3.2549 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx\)

Optimal. Leaf size=209 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^5}+\frac{426781 \sqrt{1-2 x} \sqrt{5 x+3}}{6453888 (3 x+2)}-\frac{55277 \sqrt{1-2 x} \sqrt{5 x+3}}{460992 (3 x+2)^2}-\frac{29297 \sqrt{1-2 x} \sqrt{5 x+3}}{82320 (3 x+2)^3}-\frac{42863 \sqrt{1-2 x} \sqrt{5 x+3}}{41160 (3 x+2)^4}+\frac{164 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^5}-\frac{3474273 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2151296 \sqrt{7}} \]

[Out]

(164*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^5) - (42863*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41160*(2 + 3*x)^4)
- (29297*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(82320*(2 + 3*x)^3) - (55277*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(460992*(2 + 3
*x)^2) + (426781*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6453888*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 +
3*x)^5) - (3474273*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2151296*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0817854, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {98, 149, 151, 12, 93, 204} \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^5}+\frac{426781 \sqrt{1-2 x} \sqrt{5 x+3}}{6453888 (3 x+2)}-\frac{55277 \sqrt{1-2 x} \sqrt{5 x+3}}{460992 (3 x+2)^2}-\frac{29297 \sqrt{1-2 x} \sqrt{5 x+3}}{82320 (3 x+2)^3}-\frac{42863 \sqrt{1-2 x} \sqrt{5 x+3}}{41160 (3 x+2)^4}+\frac{164 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^5}-\frac{3474273 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2151296 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^6),x]

[Out]

(164*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^5) - (42863*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41160*(2 + 3*x)^4)
- (29297*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(82320*(2 + 3*x)^3) - (55277*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(460992*(2 + 3
*x)^2) + (426781*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6453888*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 +
3*x)^5) - (3474273*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2151296*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 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)^{3/2} (2+3 x)^6} \, dx &=\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^5}-\frac{1}{7} \int \frac{\left (-327-\frac{1145 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^6} \, dx\\ &=\frac{164 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^5}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^5}-\frac{1}{735} \int \frac{-\frac{110549}{2}-\frac{187255 x}{2}}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx\\ &=\frac{164 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^5}-\frac{42863 \sqrt{1-2 x} \sqrt{3+5 x}}{41160 (2+3 x)^4}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^5}-\frac{\int \frac{-\frac{1509441}{4}-642945 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{20580}\\ &=\frac{164 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^5}-\frac{42863 \sqrt{1-2 x} \sqrt{3+5 x}}{41160 (2+3 x)^4}-\frac{29297 \sqrt{1-2 x} \sqrt{3+5 x}}{82320 (2+3 x)^3}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^5}-\frac{\int \frac{-\frac{14471625}{8}-3076185 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{432180}\\ &=\frac{164 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^5}-\frac{42863 \sqrt{1-2 x} \sqrt{3+5 x}}{41160 (2+3 x)^4}-\frac{29297 \sqrt{1-2 x} \sqrt{3+5 x}}{82320 (2+3 x)^3}-\frac{55277 \sqrt{1-2 x} \sqrt{3+5 x}}{460992 (2+3 x)^2}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^5}-\frac{\int \frac{-\frac{92325135}{16}-\frac{29020425 x}{4}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{6050520}\\ &=\frac{164 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^5}-\frac{42863 \sqrt{1-2 x} \sqrt{3+5 x}}{41160 (2+3 x)^4}-\frac{29297 \sqrt{1-2 x} \sqrt{3+5 x}}{82320 (2+3 x)^3}-\frac{55277 \sqrt{1-2 x} \sqrt{3+5 x}}{460992 (2+3 x)^2}+\frac{426781 \sqrt{1-2 x} \sqrt{3+5 x}}{6453888 (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^5}-\frac{\int -\frac{1094395995}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{42353640}\\ &=\frac{164 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^5}-\frac{42863 \sqrt{1-2 x} \sqrt{3+5 x}}{41160 (2+3 x)^4}-\frac{29297 \sqrt{1-2 x} \sqrt{3+5 x}}{82320 (2+3 x)^3}-\frac{55277 \sqrt{1-2 x} \sqrt{3+5 x}}{460992 (2+3 x)^2}+\frac{426781 \sqrt{1-2 x} \sqrt{3+5 x}}{6453888 (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^5}+\frac{3474273 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{4302592}\\ &=\frac{164 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^5}-\frac{42863 \sqrt{1-2 x} \sqrt{3+5 x}}{41160 (2+3 x)^4}-\frac{29297 \sqrt{1-2 x} \sqrt{3+5 x}}{82320 (2+3 x)^3}-\frac{55277 \sqrt{1-2 x} \sqrt{3+5 x}}{460992 (2+3 x)^2}+\frac{426781 \sqrt{1-2 x} \sqrt{3+5 x}}{6453888 (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^5}+\frac{3474273 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2151296}\\ &=\frac{164 \sqrt{1-2 x} \sqrt{3+5 x}}{735 (2+3 x)^5}-\frac{42863 \sqrt{1-2 x} \sqrt{3+5 x}}{41160 (2+3 x)^4}-\frac{29297 \sqrt{1-2 x} \sqrt{3+5 x}}{82320 (2+3 x)^3}-\frac{55277 \sqrt{1-2 x} \sqrt{3+5 x}}{460992 (2+3 x)^2}+\frac{426781 \sqrt{1-2 x} \sqrt{3+5 x}}{6453888 (2+3 x)}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^5}-\frac{3474273 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2151296 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0791432, size = 100, normalized size = 0.48 \[ \frac{7 \sqrt{5 x+3} \left (-115230870 x^5-180017865 x^4+19738914 x^3+164918884 x^2+95331368 x+16456032\right )-17371365 \sqrt{7-14 x} (3 x+2)^5 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{75295360 \sqrt{1-2 x} (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^6),x]

