3.2298 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=238 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}-\frac{59 \sqrt{1-2 x} (5 x+3)^{3/2}}{1764 (3 x+2)^6}+\frac{8818415317 \sqrt{1-2 x} \sqrt{5 x+3}}{3252759552 (3 x+2)}+\frac{84539611 \sqrt{1-2 x} \sqrt{5 x+3}}{232339968 (3 x+2)^2}+\frac{2524471 \sqrt{1-2 x} \sqrt{5 x+3}}{41489280 (3 x+2)^3}+\frac{369409 \sqrt{1-2 x} \sqrt{5 x+3}}{20744640 (3 x+2)^4}-\frac{6577 \sqrt{1-2 x} \sqrt{5 x+3}}{370440 (3 x+2)^5}-\frac{3735929329 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{120472576 \sqrt{7}} \]

[Out]

(-6577*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(370440*(2 + 3*x)^5) + (369409*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(20744640*(2 +
3*x)^4) + (2524471*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41489280*(2 + 3*x)^3) + (84539611*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(232339968*(2 + 3*x)^2) + (8818415317*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3252759552*(2 + 3*x)) - (59*Sqrt[1 - 2*
x]*(3 + 5*x)^(3/2))/(1764*(2 + 3*x)^6) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(21*(2 + 3*x)^7) - (3735929329*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(120472576*Sqrt[7])

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Rubi [A]  time = 0.0999778, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {97, 149, 151, 12, 93, 204} \[ -\frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}-\frac{59 \sqrt{1-2 x} (5 x+3)^{3/2}}{1764 (3 x+2)^6}+\frac{8818415317 \sqrt{1-2 x} \sqrt{5 x+3}}{3252759552 (3 x+2)}+\frac{84539611 \sqrt{1-2 x} \sqrt{5 x+3}}{232339968 (3 x+2)^2}+\frac{2524471 \sqrt{1-2 x} \sqrt{5 x+3}}{41489280 (3 x+2)^3}+\frac{369409 \sqrt{1-2 x} \sqrt{5 x+3}}{20744640 (3 x+2)^4}-\frac{6577 \sqrt{1-2 x} \sqrt{5 x+3}}{370440 (3 x+2)^5}-\frac{3735929329 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{120472576 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6577*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(370440*(2 + 3*x)^5) + (369409*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(20744640*(2 +
3*x)^4) + (2524471*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(41489280*(2 + 3*x)^3) + (84539611*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(232339968*(2 + 3*x)^2) + (8818415317*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3252759552*(2 + 3*x)) - (59*Sqrt[1 - 2*
x]*(3 + 5*x)^(3/2))/(1764*(2 + 3*x)^6) - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(21*(2 + 3*x)^7) - (3735929329*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(120472576*Sqrt[7])

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 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{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{1}{21} \int \frac{\left (\frac{19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^7} \, dx\\ &=-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{\int \frac{\left (-\frac{783}{4}-2760 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^6} \, dx}{2646}\\ &=-\frac{6577 \sqrt{1-2 x} \sqrt{3+5 x}}{370440 (2+3 x)^5}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{\int \frac{-\frac{1154271}{8}-285690 x}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx}{277830}\\ &=-\frac{6577 \sqrt{1-2 x} \sqrt{3+5 x}}{370440 (2+3 x)^5}+\frac{369409 \sqrt{1-2 x} \sqrt{3+5 x}}{20744640 (2+3 x)^4}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{\int \frac{\frac{8684811}{16}-\frac{16623405 x}{4}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{7779240}\\ &=-\frac{6577 \sqrt{1-2 x} \sqrt{3+5 x}}{370440 (2+3 x)^5}+\frac{369409 \sqrt{1-2 x} \sqrt{3+5 x}}{20744640 (2+3 x)^4}+\frac{2524471 \sqrt{1-2 x} \sqrt{3+5 x}}{41489280 (2+3 x)^3}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{\int \frac{\frac{4635547875}{32}-\frac{795208365 x}{4}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{163364040}\\ &=-\frac{6577 \sqrt{1-2 x} \sqrt{3+5 x}}{370440 (2+3 x)^5}+\frac{369409 \sqrt{1-2 x} \sqrt{3+5 x}}{20744640 (2+3 x)^4}+\frac{2524471 \sqrt{1-2 x} \sqrt{3+5 x}}{41489280 (2+3 x)^3}+\frac{84539611 \sqrt{1-2 x} \sqrt{3+5 x}}{232339968 (2+3 x)^2}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{\int \frac{\frac{570867242085}{64}-\frac{133149887325 x}{16}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{2287096560}\\ &=-\frac{6577 \sqrt{1-2 x} \sqrt{3+5 x}}{370440 (2+3 x)^5}+\frac{369409 \sqrt{1-2 x} \sqrt{3+5 x}}{20744640 (2+3 x)^4}+\frac{2524471 \sqrt{1-2 x} \sqrt{3+5 x}}{41489280 (2+3 x)^3}+\frac{84539611 \sqrt{1-2 x} \sqrt{3+5 x}}{232339968 (2+3 x)^2}+\frac{8818415317 \sqrt{1-2 x} \sqrt{3+5 x}}{3252759552 (2+3 x)}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{\int \frac{31774078943145}{128 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{16009675920}\\ &=-\frac{6577 \sqrt{1-2 x} \sqrt{3+5 x}}{370440 (2+3 x)^5}+\frac{369409 \sqrt{1-2 x} \sqrt{3+5 x}}{20744640 (2+3 x)^4}+\frac{2524471 \sqrt{1-2 x} \sqrt{3+5 x}}{41489280 (2+3 x)^3}+\frac{84539611 \sqrt{1-2 x} \sqrt{3+5 x}}{232339968 (2+3 x)^2}+\frac{8818415317 \sqrt{1-2 x} \sqrt{3+5 x}}{3252759552 (2+3 x)}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{3735929329 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{240945152}\\ &=-\frac{6577 \sqrt{1-2 x} \sqrt{3+5 x}}{370440 (2+3 x)^5}+\frac{369409 \sqrt{1-2 x} \sqrt{3+5 x}}{20744640 (2+3 x)^4}+\frac{2524471 \sqrt{1-2 x} \sqrt{3+5 x}}{41489280 (2+3 x)^3}+\frac{84539611 \sqrt{1-2 x} \sqrt{3+5 x}}{232339968 (2+3 x)^2}+\frac{8818415317 \sqrt{1-2 x} \sqrt{3+5 x}}{3252759552 (2+3 x)}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac{3735929329 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{120472576}\\ &=-\frac{6577 \sqrt{1-2 x} \sqrt{3+5 x}}{370440 (2+3 x)^5}+\frac{369409 \sqrt{1-2 x} \sqrt{3+5 x}}{20744640 (2+3 x)^4}+\frac{2524471 \sqrt{1-2 x} \sqrt{3+5 x}}{41489280 (2+3 x)^3}+\frac{84539611 \sqrt{1-2 x} \sqrt{3+5 x}}{232339968 (2+3 x)^2}+\frac{8818415317 \sqrt{1-2 x} \sqrt{3+5 x}}{3252759552 (2+3 x)}-\frac{59 \sqrt{1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}-\frac{3735929329 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{120472576 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.284043, size = 221, normalized size = 0.93 \[ \frac{1}{49} \left (\frac{267 (1-2 x)^{3/2} (5 x+3)^{7/2}}{28 (3 x+2)^6}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{(3 x+2)^7}+\frac{6344698752 (1-2 x)^{3/2} (5 x+3)^{7/2}+255169 (3 x+2) \left (115248 \sqrt{1-2 x} (5 x+3)^{7/2}-11 (3 x+2) \left (2744 \sqrt{1-2 x} (5 x+3)^{5/2}+55 (3 x+2) \left (7 \sqrt{1-2 x} \sqrt{5 x+3} (169 x+108)+363 \sqrt{7} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )\right )\right )}{258155520 (3 x+2)^5}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(2 + 3*x)^7 + (267*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(28*(2 + 3*x)^6) + (6
344698752*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2) + 255169*(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])]))))/(258155520*(2 + 3*x)^5))/49

