3.2423 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^9} \, dx\)

Optimal. Leaf size=267 \[ \frac{47365 \sqrt{1-2 x} (5 x+3)^{5/2}}{36288 (3 x+2)^6}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1008 (3 x+2)^7}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}-\frac{720833 \sqrt{1-2 x} (5 x+3)^{3/2}}{508032 (3 x+2)^5}+\frac{6796051494355 \sqrt{1-2 x} \sqrt{5 x+3}}{200741732352 (3 x+2)}+\frac{64983635965 \sqrt{1-2 x} \sqrt{5 x+3}}{14338695168 (3 x+2)^2}+\frac{372439373 \sqrt{1-2 x} \sqrt{5 x+3}}{512096256 (3 x+2)^3}-\frac{75045071 \sqrt{1-2 x} \sqrt{5 x+3}}{85349376 (3 x+2)^4}-\frac{106656830005 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{275365888 \sqrt{7}} \]

[Out]

(-75045071*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(85349376*(2 + 3*x)^4) + (372439373*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5120
96256*(2 + 3*x)^3) + (64983635965*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14338695168*(2 + 3*x)^2) + (6796051494355*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(200741732352*(2 + 3*x)) - (720833*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(508032*(2 + 3*x)^5
) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(24*(2 + 3*x)^8) + (185*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(1008*(2 + 3*x)
^7) + (47365*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(36288*(2 + 3*x)^6) - (106656830005*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*
Sqrt[3 + 5*x])])/(275365888*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.115224, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{47365 \sqrt{1-2 x} (5 x+3)^{5/2}}{36288 (3 x+2)^6}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1008 (3 x+2)^7}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{24 (3 x+2)^8}-\frac{720833 \sqrt{1-2 x} (5 x+3)^{3/2}}{508032 (3 x+2)^5}+\frac{6796051494355 \sqrt{1-2 x} \sqrt{5 x+3}}{200741732352 (3 x+2)}+\frac{64983635965 \sqrt{1-2 x} \sqrt{5 x+3}}{14338695168 (3 x+2)^2}+\frac{372439373 \sqrt{1-2 x} \sqrt{5 x+3}}{512096256 (3 x+2)^3}-\frac{75045071 \sqrt{1-2 x} \sqrt{5 x+3}}{85349376 (3 x+2)^4}-\frac{106656830005 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{275365888 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^9,x]

[Out]

