3.11.25 \(\int \frac {1+4 x}{\sqrt {9+120 x+64 x^2+64 x^3+64 x^4}} \, dx\) [1025]
Optimal. Leaf size=100 \[ \frac {1}{16} \log \left (921+2864 x+9280 x^2+13440 x^3+17024 x^4+19456 x^5+12288 x^6+8192 x^7+4096 x^8+\sqrt {9+120 x+64 x^2+64 x^3+64 x^4} \left (179+444 x+744 x^2+1280 x^3+960 x^4+768 x^5+512 x^6\right )\right ) \]
[Out]
1/16*ln(921+2864*x+9280*x^2+13440*x^3+17024*x^4+19456*x^5+12288*x^6+8192*x^7+4096*x^8+(512*x^6+768*x^5+960*x^4
+1280*x^3+744*x^2+444*x+179)*(64*x^4+64*x^3+64*x^2+120*x+9)^(1/2))
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(243\) vs. \(2(100)=200\).
time = 0.09, 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 = {2108, 2107}
\begin {gather*} \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+960 \sqrt {64 x^4+64 x^3+64 x^2+120 x+9} x^4+1280 \sqrt {64 x^4+64 x^3+64 x^2+120 x+9} x^3+744 \sqrt {64 x^4+64 x^3+64 x^2+120 x+9} 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}+512 \sqrt {64 x^4+64 x^3+64 x^2+120 x+9} x^6+768 \sqrt {64 x^4+64 x^3+64 x^2+120 x+9} x^5+2864 x+921\right ) \end {gather*}
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 2107
Int[(x_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (e_.)*(x_)^4], x_Symbol] :> With[{Px = (1/320)*(33*b^2*c + 6*
a*c^2 + 40*a^2*e) - (22/5)*a*c*e*x^2 + (22/15)*b*c*e*x^3 + (1/4)*e*(5*c^2 + 4*a*e)*x^4 + (4/3)*b*e^2*x^5 + 2*c
*e^2*x^6 + e^3*x^8}, Simp[(1/(8*Rt[e, 2]))*Log[Px + Dist[1/(8*Rt[e, 2]*x), D[Px, x], x]*Sqrt[a + b*x + c*x^2 +
e*x^4]], x]] /; FreeQ[{a, b, c, e}, x] && EqQ[71*c^2 + 100*a*e, 0] && EqQ[1152*c^3 - 125*b^2*e, 0]
Rule 2108
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]
Rubi steps
\begin {align*} \int \frac {1+4 x}{\sqrt {9+120 x+64 x^2+64 x^3+64 x^4}} \, dx &=4 \text {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*}
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Mathematica [A]
time = 5.04, size = 100, normalized size = 1.00 \begin {gather*} -\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+\sqrt {9+120 x+64 x^2+64 x^3+64 x^4} \left (179+444 x+744 x^2+1280 x^3+960 x^4+768 x^5+512 x^6\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(1 + 4*x)/Sqrt[9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4],x]
[Out]
-1/16*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 + Sqr
t[9 + 120*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)]
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.04, size = 2992, normalized size = 29.92
| | |
| method |
result |
size |
| | |
| trager |
\(\frac {\ln \left (-4096 x^{8}-512 x^{6} \sqrt {64 x^{4}+64 x^{3}+64 x^{2}+120 x +9}-8192 x^{7}-768 x^{5} \sqrt {64 x^{4}+64 x^{3}+64 x^{2}+120 x +9}-12288 x^{6}-960 x^{4} \sqrt {64 x^{4}+64 x^{3}+64 x^{2}+120 x +9}-19456 x^{5}-1280 x^{3} \sqrt {64 x^{4}+64 x^{3}+64 x^{2}+120 x +9}-17024 x^{4}-744 x^{2} \sqrt {64 x^{4}+64 x^{3}+64 x^{2}+120 x +9}-13440 x^{3}-444 x \sqrt {64 x^{4}+64 x^{3}+64 x^{2}+120 x +9}-9280 x^{2}-179 \sqrt {64 x^{4}+64 x^{3}+64 x^{2}+120 x +9}-2864 x -921\right )}{16}\) |
\(228\) |
| default |
\(\text {Expression too large to display}\) |
\(2992\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(2992\) |
| | |
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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,method=_RETURNVERBOSE)
[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...
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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 = 0.42, size = 97, normalized size = 0.97 \begin {gather*} \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 {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 x + 1}{\sqrt {64 x^{4} + 64 x^{3} + 64 x^{2} + 120 x + 9}}\, dx \end {gather*}
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {4\,x+1}{\sqrt {64\,x^4+64\,x^3+64\,x^2+120\,x+9}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((4*x + 1)/(120*x + 64*x^2 + 64*x^3 + 64*x^4 + 9)^(1/2),x)
[Out]
int((4*x + 1)/(120*x + 64*x^2 + 64*x^3 + 64*x^4 + 9)^(1/2), x)
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