3.2467 \(\int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx\)
Optimal. Leaf size=151 \[ \frac {32735 \sqrt {1-2 x} \sqrt {5 x+3}}{21952 (3 x+2)}+\frac {305 \sqrt {1-2 x} \sqrt {5 x+3}}{1568 (3 x+2)^2}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{56 (3 x+2)^3}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}-\frac {375265 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \]
[Out]
-375265/153664*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1/28*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x
)^4+1/56*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+305/1568*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+32735/21952*(1-2
*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 151, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used =
{99, 151, 12, 93, 204} \[ \frac {32735 \sqrt {1-2 x} \sqrt {5 x+3}}{21952 (3 x+2)}+\frac {305 \sqrt {1-2 x} \sqrt {5 x+3}}{1568 (3 x+2)^2}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{56 (3 x+2)^3}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{28 (3 x+2)^4}-\frac {375265 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[3 + 5*x]/(Sqrt[1 - 2*x]*(2 + 3*x)^5),x]
[Out]
-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28*(2 + 3*x)^4) + (Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(56*(2 + 3*x)^3) + (305*Sqrt[1
- 2*x]*Sqrt[3 + 5*x])/(1568*(2 + 3*x)^2) + (32735*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (375265*Ar
cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*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 99
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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])
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 {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {1}{28} \int \frac {\frac {47}{2}+30 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {1}{588} \int \frac {\frac {1575}{4}-210 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {\int \frac {\frac {143745}{8}-\frac {32025 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{8232}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {32735 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}+\frac {\int \frac {7880565}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{57624}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {32735 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}+\frac {375265 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{43904}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {32735 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}+\frac {375265 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{21952}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)^4}+\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {305 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {32735 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {375265 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 79, normalized size = 0.52 \[ \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (883845 x^3+1806120 x^2+1230876 x+278960\right )}{(3 x+2)^4}-375265 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[3 + 5*x]/(Sqrt[1 - 2*x]*(2 + 3*x)^5),x]
[Out]
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(278960 + 1230876*x + 1806120*x^2 + 883845*x^3))/(2 + 3*x)^4 - 375265*Sqrt[7]*
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/153664
________________________________________________________________________________________
fricas [A] time = 1.16, size = 116, normalized size = 0.77 \[ -\frac {375265 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (883845 \, x^{3} + 1806120 \, x^{2} + 1230876 \, x + 278960\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{307328 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((3+5*x)^(1/2)/(2+3*x)^5/(1-2*x)^(1/2),x, algorithm="fricas")
[Out]
-1/307328*(375265*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(883845*x^3 + 1806120*x^2 + 1230876*x + 278960)*sqrt(5*x + 3)*sqrt(-2
*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
________________________________________________________________________________________
giac [B] time = 3.23, size = 368, normalized size = 2.44 \[ \frac {75053}{614656} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {55 \, \sqrt {10} {\left (6823 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 7629720 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 1915892160 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {149136243200 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {596544972800 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{10976 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((3+5*x)^(1/2)/(2+3*x)^5/(1-2*x)^(1/2),x, algorithm="giac")
[Out]
75053/614656*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 55/10976*sqrt(10)*(6823*((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 - 7629720*((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 - 1915892160*(
(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 -
149136243200*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 596544972800*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(-10
*x + 5) - sqrt(22)))^2 + 280)^4
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maple [B] time = 0.02, size = 250, normalized size = 1.66 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (30396465 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+81057240 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+12373830 \sqrt {-10 x^{2}-x +3}\, x^{3}+81057240 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+25285680 \sqrt {-10 x^{2}-x +3}\, x^{2}+36025440 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+17232264 \sqrt {-10 x^{2}-x +3}\, x +6004240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3905440 \sqrt {-10 x^{2}-x +3}\right )}{307328 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((5*x+3)^(1/2)/(3*x+2)^5/(-2*x+1)^(1/2),x)
[Out]
1/307328*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(30396465*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
