3.2422 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=238 \[ \frac{4477 \sqrt{1-2 x} (5 x+3)^{7/2}}{448 (3 x+2)^4}+\frac{407 (1-2 x)^{3/2} (5 x+3)^{7/2}}{168 (3 x+2)^5}+\frac{37 (1-2 x)^{5/2} (5 x+3)^{7/2}}{84 (3 x+2)^6}+\frac{3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}-\frac{49247 \sqrt{1-2 x} (5 x+3)^{5/2}}{18816 (3 x+2)^3}-\frac{2708585 \sqrt{1-2 x} (5 x+3)^{3/2}}{526848 (3 x+2)^2}-\frac{29794435 \sqrt{1-2 x} \sqrt{5 x+3}}{2458624 (3 x+2)}-\frac{327738785 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2458624 \sqrt{7}} \]

[Out]

(-29794435*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2458624*(2 + 3*x)) - (2708585*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(526848*
(2 + 3*x)^2) - (49247*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(18816*(2 + 3*x)^3) + (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))
/(49*(2 + 3*x)^7) + (37*(1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(84*(2 + 3*x)^6) + (407*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/
2))/(168*(2 + 3*x)^5) + (4477*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(448*(2 + 3*x)^4) - (327738785*ArcTan[Sqrt[1 - 2*
x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2458624*Sqrt[7])

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Rubi [A]  time = 0.0829556, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac{4477 \sqrt{1-2 x} (5 x+3)^{7/2}}{448 (3 x+2)^4}+\frac{407 (1-2 x)^{3/2} (5 x+3)^{7/2}}{168 (3 x+2)^5}+\frac{37 (1-2 x)^{5/2} (5 x+3)^{7/2}}{84 (3 x+2)^6}+\frac{3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}-\frac{49247 \sqrt{1-2 x} (5 x+3)^{5/2}}{18816 (3 x+2)^3}-\frac{2708585 \sqrt{1-2 x} (5 x+3)^{3/2}}{526848 (3 x+2)^2}-\frac{29794435 \sqrt{1-2 x} \sqrt{5 x+3}}{2458624 (3 x+2)}-\frac{327738785 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2458624 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^8,x]

[Out]

(-29794435*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2458624*(2 + 3*x)) - (2708585*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(526848*
(2 + 3*x)^2) - (49247*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(18816*(2 + 3*x)^3) + (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))
/(49*(2 + 3*x)^7) + (37*(1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(84*(2 + 3*x)^6) + (407*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/
2))/(168*(2 + 3*x)^5) + (4477*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(448*(2 + 3*x)^4) - (327738785*ArcTan[Sqrt[1 - 2*
x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2458624*Sqrt[7])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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)^8} \, dx &=\frac{3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac{37}{14} \int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\\ &=\frac{3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac{37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac{2035}{168} \int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=\frac{3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac{37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac{407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac{4477}{112} \int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac{3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac{37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac{407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac{4477 \sqrt{1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}+\frac{49247}{896} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=-\frac{49247 \sqrt{1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac{3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac{37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac{407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac{4477 \sqrt{1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}+\frac{2708585 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{37632}\\ &=-\frac{2708585 \sqrt{1-2 x} (3+5 x)^{3/2}}{526848 (2+3 x)^2}-\frac{49247 \sqrt{1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac{3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac{37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac{407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac{4477 \sqrt{1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}+\frac{29794435 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{351232}\\ &=-\frac{29794435 \sqrt{1-2 x} \sqrt{3+5 x}}{2458624 (2+3 x)}-\frac{2708585 \sqrt{1-2 x} (3+5 x)^{3/2}}{526848 (2+3 x)^2}-\frac{49247 \sqrt{1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac{3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac{37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac{407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac{4477 \sqrt{1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}+\frac{327738785 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{4917248}\\ &=-\frac{29794435 \sqrt{1-2 x} \sqrt{3+5 x}}{2458624 (2+3 x)}-\frac{2708585 \sqrt{1-2 x} (3+5 x)^{3/2}}{526848 (2+3 x)^2}-\frac{49247 \sqrt{1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac{3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac{37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac{407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac{4477 \sqrt{1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}+\frac{327738785 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2458624}\\ &=-\frac{29794435 \sqrt{1-2 x} \sqrt{3+5 x}}{2458624 (2+3 x)}-\frac{2708585 \sqrt{1-2 x} (3+5 x)^{3/2}}{526848 (2+3 x)^2}-\frac{49247 \sqrt{1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac{3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac{37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac{407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac{4477 \sqrt{1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}-\frac{327738785 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2458624 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.233891, size = 219, normalized size = 0.92 \[ \frac{37 (1-2 x)^{5/2} (5 x+3)^{7/2}}{84 (3 x+2)^6}+\frac{3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}+\frac{407 \left (307328 (1-2 x)^{3/2} (5 x+3)^{7/2}+11 (3 x+2) \left (115248 \sqrt{1-2 x} (5 x+3)^{7/2}-11 (3 x+2) \left (2744 \sqrt{1-2 x} (5 x+3)^{5/2}+55 (3 x+2) \left (7 \sqrt{1-2 x} \sqrt{5 x+3} (169 x+108)+363 \sqrt{7} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )\right )\right )\right )}{51631104 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^8,x]

