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]

-1/24*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8+185/1008*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^7-106656830005/192756
1216*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-720833/508032*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)
^5+47365/36288*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^6-75045071/85349376*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+3
72439373/512096256*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+64983635965/14338695168*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(
2+3*x)^2+6796051494355/200741732352*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 267, normalized size of antiderivative = 1.00, 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 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 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 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 = 0.30, 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

________________________________________________________________________________________

fricas [A]  time = 1.08, size = 176, normalized size = 0.66 \[ -\frac {319970490015 \, \sqrt {7} {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (61164463449195 \, x^{7} + 288163475473440 \, x^{6} + 581931572602156 \, x^{5} + 652979564561296 \, x^{4} + 439702534402320 \, x^{3} + 177688060285568 \, x^{2} + 39899303549504 \, x + 3840133416192\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{11565367296 \, {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )}} \]

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)

________________________________________________________________________________________

giac [B]  time = 8.61, size = 600, normalized size = 2.25 \[ \frac {21331366001}{7710244864} \, \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 \, \sqrt {10} {\left (36123 \, {\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 )}^{15} + 77544040 \, {\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} + 72311503040 \, {\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} - 37368091174400 \, {\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} - 10615979648512000 \, {\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} - 1587382114734080000 \, {\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} - 133456146460672000000 \, {\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} - \frac {4874050566389760000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {19496202265559040000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{413048832 \, {\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 )}^{8}} \]

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*sqrt(10)*(36123*((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(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(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(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(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 - 1587382114734080000*((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*((s
qrt(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 - 4
874050566389760000000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 19496202265559040000000*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

________________________________________________________________________________________

maple [B]  time = 0.03, size = 442, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (2099326384988415 \sqrt {7}\, x^{8} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+11196407386604880 \sqrt {7}\, x^{7} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{7}+26124950568744720 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4034288656628160 \sqrt {-10 x^{2}-x +3}\, x^{6}+34833267424992960 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8147042016430184 \sqrt {-10 x^{2}-x +3}\, x^{5}+29027722854160800 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+9141713903858144 \sqrt {-10 x^{2}-x +3}\, x^{4}+15481452188885760 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+6155835481632480 \sqrt {-10 x^{2}-x +3}\, x^{3}+5160484062961920 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2487632843997952 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{11565367296 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)*(5*x+3)^(5/2)/(3*x+2)^9,x)

[Out]

1/11565367296*(-2*x+1)^(1/2)*(5*x+3)^(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)*x^7*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]
20*7^(1/2)*x^6*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^7+3483
3267424992960*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4034288656628160*(-10*x^2-x+3)^(1
/2)*x^6+29027722854160800*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+8147042016430184*(-10
*x^2-x+3)^(1/2)*x^5+15481452188885760*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]
3858144*(-10*x^2-x+3)^(1/2)*x^4+5160484062961920*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)+6155835481632480*(-10*x^2-x+3)^(1/2)*x^3+982949345326080*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))+2487632843997952*(-10*x^2-x+3)^(1/2)*x^2+81912445443840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+53761867826688*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(
3*x+2)^8

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 409, normalized size = 1.53 \[ \frac {39793036595}{30359089152} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{56 \, {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )}} + \frac {999 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{5488 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {12041 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{21952 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {445517 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{307328 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {52823867 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{17210368 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {984147053 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{240945152 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {7958607319 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{6746464256 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {712927441325}{20239392768} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1368574460935}{40478785536} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {1321083986311 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{121436356608 \, {\left (3 \, x + 2\right )}} + \frac {163070359925}{963780608} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {106656830005}{3855122432} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {143678209015}{1927561216} \, \sqrt {-10 \, x^{2} - x + 3} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^9} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^9,x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^9, x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________