(-75045071*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(85349376*(2 + 3*x)^4) + (372439373*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5120
96256*(2 + 3*x)^3) + (64983635965*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14338695168*(2 + 3*x)^2) + (6796051494355*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(200741732352*(2 + 3*x)) - (720833*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(508032*(2 + 3*x)^5
) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(24*(2 + 3*x)^8) + (185*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(1008*(2 + 3*x)
^7) + (47365*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(36288*(2 + 3*x)^6) - (106656830005*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*
Sqrt[3 + 5*x])])/(275365888*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{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^9} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{1}{24} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}-\frac{1}{504} \int \frac{\sqrt{1-2 x} (3+5 x)^{3/2} \left (-\frac{10255}{4}+2075 x\right )}{(2+3 x)^7} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}+\frac{47365 \sqrt{1-2 x} (3+5 x)^{5/2}}{36288 (2+3 x)^6}+\frac{\int \frac{\left (\frac{1842365}{8}-\frac{660675 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^6} \, dx}{9072}\\ &=-\frac{720833 \sqrt{1-2 x} (3+5 x)^{3/2}}{508032 (2+3 x)^5}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}+\frac{47365 \sqrt{1-2 x} (3+5 x)^{5/2}}{36288 (2+3 x)^6}+\frac{\int \frac{\left (\frac{191095155}{16}-\frac{69048825 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^5} \, dx}{952560}\\ &=-\frac{75045071 \sqrt{1-2 x} \sqrt{3+5 x}}{85349376 (2+3 x)^4}-\frac{720833 \sqrt{1-2 x} (3+5 x)^{3/2}}{508032 (2+3 x)^5}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}+\frac{47365 \sqrt{1-2 x} (3+5 x)^{5/2}}{36288 (2+3 x)^6}+\frac{\int \frac{\frac{6505964655}{32}-\frac{2448530025 x}{8}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{80015040}\\ &=-\frac{75045071 \sqrt{1-2 x} \sqrt{3+5 x}}{85349376 (2+3 x)^4}+\frac{372439373 \sqrt{1-2 x} \sqrt{3+5 x}}{512096256 (2+3 x)^3}-\frac{720833 \sqrt{1-2 x} (3+5 x)^{3/2}}{508032 (2+3 x)^5}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}+\frac{47365 \sqrt{1-2 x} (3+5 x)^{5/2}}{36288 (2+3 x)^6}+\frac{\int \frac{\frac{1231597014375}{64}-\frac{195530670825 x}{8}}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{1680315840}\\ &=-\frac{75045071 \sqrt{1-2 x} \sqrt{3+5 x}}{85349376 (2+3 x)^4}+\frac{372439373 \sqrt{1-2 x} \sqrt{3+5 x}}{512096256 (2+3 x)^3}+\frac{64983635965 \sqrt{1-2 x} \sqrt{3+5 x}}{14338695168 (2+3 x)^2}-\frac{720833 \sqrt{1-2 x} (3+5 x)^{3/2}}{508032 (2+3 x)^5}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}+\frac{47365 \sqrt{1-2 x} (3+5 x)^{5/2}}{36288 (2+3 x)^6}+\frac{\int \frac{\frac{146884711951425}{128}-\frac{34116408881625 x}{32}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{23524421760}\\ &=-\frac{75045071 \sqrt{1-2 x} \sqrt{3+5 x}}{85349376 (2+3 x)^4}+\frac{372439373 \sqrt{1-2 x} \sqrt{3+5 x}}{512096256 (2+3 x)^3}+\frac{64983635965 \sqrt{1-2 x} \sqrt{3+5 x}}{14338695168 (2+3 x)^2}+\frac{6796051494355 \sqrt{1-2 x} \sqrt{3+5 x}}{200741732352 (2+3 x)}-\frac{720833 \sqrt{1-2 x} (3+5 x)^{3/2}}{508032 (2+3 x)^5}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}+\frac{47365 \sqrt{1-2 x} (3+5 x)^{5/2}}{36288 (2+3 x)^6}+\frac{\int \frac{8164047052732725}{256 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{164670952320}\\ &=-\frac{75045071 \sqrt{1-2 x} \sqrt{3+5 x}}{85349376 (2+3 x)^4}+\frac{372439373 \sqrt{1-2 x} \sqrt{3+5 x}}{512096256 (2+3 x)^3}+\frac{64983635965 \sqrt{1-2 x} \sqrt{3+5 x}}{14338695168 (2+3 x)^2}+\frac{6796051494355 \sqrt{1-2 x} \sqrt{3+5 x}}{200741732352 (2+3 x)}-\frac{720833 \sqrt{1-2 x} (3+5 x)^{3/2}}{508032 (2+3 x)^5}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}+\frac{47365 \sqrt{1-2 x} (3+5 x)^{5/2}}{36288 (2+3 x)^6}+\frac{106656830005 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{550731776}\\ &=-\frac{75045071 \sqrt{1-2 x} \sqrt{3+5 x}}{85349376 (2+3 x)^4}+\frac{372439373 \sqrt{1-2 x} \sqrt{3+5 x}}{512096256 (2+3 x)^3}+\frac{64983635965 \sqrt{1-2 x} \sqrt{3+5 x}}{14338695168 (2+3 x)^2}+\frac{6796051494355 \sqrt{1-2 x} \sqrt{3+5 x}}{200741732352 (2+3 x)}-\frac{720833 \sqrt{1-2 x} (3+5 x)^{3/2}}{508032 (2+3 x)^5}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}+\frac{47365 \sqrt{1-2 x} (3+5 x)^{5/2}}{36288 (2+3 x)^6}+\frac{106656830005 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{275365888}\\ &=-\frac{75045071 \sqrt{1-2 x} \sqrt{3+5 x}}{85349376 (2+3 x)^4}+\frac{372439373 \sqrt{1-2 x} \sqrt{3+5 x}}{512096256 (2+3 x)^3}+\frac{64983635965 \sqrt{1-2 x} \sqrt{3+5 x}}{14338695168 (2+3 x)^2}+\frac{6796051494355 \sqrt{1-2 x} \sqrt{3+5 x}}{200741732352 (2+3 x)}-\frac{720833 \sqrt{1-2 x} (3+5 x)^{3/2}}{508032 (2+3 x)^5}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{24 (2+3 x)^8}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1008 (2+3 x)^7}+\frac{47365 \sqrt{1-2 x} (3+5 x)^{5/2}}{36288 (2+3 x)^6}-\frac{106656830005 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{275365888 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.289434, size = 249, normalized size = 0.93 \[ \frac{1}{56} \left (\frac{999 (1-2 x)^{7/2} (5 x+3)^{7/2}}{98 (3 x+2)^7}+\frac{3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{(3 x+2)^8}+\frac{12041 \left (614656 (1-2 x)^{5/2} (5 x+3)^{7/2}+11 (3 x+2) \left (307328 (1-2 x)^{3/2} (5 x+3)^{7/2}+11 (3 x+2) \left (115248 \sqrt{1-2 x} (5 x+3)^{7/2}-11 (3 x+2) \left (2744 \sqrt{1-2 x} (5 x+3)^{5/2}+55 (3 x+2) \left (7 \sqrt{1-2 x} \sqrt{5 x+3} (169 x+108)+363 \sqrt{7} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )\right )\right )\right )\right )}{103262208 (3 x+2)^6}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^9,x]

[Out]