[Out]

(7*Sqrt[3 + 5*x]*(16456032 + 95331368*x + 164918884*x^2 + 19738914*x^3 - 180017865*x^4 - 115230870*x^5) - 1737
1365*Sqrt[7 - 14*x]*(2 + 3*x)^5*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(75295360*Sqrt[1 - 2*x]*(2 + 3*
x)^5)

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 353, normalized size = 1.7 \begin{align*}{\frac{1}{150590720\, \left ( 2+3\,x \right ) ^{5} \left ( 2\,x-1 \right ) } \left ( 8442483390\,\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}[phone]\,{x}^{5}\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}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-4169127600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-276344796\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-3057360240\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-2308864376\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-555883680\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -1334639152\,x\sqrt{-10\,{x}^{2}-x+3}-230384448\,\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)^(3/2)/(2+3*x)^6,x)

[Out]

1/150590720*(8442483390*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+23920369605*arctan(1/14
*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+23451342750*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^4+1613232180*x^5*(-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^3+2520250110*x^4*(-10*x^2-x+3)^(1/2)-4169127600*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x
+3)^(1/2))*x^2-276344796*x^3*(-10*x^2-x+3)^(1/2)-3057360240*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))*x-2308864376*x^2*(-10*x^2-x+3)^(1/2)-555883680*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1
/2))-1334639152*x*(-10*x^2-x+3)^(1/2)-230384448*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^5/(2*
x-1)/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [B]  time = 1.85168, size = 537, normalized size = 2.57 \begin{align*} \frac{3474273}{30118144} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2133905 \, x}{9680832 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{4998019}{19361664 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{945 \,{\left (243 \, \sqrt{-10 \, x^{2} - x + 3} x^{5} + 810 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 1080 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 720 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 240 \, \sqrt{-10 \, x^{2} - x + 3} x + 32 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{331}{17640 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{83537}{740880 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{23353}{109760 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{137335}{921984 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^6,x, algorithm="maxima")

[Out]

3474273/30118144*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2133905/9680832*x/sqrt(-10*x^2 -
x + 3) + 4998019/19361664/sqrt(-10*x^2 - x + 3) + 1/945/(243*sqrt(-10*x^2 - x + 3)*x^5 + 810*sqrt(-10*x^2 - x
+ 3)*x^4 + 1080*sqrt(-10*x^2 - x + 3)*x^3 + 720*sqrt(-10*x^2 - x + 3)*x^2 + 240*sqrt(-10*x^2 - x + 3)*x + 32*s
qrt(-10*x^2 - x + 3)) - 331/17640/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3 + 216*sqrt(-10
*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10*x^2 - x + 3)) + 83537/740880/(27*sqrt(-10*x^2 - x
+ 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) - 23353/10976
0/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) - 137335/921984/(3*sqrt
(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

________________________________________________________________________________________

Fricas [A]  time = 1.87797, size = 490, normalized size = 2.34 \begin{align*} -\frac{17371365 \, \sqrt{7}{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\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 (115230870 \, x^{5} + 180017865 \, x^{4} - 19738914 \, x^{3} - 164918884 \, x^{2} - 95331368 \, x - 16456032\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{150590720 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/150590720*(17371365*sqrt(7)*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*arctan(1/14*sq
rt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(115230870*x^5 + 180017865*x^4 - 1973891
4*x^3 - 164918884*x^2 - 95331368*x - 16456032)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(486*x^6 + 1377*x^5 + 1350*x^4 +
360*x^3 - 240*x^2 - 176*x - 32)

________________________________________________________________________________________

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)**(3/2)/(2+3*x)**6,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 5.21297, size = 629, normalized size = 3.01 \begin{align*} \frac{3474273}{301181440} \, \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{1936 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{588245 \,{\left (2 \, x - 1\right )}} - \frac{121 \,{\left (203039 \, \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 )}^{9} + 265495440 \, \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} + 136071290880 \, \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} - 774949504000 \, \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} - 650054039040000 \, \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 )}}{7529536 \,{\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 )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

3474273/301181440*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 1936/588245*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*
x + 5)/(2*x - 1) - 121/7529536*(203039*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)))^9 + 265495440*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 136071290880*sqrt(10)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 774949504000*sq
rt(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(2
2)))^3 - 650054039040000*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)^5