left (518400 x^5-688320 x^4+93864 x^3+206046 x^2-164165 x+45658\right )+660959 \sqrt{10-20 x} (3 x+2) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-1657600 \sqrt{7-14 x} (3 x+2) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{933120 \sqrt{1-2 x} (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*Sqrt[3 + 5*x]*(45658 - 164165*x + 206046*x^2 + 93864*x^3 - 688320*x^4 + 518400*x^5) + 660959*Sqrt[10 - 20
*x]*(2 + 3*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]] - 1657600*Sqrt[7 - 14*x]*(2 + 3*x)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7
]*Sqrt[3 + 5*x])])/(933120*Sqrt[1 - 2*x]*(2 + 3*x))

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Maple [A]  time = 0.01, size = 197, normalized size = 1.1 \begin{align*} -{\frac{1}{3732480+5598720\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -15552000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1982877\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-4972800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3620880\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1321918\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3315200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -4370940\,x\sqrt{-10\,{x}^{2}-x+3}+2739480\,\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)^(5/2)/(2+3*x)^2,x)

[Out]

-1/1866240*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-15552000*x^4*(-10*x^2-x+3)^(1/2)[phone]*x^3*(-10*x^2-x+3)^(1/2)+198
2877*10^(1/2)*arcsin(20/11*x+1/11)*x-4972800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+3620
880*x^2*(-10*x^2-x+3)^(1/2)+1321918*10^(1/2)*arcsin(20/11*x+1/11)-3315200*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))-4370940*x*(-10*x^2-x+3)^(1/2)+2739480*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 3.34024, size = 161, normalized size = 0.9 \begin{align*} -\frac{25}{18} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{695}{648} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3 \,{\left (3 \, x + 2\right )}} + \frac{11045}{2592} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{660959}{1866240} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1295}{1458} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{76253}{31104} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/18*(-10*x^2 - x + 3)^(3/2)*x + 695/648*(-10*x^2 - x + 3)^(3/2) - 1/3*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 1
1045/2592*sqrt(-10*x^2 - x + 3)*x - 660959/1866240*sqrt(10)*arcsin(20/11*x + 1/11) + 1295/1458*sqrt(7)*arcsin(
37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 76253/31104*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.81705, size = 440, normalized size = 2.46 \begin{align*} -\frac{1657600 \, \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 ) - 660959 \, \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 (259200 \, x^{4} - 214560 \, x^{3} - 60348 \, x^{2} + 72849 \, x - 45658\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1866240 \,{\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)^(5/2)/(2+3*x)^2,x, algorithm="fricas")

[Out]

-1/1866240*(1657600*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)) - 660959*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*(259200*x^4 - 214560*x^3 - 60348*x^2 + 72849*x - 45658)*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)**(5/2)/(2+3*x)**2,x)

[Out]

Timed out

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Giac [B]  time = 3.50533, size = 429, normalized size = 2.4 \begin{align*} \frac{259}{2916} \, \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}{777600} \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} - 593 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 26185 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 622085 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{660959}{1866240} \, \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 )}}{243 \,{\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)^(5/2)/(2+3*x)^2,x, algorithm="giac")

[Out]

259/2916*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)))) + 1/777600*(12*(8*(36*sqrt(5)*(5*x + 3) - 593*sqrt(5))
*(5*x + 3) + 26185*sqrt(5))*(5*x + 3) - 622085*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 660959/1866240*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)))) - 1078/243*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)