3.2607 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=202 \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac{139745 \sqrt{5 x+3}}{1613472 \sqrt{1-2 x}}-\frac{14135 \sqrt{5 x+3}}{153664 \sqrt{1-2 x} (3 x+2)}-\frac{2013 \sqrt{5 x+3}}{10976 \sqrt{1-2 x} (3 x+2)^2}-\frac{2717 \sqrt{5 x+3}}{8232 \sqrt{1-2 x} (3 x+2)^3}+\frac{43 \sqrt{5 x+3}}{588 \sqrt{1-2 x} (3 x+2)^4}-\frac{547745 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1075648 \sqrt{7}} \]

[Out]

(139745*Sqrt[3 + 5*x])/(1613472*Sqrt[1 - 2*x]) + (43*Sqrt[3 + 5*x])/(588*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (2717*Sq
rt[3 + 5*x])/(8232*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (2013*Sqrt[3 + 5*x])/(10976*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (1413
5*Sqrt[3 + 5*x])/(153664*Sqrt[1 - 2*x]*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - (5
47745*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1075648*Sqrt[7])

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Rubi [A]  time = 0.0775794, antiderivative size = 202, 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 = {98, 149, 151, 152, 12, 93, 204} \[ \frac{11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}+\frac{139745 \sqrt{5 x+3}}{1613472 \sqrt{1-2 x}}-\frac{14135 \sqrt{5 x+3}}{153664 \sqrt{1-2 x} (3 x+2)}-\frac{2013 \sqrt{5 x+3}}{10976 \sqrt{1-2 x} (3 x+2)^2}-\frac{2717 \sqrt{5 x+3}}{8232 \sqrt{1-2 x} (3 x+2)^3}+\frac{43 \sqrt{5 x+3}}{588 \sqrt{1-2 x} (3 x+2)^4}-\frac{547745 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1075648 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(139745*Sqrt[3 + 5*x])/(1613472*Sqrt[1 - 2*x]) + (43*Sqrt[3 + 5*x])/(588*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (2717*Sq
rt[3 + 5*x])/(8232*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (2013*Sqrt[3 + 5*x])/(10976*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (1413
5*Sqrt[3 + 5*x])/(153664*Sqrt[1 - 2*x]*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - (5
47745*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1075648*Sqrt[7])

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 151

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 152

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

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

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{(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^5} \, dx &=\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac{1}{21} \int \frac{\left (-222-\frac{795 x}{2}\right ) \sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\\ &=\frac{43 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^4}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac{\int \frac{-\frac{59169}{2}-50490 x}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{1764}\\ &=\frac{43 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^4}-\frac{2717 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)^3}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac{\int \frac{-\frac{851301}{4}-366795 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{37044}\\ &=\frac{43 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^4}-\frac{2717 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)^3}-\frac{2013 \sqrt{3+5 x}}{10976 \sqrt{1-2 x} (2+3 x)^2}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac{\int \frac{-\frac{9255015}{8}-1902285 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{518616}\\ &=\frac{43 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^4}-\frac{2717 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)^3}-\frac{2013 \sqrt{3+5 x}}{10976 \sqrt{1-2 x} (2+3 x)^2}-\frac{14135 \sqrt{3+5 x}}{153664 \sqrt{1-2 x} (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac{\int \frac{-\frac{70128135}{16}-\frac{13357575 x}{4}}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx}{3630312}\\ &=\frac{139745 \sqrt{3+5 x}}{1613472 \sqrt{1-2 x}}+\frac{43 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^4}-\frac{2717 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)^3}-\frac{2013 \sqrt{3+5 x}}{10976 \sqrt{1-2 x} (2+3 x)^2}-\frac{14135 \sqrt{3+5 x}}{153664 \sqrt{1-2 x} (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac{\int \frac{1138761855}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{139767012}\\ &=\frac{139745 \sqrt{3+5 x}}{1613472 \sqrt{1-2 x}}+\frac{43 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^4}-\frac{2717 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)^3}-\frac{2013 \sqrt{3+5 x}}{10976 \sqrt{1-2 x} (2+3 x)^2}-\frac{14135 \sqrt{3+5 x}}{153664 \sqrt{1-2 x} (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac{547745 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2151296}\\ &=\frac{139745 \sqrt{3+5 x}}{1613472 \sqrt{1-2 x}}+\frac{43 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^4}-\frac{2717 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)^3}-\frac{2013 \sqrt{3+5 x}}{10976 \sqrt{1-2 x} (2+3 x)^2}-\frac{14135 \sqrt{3+5 x}}{153664 \sqrt{1-2 x} (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}+\frac{547745 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1075648}\\ &=\frac{139745 \sqrt{3+5 x}}{1613472 \sqrt{1-2 x}}+\frac{43 \sqrt{3+5 x}}{588 \sqrt{1-2 x} (2+3 x)^4}-\frac{2717 \sqrt{3+5 x}}{8232 \sqrt{1-2 x} (2+3 x)^3}-\frac{2013 \sqrt{3+5 x}}{10976 \sqrt{1-2 x} (2+3 x)^2}-\frac{14135 \sqrt{3+5 x}}{153664 \sqrt{1-2 x} (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac{547745 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1075648 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0847261, size = 105, normalized size = 0.52 \[ -\frac{7 \sqrt{5 x+3} \left (45277380 x^5+82071900 x^4+25673409 x^3-27318504 x^2-18627988 x-2906640\right )-1643235 \sqrt{7-14 x} (2 x-1) (3 x+2)^4 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{22588608 (1-2 x)^{3/2} (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(7*Sqrt[3 + 5*x]*(-2906640 - 18627988*x - 27318504*x^2 + 25673409*x^3 + 82071900*x^4 + 45277380*x^5) - 164323
5*Sqrt[7 - 14*x]*(-1 + 2*x)*(2 + 3*x)^4*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(22588608*(1 - 2*x)^(3/
2)*(2 + 3*x)^4)

