3.1025 \(\int \frac{1+4 x}{\sqrt{9+120 x+64 x^2+64 x^3+64 x^4}} \, dx\)
Optimal. Leaf size=100 \[ \frac{1}{16} \log \left (4096 x^8+8192 x^7+12288 x^6+19456 x^5+17024 x^4+13440 x^3+9280 x^2+\sqrt{64 x^4+64 x^3+64 x^2+120 x+9} \left (512 x^6+768 x^5+960 x^4+1280 x^3+744 x^2+444 x+179\right )+2864 x+921\right ) \]
[Out]
Log[921 + 2864*x + 9280*x^2 + 13440*x^3 + 17024*x^4 + 19456*x^5 + 12288*x^6 + 8192*x^7 + 4096*x^8 + Sqrt[9 + 1
20*x + 64*x^2 + 64*x^3 + 64*x^4]*(179 + 444*x + 744*x^2 + 1280*x^3 + 960*x^4 + 768*x^5 + 512*x^6)]/16
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Rubi [B] time = 0.138549, antiderivative size = 243, normalized size of antiderivative =
2.43, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules
used = {2083, 2082} \[ \frac{1}{16} \log \left (4096 x^8+8192 x^7+512 \sqrt{64 x^4+64 x^3+64 x^2+120 x+9} x^6+12288 x^6+768 \sqrt{64 x^4+64 x^3+64 x^2+120 x+9} x^5+19456 x^5+960 \sqrt{64 x^4+64 x^3+64 x^2+120 x+9} x^4+17024 x^4+1280 \sqrt{64 x^4+64 x^3+64 x^2+120 x+9} x^3+13440 x^3+744 \sqrt{64 x^4+64 x^3+64 x^2+120 x+9} x^2+9280 x^2+444 \sqrt{64 x^4+64 x^3+64 x^2+120 x+9} x+179 \sqrt{64 x^4+64 x^3+64 x^2+120 x+9}+2864 x+921\right ) \]
Antiderivative was successfully verified.
[In]
Int[(1 + 4*x)/Sqrt[9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4],x]
[Out]
Log[921 + 2864*x + 9280*x^2 + 13440*x^3 + 17024*x^4 + 19456*x^5 + 12288*x^6 + 8192*x^7 + 4096*x^8 + 179*Sqrt[9
+ 120*x + 64*x^2 + 64*x^3 + 64*x^4] + 444*x*Sqrt[9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4] + 744*x^2*Sqrt[9 + 120
*x + 64*x^2 + 64*x^3 + 64*x^4] + 1280*x^3*Sqrt[9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4] + 960*x^4*Sqrt[9 + 120*x
+ 64*x^2 + 64*x^3 + 64*x^4] + 768*x^5*Sqrt[9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4] + 512*x^6*Sqrt[9 + 120*x + 64
*x^2 + 64*x^3 + 64*x^4]]/16
Rule 2083
Int[((A_) + (B_.)*(x_))/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4], x_Symbol] :> Dis
t[B, Subst[Int[x/Sqrt[(-3*d^4 + 16*c*d^2*e - 64*b*d*e^2 + 256*a*e^3)/(256*e^3) + ((d^3 - 4*c*d*e + 8*b*e^2)*x)
/(8*e^2) - ((3*d^2 - 8*c*e)*x^2)/(8*e) + e*x^4], x], x, d/(4*e) + x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] &&
EqQ[B*d - 4*A*e, 0] && EqQ[d*(141*d^3 - 752*c*d*e - 400*b*e^2) + 16*e^2*(71*c^2 + 100*a*e), 0] && EqQ[144*(3*
d^2 - 8*c*e)^3 + 125*(d^3 - 4*c*d*e + 8*b*e^2)^2, 0]
Rule 2082
Int[(x_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (e_.)*(x_)^4], x_Symbol] :> With[{Px = (1*(33*b^2*c + 6*a*c^2
+ 40*a^2*e))/320 - (22*a*c*e*x^2)/5 + (22*b*c*e*x^3)/15 + (1*e*(5*c^2 + 4*a*e)*x^4)/4 + (4*b*e^2*x^5)/3 + 2*c
*e^2*x^6 + e^3*x^8}, Simp[(1*Log[Px + Dist[1/(8*Rt[e, 2]*x), D[Px, x], x]*Sqrt[a + b*x + c*x^2 + e*x^4]])/(8*R
t[e, 2]), x]] /; FreeQ[{a, b, c, e}, x] && EqQ[71*c^2 + 100*a*e, 0] && EqQ[1152*c^3 - 125*b^2*e, 0]
Rubi steps
\begin{align*} \int \frac{1+4 x}{\sqrt{9+120 x+64 x^2+64 x^3+64 x^4}} \, dx &=4 \operatorname{Subst}\left (\int \frac{x}{\sqrt{-\frac{71}{4}+96 x+40 x^2+64 x^4}} \, dx,x,\frac{1}{4}+x\right )\\ &=\frac{1}{16} \log \left (921+2864 x+9280 x^2+13440 x^3+17024 x^4+19456 x^5+12288 x^6+8192 x^7+4096 x^8+179 \sqrt{9+120 x+64 x^2+64 x^3+64 x^4}+444 x \sqrt{9+120 x+64 x^2+64 x^3+64 x^4}+744 x^2 \sqrt{9+120 x+64 x^2+64 x^3+64 x^4}+1280 x^3 \sqrt{9+120 x+64 x^2+64 x^3+64 x^4}+960 x^4 \sqrt{9+120 x+64 x^2+64 x^3+64 x^4}+768 x^5 \sqrt{9+120 x+64 x^2+64 x^3+64 x^4}+512 x^6 \sqrt{9+120 x+64 x^2+64 x^3+64 x^4}\right )\\ \end{align*}
Mathematica [C] time = 6.10326, size = 2787, normalized size = 27.87 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(1 + 4*x)/Sqrt[9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4],x]
[Out]
(8*(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])^2*(-(EllipticF[ArcSin[Sqrt[((x - Root[9 + 120
*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9
+ 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2,
0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4
& , 4, 0]))]], -(((Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1
^3 + 64*#1^4 & , 3, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 +
64*#1^3 + 64*#1^4 & , 4, 0]))/((-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 +
64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120
*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])))]*Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0]) + Ell
ipticPi[(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#
1^4 & , 4, 0])/(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3
+ 64*#1^4 & , 4, 0]), ArcSin[Sqrt[((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0])*(Root[9 + 12
0*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((x -
Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1
, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))]], -(((Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 +
64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#
1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/((-Root[9 + 120*#1 + 64*#1
^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])*(Root[9 + 120*#1 +
64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])))]*(-Root[9
+ 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0]))*S
qrt[(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 +
64*#1^4 & , 2, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*
#1^3 + 64*#1^4 & , 3, 0]))]*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1
^2 + 64*#1^3 + 64*#1^4 & , 4, 0])*Sqrt[((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0])*(Root[9
+ 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))/(
(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 &
, 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))]*Sqrt[(x - Root[9 + 120*#1 + 64*#1^2 + 64
*#1^3 + 64*#1^4 & , 4, 0])/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(-Root[9 + 120*#1 +
64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))])/(Sqrt[9 +
120*x + 64*x^2 + 64*x^3 + 64*x^4]*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 +
64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])) + (2*EllipticF[ArcSin[Sqrt[((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3
+ 64*#1^4 & , 1, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] + Root[9 + 120*#1 + 64*#1^2 + 6
4*#1^3 + 64*#1^4 & , 4, 0]))/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(-Root[9 + 120*#1
+ 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))]], ((Root[
9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])*
(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4
, 0]))/((Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1
^4 & , 3, 0])*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 +
64*#1^4 & , 4, 0]))]*(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])^2*Sqrt[(x - Root[9 + 120*#
1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0])/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(-Ro
ot[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 3, 0
]))]*(Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4
& , 4, 0])*Sqrt[(x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0])/((x - Root[9 + 120*#1 + 64*#1^2
+ 64*#1^3 + 64*#1^4 & , 2, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0] + Root[9 + 120*#1 + 6
4*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))]*Sqrt[((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 1, 0])*(-
Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4,
0]))/((x - Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0])*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 6
4*#1^4 & , 1, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 4, 0]))])/(Sqrt[9 + 120*x + 64*x^2 + 64*x
^3 + 64*x^4]*(-Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 + 64*#1^4 & , 2, 0] + Root[9 + 120*#1 + 64*#1^2 + 64*#1^3 +
64*#1^4 & , 4, 0]))
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Maple [C] time = 0.816, size = 2992, normalized size = 29.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1+4*x)/(64*x^4+64*x^3+64*x^2+120*x+9)^(1/2),x)
[Out]
1/4*(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4))*((1/
2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2))*(x-1/2*Root
Of(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^
4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)))^(1/2)*(x-1/2*RootOf(
4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2))^2*((1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^
4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=3))/(1/2*RootOf(4*_Z^4+8*
_Z^3+16*_Z^2+60*_Z+9,index=3)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(x-1/2*RootOf(4*_Z^4+8*_Z^3+1
6*_Z^2+60*_Z+9,index=2)))^(1/2)*((1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^4+8*_Z^3+1
6*_Z^2+60*_Z+9,index=1))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z
^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*
_Z+9,index=2)))^(1/2)/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_
Z+9,index=2))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,inde
x=1))/((x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index
=2))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=3))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4
)))^(1/2)*EllipticF(((1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z
+9,index=2))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,i
ndex=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2
)))^(1/2),((1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=3