[Out]

(3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))/(49*(2 + 3*x)^7) + (37*(1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(84*(2 + 3*x)^6) +
(407*(307328*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2) + 11*(2 + 3*x)*(115248*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2) - 11*(2 + 3
*x)*(2744*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2) + 55*(2 + 3*x)*(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(108 + 169*x) + 363*Sqrt
[7]*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])))))/(51631104*(2 + 3*x)^5)

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Maple [B]  time = 0.013, size = 394, normalized size = 1.7 \begin{align*}{\frac{1}{103262208\, \left ( 2+3\,x \right ) ^{7}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2150294168385\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{7}[phone]\,\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{-10\,{x}^{2}-x+3}{x}^{6}[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}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{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+2720742382624\,{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 ){\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)^8,x)

[Out]

1/103262208*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2150294168385*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))*x^7+10034706119130*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+20069412238260*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+877238952870*(-10*x^2-x+3)^(1/2)*x^6+22299346931400*7^(1
/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+3548184526460*x^5*(-10*x^2-x+3)^(1/2)[phone]
0*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+5979472745456*x^4*(-10*x^2-x+3)^(1/2)+5946492
515040*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+5376679039872*x^3*(-10*x^2-x+3)^(1/2)+13
21442781120*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+2720742382624*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]*x*(-10*x^2-x+3)^(1/2)+82
570989312*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^7

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Maxima [A]  time = 4.27453, size = 477, normalized size = 2. \begin{align*} \frac{122277415}{271063296} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{49 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{196 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{1369 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{2744 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{162319 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{153664 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{3024121 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{2151296 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{24455483 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{60236288 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{2190708025}{180708864} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{4205402795}{361417728} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{4059472427 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1084253184 \,{\left (3 \, x + 2\right )}} + \frac{501088225}{8605184} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{327738785}{34420736} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{441499355}{17210368} \, \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)^8,x, algorithm="maxima")

[Out]

122277415/271063296*(-10*x^2 - x + 3)^(5/2) + 3/49*(-10*x^2 - x + 3)^(7/2)/(2187*x^7 + 10206*x^6 + 20412*x^5 +
22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 37/196*(-10*x^2 - x + 3)^(7/2)/(729*x^6 + 2916*x^5 + 4860*
x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 1369/2744*(-10*x^2 - x + 3)^(7/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 72
0*x^2 + 240*x + 32) + 162319/153664*(-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 3024121
/2151296*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 24455483/60236288*(-10*x^2 - x + 3)^(7/2)/(9*x
^2 + 12*x + 4) - 2190708025/180708864*(-10*x^2 - x + 3)^(3/2)*x + 4205402795/361417728*(-10*x^2 - x + 3)^(3/2)
- 4059472427/1084253184*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 501088225/8605184*sqrt(-10*x^2 - x + 3)*x + 32773
8785/34420736*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 441499355/17210368*sqrt(-10*x^2 - x
+ 3)

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Fricas [A]  time = 2.1984, size = 594, normalized size = 2.5 \begin{align*} -\frac{983216355 \, \sqrt{7}{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\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 (62659925205 \, x^{6} + 253441751890 \, x^{5} + 427105196104 \, x^{4} + 384048502848 \, x^{3} + 194338741616 \, x^{2} + 52456780256 \, x + 5897927808\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{103262208 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\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)^8,x, algorithm="fricas")

[Out]

-1/103262208*(983216355*sqrt(7)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x
+ 128)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(62659925205*x^6 +
253441751890*x^5 + 427105196104*x^4 + 384048502848*x^3 + 194338741616*x^2 + 52456780256*x + 5897927808)*sqrt(5
*x + 3)*sqrt(-2*x + 1))/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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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)**8,x)

[Out]

Timed out

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Giac [B]  time = 6.38316, size = 759, normalized size = 3.19 \begin{align*} \frac{65547757}{68841472} \, \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{8857805 \,{\left (111 \, \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 )}^{13} + 207200 \, \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 )}^{11} + 164185280 \, \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} - 63583027200 \, \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} - 12872125952000 \, \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} - 1273567232000000 \, \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} - 53489823744000000 \, \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 )}}{3687936 \,{\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 )}^{7}} \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)^8,x, algorithm="giac")

[Out]

65547757/68841472*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)))) - 8857805/3687936*(111*sqrt(10)*((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^13 + 207200*sqr
t(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
)))^11 + 164185280*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^9 - 63583027200*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 - 12872125952000*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 - 1273567232000000*sqrt(10)*((sqr
t(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 - 534
89823744000000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22)))^2 + 280)^7