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Maple [B]  time = 0.015, size = 353, normalized size = 1.8 \begin{align*}{\frac{1}{45177216\, \left ( 2+3\,x \right ) ^{4} \left ( 2\,x-1 \right ) ^{2}} \left ( 532408140\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-633883320\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-433814040\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-1149006600\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-170896440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-359427726\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+382459056\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45177216*(532408140*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+887346900*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+133102035*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^
(1/2))*x^4-633883320*x^5*(-10*x^2-x+3)^(1/2)-433814040*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))*x^3-1149006600*x^4*(-10*x^2-x+3)^(1/2)-170896440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x^2-359427726*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+
382459056*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]
2*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4/(2*x-1)^2/(-10*x^2
-x+3)^(1/2)

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Maxima [B]  time = 3.01489, size = 439, normalized size = 2.17 \begin{align*} \frac{547745}{15059072} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{698725 \, x}{1613472 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343745}{3226944 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{633875 \, x}{691488 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{2268 \,{\left (81 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + 216 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 216 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 96 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 16 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{331}{31752 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{9313}{98784 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{659891}{1778112 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{296615}{12446784 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

547745/15059072*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 698725/1613472*x/sqrt(-10*x^2 - x
+ 3) + 343745/3226944/sqrt(-10*x^2 - x + 3) + 633875/691488*x/(-10*x^2 - x + 3)^(3/2) - 1/2268/(81*(-10*x^2 -
x + 3)^(3/2)*x^4 + 216*(-10*x^2 - x + 3)^(3/2)*x^3 + 216*(-10*x^2 - x + 3)^(3/2)*x^2 + 96*(-10*x^2 - x + 3)^(3
/2)*x + 16*(-10*x^2 - x + 3)^(3/2)) + 331/31752/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x
^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) - 9313/98784/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 1
2*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 659891/1778112/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-1
0*x^2 - x + 3)^(3/2)) + 296615/12446784/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.61105, size = 471, normalized size = 2.33 \begin{align*} -\frac{1643235 \, \sqrt{7}{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\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 ) + 14 \,{\left (45277380 \, x^{5} + 82071900 \, x^{4} + 25673409 \, x^{3} - 27318504 \, x^{2} - 18627988 \, x - 2906640\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{45177216 \,{\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/45177216*(1643235*sqrt(7)*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*arctan(1/14*sqrt(7)*
(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(45277380*x^5 + 82071900*x^4 + 25673409*x^3 -
27318504*x^2 - 18627988*x - 2906640)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104
*x^2 + 32*x + 16)

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

[Out]

Timed out

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Giac [B]  time = 5.4257, size = 564, normalized size = 2.79 \begin{align*} \frac{109549}{30118144} \, \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{88 \,{\left (100 \, \sqrt{5}{\left (5 \, x + 3\right )} - 627 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1764735 \,{\left (2 \, x - 1\right )}^{2}} - \frac{55 \,{\left (79441 \, \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 )}^{7} + 82486488 \, \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 )}^{5} + 31196222400 \, \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 )}^{3} + 1487445568000 \, \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 )}\right )}}{3764768 \,{\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 )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

109549/30118144*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 88/1764735*(100*sqrt(5)*(5*x + 3) - 627*sqrt(
5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 55/3764768*(79441*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)))^7 + 82486488*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)))^5 + 31196222400*s
qrt(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)))^3 + 1487445568000*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)^4