))*(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4))/(1/2*
RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=3))/(1/2*RootOf(4
*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)))^(1/2))+(1/2*RootOf(4
*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4))*((1/2*RootOf(4*_Z^4+8
*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+
16*_Z^2+60*_Z+9,index=1))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+
60*_Z+9,index=1))/(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)))^(1/2)*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*
_Z^2+60*_Z+9,index=2))^2*((1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+
60*_Z+9,index=1))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=3))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_
Z+9,index=3)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,in
dex=2)))^(1/2)*((1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,in
dex=1))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=
4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)))^(
1/2)/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2))/(1/
2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/((x-1/2*Roo
tOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2))*(x-1/2*RootO
f(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=3))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)))^(1/2)*(1/2*Ro
otOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)*EllipticF(((1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*
RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(1/2*Root
Of(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(x-1/2*RootOf(4*_
Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)))^(1/2),((1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_
Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=3))*(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1)-1/2*RootOf(4*_Z^4+8*_Z
^3+16*_Z^2+60*_Z+9,index=4))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z
^2+60*_Z+9,index=3))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z
+9,index=4)))^(1/2))+(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z
+9,index=2))*EllipticPi(((1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+6
0*_Z+9,index=2))*(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z
+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1))/(x-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,ind
ex=2)))^(1/2),(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,inde
x=1))/(1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)),((
1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=3))*(1/2*Root
Of(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=1)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4))/(1/2*RootOf(4*_Z^
4+8*_Z^3+16*_Z^2+60*_Z+9,index=1)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=3))/(1/2*RootOf(4*_Z^4+8*_Z^3
+16*_Z^2+60*_Z+9,index=2)-1/2*RootOf(4*_Z^4+8*_Z^3+16*_Z^2+60*_Z+9,index=4)))^(1/2)))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4 \, x + 1}{\sqrt{64 \, x^{4} + 64 \, x^{3} + 64 \, x^{2} + 120 \, x + 9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+4*x)/(64*x^4+64*x^3+64*x^2+120*x+9)^(1/2),x, algorithm="maxima")
[Out]
integrate((4*x + 1)/sqrt(64*x^4 + 64*x^3 + 64*x^2 + 120*x + 9), x)
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Fricas [A] time = 1.89448, size = 292, normalized size = 2.92 \begin{align*} \frac{1}{16} \, \log \left (-4096 \, x^{8} - 8192 \, x^{7} - 12288 \, x^{6} - 19456 \, x^{5} - 17024 \, x^{4} - 13440 \, x^{3} - 9280 \, x^{2} -{\left (512 \, x^{6} + 768 \, x^{5} + 960 \, x^{4} + 1280 \, x^{3} + 744 \, x^{2} + 444 \, x + 179\right )} \sqrt{64 \, x^{4} + 64 \, x^{3} + 64 \, x^{2} + 120 \, x + 9} - 2864 \, x - 921\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+4*x)/(64*x^4+64*x^3+64*x^2+120*x+9)^(1/2),x, algorithm="fricas")
[Out]
1/16*log(-4096*x^8 - 8192*x^7 - 12288*x^6 - 19456*x^5 - 17024*x^4 - 13440*x^3 - 9280*x^2 - (512*x^6 + 768*x^5
+ 960*x^4 + 1280*x^3 + 744*x^2 + 444*x + 179)*sqrt(64*x^4 + 64*x^3 + 64*x^2 + 120*x + 9) - 2864*x - 921)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4 x + 1}{\sqrt{64 x^{4} + 64 x^{3} + 64 x^{2} + 120 x + 9}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+4*x)/(64*x**4+64*x**3+64*x**2+120*x+9)**(1/2),x)
[Out]
Integral((4*x + 1)/sqrt(64*x**4 + 64*x**3 + 64*x**2 + 120*x + 9), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4 \, x + 1}{\sqrt{64 \, x^{4} + 64 \, x^{3} + 64 \, x^{2} + 120 \, x + 9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+4*x)/(64*x^4+64*x^3+64*x^2+120*x+9)^(1/2),x, algorithm="giac")
[Out]
integrate((4*x + 1)/sqrt(64*x^4 + 64*x^3 + 64*x^2 + 120*x + 9), x)