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Maple [B]  time = 0.02, size = 394, normalized size = 1.7 \begin{align*}{\frac{1}{25299240960\, \left ( 2+3\,x \right ) ^{7}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 122557161637845\,\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}+1143866841953220\,\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}+1270963157725800\,\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+155087260368544\,{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((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^8,x)

[Out]

1/25299240960*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(122557161637845*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)
^(1/2))*x^7+571933420976610*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+1143866841953220*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+50000414847390*(-10*x^2-x+3)^(1/2)*x^6+1270963157
725800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+202238577496620*x^5*(-10*x^2-x+3)^(1/2)+
847308771817200*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+340917181344432*x^4*(-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^2+306585279928704*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+155087260368544*
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))/(-10*x^2-x+3)^(1/2)/(2+3*x)^7

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Maxima [A]  time = 2.99377, size = 398, normalized size = 1.67 \begin{align*} \frac{3735929329}{1686616064} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{154377245}{90354432} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{147 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} - \frac{191 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4116 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{919 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{96040 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{72203 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{768320 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{2612695 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{6453888 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{92626347 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{60236288 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{1142391613 \, \sqrt{-10 \, x^{2} - x + 3}}{361417728 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^8,x, algorithm="maxima")

[Out]

3735929329/1686616064*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 154377245/90354432*sqrt(-10*
x^2 - x + 3) + 1/147*(-10*x^2 - x + 3)^(3/2)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*
x^2 + 1344*x + 128) - 191/4116*(-10*x^2 - x + 3)^(3/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 +
576*x + 64) + 919/96040*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 72203/
768320*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 2612695/6453888*(-10*x^2 - x + 3)^(3
/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 92626347/60236288*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 1142391613/3
61417728*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.62324, size = 618, normalized size = 2.6 \begin{align*} -\frac{56038939935 \, \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 (3571458203385 \, x^{6} + 14445612678330 \, x^{5} + 24351227238888 \, x^{4} + 21898948566336 \, x^{3} + 11077661454896 \, x^{2} + 2987299350368 \, x + 335335888512\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{25299240960 \,{\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((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/25299240960*(56038939935*sqrt(7)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 134
4*x + 128)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(3571458203385*
x^6 + 14445612678330*x^5 + 24351227238888*x^4 + 21898948566336*x^3 + 11077661454896*x^2 + 2987299350368*x + 33
5335888512)*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((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**8,x)

[Out]

Timed out

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Giac [B]  time = 4.64791, size = 759, normalized size = 3.19 \begin{align*} \frac{3735929329}{16866160640} \, \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{14641 \,{\left (765507 \, \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} + 1428946400 \, \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} + 1132297127360 \, \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} - 334448649830400 \, \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} - 85378328229376000 \, \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} - 8754907317452800000 \, \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} - 368890400944128000000 \, \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 )}}{180708864 \,{\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((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^8,x, algorithm="giac")

[Out]

3735929329/16866160640*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)))) - 14641/180708864*(765507*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)))^13 + 14
28946400*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)))^11 + 1132297127360*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 - 334448649830400*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 - 85378328229376000*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 - 87549073
17452800000*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)))^3 - 368890400944128000000*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)^7