((3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))/(2 + 3*x)^8 + (999*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))/(98*(2 + 3*x)^7) + (1
2041*(614656*(1 - 2*x)^(5/2)*(3 + 5*x)^(7/2) + 11*(2 + 3*x)*(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])]))))))/(103262208*(2 + 3*x)^6))/56

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Maple [B]  time = 0.022, size = 442, normalized size = 1.7 \begin{align*}{\frac{1}{11565367296\, \left ( 2+3\,x \right ) ^{8}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2099326384988415\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{8}+11196407386604880\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{7}+26124950568744720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{-10\,{x}^{2}-x+3}{x}^{7}+34833267424992960\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+4034288656628160\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}+29027722854160800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+8147042016430184\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+15481452188885760\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+9141713903858144\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+5160484062961920\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+6155835481632480\,{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+2487632843997952\,{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)^9,x)

[Out]

1/11565367296*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2099326384988415*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))*x^8+11196407386604880*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^7+2612495056874472
0*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+856302488288730*(-10*x^2-x+3)^(1/2)*x^7+34833
267424992960*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+4034288656628160*(-10*x^2-x+3)^(1/
2)*x^6+29027722854160800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+8147042016430184*x^5*(
-10*x^2-x+3)^(1/2)+15481452188885760*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+9141713903
858144*x^4*(-10*x^2-x+3)^(1/2)+5160484062961920*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2
+6155835481632480*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+2487632843997952*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)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2
+3*x)^8

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Maxima [A]  time = 3.61573, size = 552, normalized size = 2.07 \begin{align*} \frac{39793036595}{30359089152} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{56 \,{\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )}} + \frac{999 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{5488 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac{12041 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{21952 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{445517 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{307328 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{52823867 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{17210368 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{984147053 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{240945152 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{7958607319 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{6746464256 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{712927441325}{20239392768} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1368574460935}{40478785536} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{1321083986311 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{121436356608 \,{\left (3 \, x + 2\right )}} + \frac{163070359925}{963780608} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{106656830005}{3855122432} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{143678209015}{1927561216} \, \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)^9,x, algorithm="maxima")

[Out]

39793036595/30359089152*(-10*x^2 - x + 3)^(5/2) + 3/56*(-10*x^2 - x + 3)^(7/2)/(6561*x^8 + 34992*x^7 + 81648*x
^6 + 108864*x^5 + 90720*x^4 + 48384*x^3 + 16128*x^2 + 3072*x + 256) + 999/5488*(-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) + 12041/21952*(-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) + 445517/307328*(-10*x^2 - x + 3)^(7/
2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 52823867/17210368*(-10*x^2 - x + 3)^(7/2)/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16) + 984147053/240945152*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 7
958607319/6746464256*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 712927441325/20239392768*(-10*x^2 - x + 3)^(
3/2)*x + 1368574460935/40478785536*(-10*x^2 - x + 3)^(3/2) - 1321083986311/121436356608*(-10*x^2 - x + 3)^(5/2
)/(3*x + 2) + 163070359925/963780608*sqrt(-10*x^2 - x + 3)*x + 106656830005/3855122432*sqrt(7)*arcsin(37/11*x/
abs(3*x + 2) + 20/11/abs(3*x + 2)) - 143678209015/1927561216*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 2.23241, size = 697, normalized size = 2.61 \begin{align*} -\frac{319970490015 \, \sqrt{7}{\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\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 (61164463449195 \, x^{7} + 288163475473440 \, x^{6} + 581931572602156 \, x^{5} + 652979564561296 \, x^{4} + 439702534402320 \, x^{3} + 177688060285568 \, x^{2} + 39899303549504 \, x + 3840133416192\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{11565367296 \,{\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\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)^9,x, algorithm="fricas")

[Out]

-1/11565367296*(319970490015*sqrt(7)*(6561*x^8 + 34992*x^7 + 81648*x^6 + 108864*x^5 + 90720*x^4 + 48384*x^3 +
16128*x^2 + 3072*x + 256)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*
(61164463449195*x^7 + 288163475473440*x^6 + 581931572602156*x^5 + 652979564561296*x^4 + 439702534402320*x^3 +
177688060285568*x^2 + 39899303549504*x + 3840133416192)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(6561*x^8 + 34992*x^7 +
81648*x^6 + 108864*x^5 + 90720*x^4 + 48384*x^3 + 16128*x^2 + 3072*x + 256)

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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)**9,x)

[Out]

Timed out

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Giac [B]  time = 6.68281, size = 841, normalized size = 3.15 \begin{align*} \text{result too large to display} \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)^9,x, algorithm="giac")

[Out]

21331366001/7710244864*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)))) - 8857805/413048832*(36123*sqrt(10)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^15 + 7
7544040*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)))^13 + 72311503040*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 - 37368091174400*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 - 10615979648512000*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)))^7 - 15873821147
34080000*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 - 133456146460672000000*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)))^3 - 4874050566389760000000*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)^8