+81057240*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+81057240*7^(1/2)*x^2*arctan(1/14*(37*
x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+12373830*(-10*x^2-x+3)^(1/2)*x^3+36025440*7^(1/2)*x*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))+25285680*(-10*x^2-x+3)^(1/2)*x^2+6004240*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))+17232264*(-10*x^2-x+3)^(1/2)*x+3905440*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^4
________________________________________________________________________________________
maxima [A] time = 1.33, size = 143, normalized size = 0.95 \[ \frac {375265}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {\sqrt {-10 \, x^{2} - x + 3}}{56 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {305 \, \sqrt {-10 \, x^{2} - x + 3}}{1568 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {32735 \, \sqrt {-10 \, x^{2} - x + 3}}{21952 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((3+5*x)^(1/2)/(2+3*x)^5/(1-2*x)^(1/2),x, algorithm="maxima")
[Out]
375265/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/28*sqrt(-10*x^2 - x + 3)/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16) + 1/56*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 305/1568*sqrt(-10*
x^2 - x + 3)/(9*x^2 + 12*x + 4) + 32735/21952*sqrt(-10*x^2 - x + 3)/(3*x + 2)
________________________________________________________________________________________
mupad [B] time = 18.54, size = 1509, normalized size = 9.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((5*x + 3)^(1/2)/((1 - 2*x)^(1/2)*(3*x + 2)^5),x)
[Out]
((229171111*((1 - 2*x)^(1/2) - 1)^7)/(15312500*(3^(1/2) - (5*x + 3)^(1/2))^7) - (1983904*((1 - 2*x)^(1/2) - 1)
^3)/(2734375*(3^(1/2) - (5*x + 3)^(1/2))^3) - (13839741*((1 - 2*x)^(1/2) - 1)^5)/(7656250*(3^(1/2) - (5*x + 3)
^(1/2))^5) - (734066*((1 - 2*x)^(1/2) - 1))/(133984375*(3^(1/2) - (5*x + 3)^(1/2))) - (229171111*((1 - 2*x)^(1
/2) - 1)^9)/(6125000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (13839741*((1 - 2*x)^(1/2) - 1)^11)/(490000*(3^(1/2) - (
5*x + 3)^(1/2))^11) + (61997*((1 - 2*x)^(1/2) - 1)^13)/(875*(3^(1/2) - (5*x + 3)^(1/2))^13) + (367033*((1 - 2*
x)^(1/2) - 1)^15)/(109760*(3^(1/2) - (5*x + 3)^(1/2))^15) + (1111291*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(1914062
5*(3^(1/2) - (5*x + 3)^(1/2))^2) + (746274*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(546875*(3^(1/2) - (5*x + 3)^(1/2)
)^4) - (7569447*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(1531250*(3^(1/2) - (5*x + 3)^(1/2))^6) + (631898231*3^(1/2)*
((1 - 2*x)^(1/2) - 1)^8)/(53593750*(3^(1/2) - (5*x + 3)^(1/2))^8) - (7569447*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)
/(245000*(3^(1/2) - (5*x + 3)^(1/2))^10) + (373137*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(7000*(3^(1/2) - (5*x + 3
)^(1/2))^12) + (1111291*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(78400*(3^(1/2) - (5*x + 3)^(1/2))^14))/((45056*((1
- 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (294784*((1 - 2*x)^(1/2) - 1)^4)/(390625*(3^(1/2
) - (5*x + 3)^(1/2))^4) - (1921024*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (5828656*
((1 - 2*x)^(1/2) - 1)^8)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^8) - (480256*((1 - 2*x)^(1/2) - 1)^10)/(15625*(3^
(1/2) - (5*x + 3)^(1/2))^10) + (18424*((1 - 2*x)^(1/2) - 1)^12)/(625*(3^(1/2) - (5*x + 3)^(1/2))^12) + (704*((
1 - 2*x)^(1/2) - 1)^14)/(25*(3^(1/2) - (5*x + 3)^(1/2))^14) + ((1 - 2*x)^(1/2) - 1)^16/(3^(1/2) - (5*x + 3)^(1
/2))^16 - (21504*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^3) + (48384*3^(1/2)*((1 -
2*x)^(1/2) - 1)^5)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^5) - (4992*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(390625*(3^
(1/2) - (5*x + 3)^(1/2))^7) + (2496*3^(1/2)*((1 - 2*x)^(1/2) - 1)^9)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^9) - (
6048*3^(1/2)*((1 - 2*x)^(1/2) - 1)^11)/(625*(3^(1/2) - (5*x + 3)^(1/2))^11) + (672*3^(1/2)*((1 - 2*x)^(1/2) -
1)^13)/(25*(3^(1/2) - (5*x + 3)^(1/2))^13) + (24*3^(1/2)*((1 - 2*x)^(1/2) - 1)^15)/(5*(3^(1/2) - (5*x + 3)^(1/
2))^15) - (3072*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(390625*(3^(1/2) - (5*x + 3)^(1/2))) + 256/390625) - (375265*7^
(1/2)*atan(((375265*7^(1/2)*((225159*3^(1/2))/68600 + (225159*((1 - 2*x)^(1/2) - 1))/(137200*(3^(1/2) - (5*x +
3)^(1/2))) - (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 -
2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*375265i)/307328 - (225159*3^(1/2)*((1 - 2*x)^(1/
2) - 1)^2)/(27440*(3^(1/2) - (5*x + 3)^(1/2))^2)))/307328 + (375265*7^(1/2)*((225159*3^(1/2))/68600 + (225159*
((1 - 2*x)^(1/2) - 1))/(137200*(3^(1/2) - (5*x + 3)^(1/2))) + (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(
1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*
375265i)/307328 - (225159*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(27440*(3^(1/2) - (5*x + 3)^(1/2))^2)))/307328)/((7
^(1/2)*((225159*3^(1/2))/68600 + (225159*((1 - 2*x)^(1/2) - 1))/(137200*(3^(1/2) - (5*x + 3)^(1/2))) - (7^(1/2
)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125
*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*375265i)/307328 - (225159*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(27440*(3^
(1/2) - (5*x + 3)^(1/2))^2))*375265i)/307328 - (7^(1/2)*((225159*3^(1/2))/68600 + (225159*((1 - 2*x)^(1/2) - 1
))/(137200*(3^(1/2) - (5*x + 3)^(1/2))) + (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/
2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*375265i)/307328 - (2
25159*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(27440*(3^(1/2) - (5*x + 3)^(1/2))^2))*375265i)/307328 + (5632952809*((
1 - 2*x)^(1/2) - 1)^2)/(240945152*(3^(1/2) - (5*x + 3)^(1/2))^2) + 5632952809/602362880)))/153664
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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((3+5*x)**(1/2)/(2+3*x)**5/(1-2*x)**(1/2),x)
[Out]
Timed out
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