3.15.24 \(\int \frac {1-x^4}{x^2 \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\) [1424]
Optimal. Leaf size=101 \[ -\frac {\sqrt {a+b x+c x^2+b x^3+a x^4}}{a x}+\frac {b \log (x)}{2 a^{3/2}}-\frac {b \log \left (-2 a-b x-2 a x^2+2 \sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{2 a^{3/2}} \]
[Out]
-(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/a/x+1/2*b*ln(x)/a^(3/2)-1/2*b*ln(-2*a-b*x-2*a*x^2+2*a^(1/2)*(a*x^4+b*x^3+c*x^
2+b*x+a)^(1/2))/a^(3/2)
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Rubi [F]
time = 0.49, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {1-x^4}{x^2 \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(1 - x^4)/(x^2*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]
[Out]
Defer[Int][1/(x^2*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]), x] - Defer[Int][x^2/Sqrt[a + b*x + c*x^2 + b*x^3 + a
*x^4], x]
Rubi steps
\begin {align*} \int \frac {1-x^4}{x^2 \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx &=\int \left (\frac {1}{x^2 \sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {x^2}{\sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\\ &=\int \frac {1}{x^2 \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx-\int \frac {x^2}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 167, normalized size = 1.65 \begin {gather*} \frac {-2 \sqrt {a} \left (a+b x+c x^2+b x^3+a x^4\right )+b x \sqrt {x \left (b+c x+b x^2\right )+a \left (1+x^4\right )} \log (x)-b x \sqrt {a+b x+c x^2+b x^3+a x^4} \log \left (a \left (b x+2 a \left (1+x^2\right )-2 \sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )\right )}{2 a^{3/2} x \sqrt {x \left (b+c x+b x^2\right )+a \left (1+x^4\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(1 - x^4)/(x^2*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]
[Out]
(-2*Sqrt[a]*(a + b*x + c*x^2 + b*x^3 + a*x^4) + b*x*Sqrt[x*(b + c*x + b*x^2) + a*(1 + x^4)]*Log[x] - b*x*Sqrt[
a + b*x + c*x^2 + b*x^3 + a*x^4]*Log[a*(b*x + 2*a*(1 + x^2) - 2*Sqrt[a]*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])
])/(2*a^(3/2)*x*Sqrt[x*(b + c*x + b*x^2) + a*(1 + x^4)])
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.24, size = 3402, normalized size = 33.68
| | |
| method |
result |
size |
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| default |
\(\text {Expression too large to display}\) |
\(3402\) |
| risch |
\(\text {Expression too large to display}\) |
\(3402\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(3402\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-x^4+1)/x^2/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
[Out]
-(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/a/x+b/a*(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z
^2*c+_Z*b+a,index=1))*((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2
))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+
_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)*(x-RootOf(_Z^4*a+_Z^3*b+
_Z^2*c+_Z*b+a,index=2))^2*((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,ind
ex=1))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3)-RootOf(_Z^
4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)*((RootOf(_Z^4*a+_Z^3
*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,i
ndex=4))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_
Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+
_Z^2*c+_Z*b+a,index=2))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=
1))/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)*(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)*EllipticF(((RootOf(_Z^4*a+_Z^
3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,
index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(
_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2),((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*
b+_Z^2*c+_Z*b+a,index=3))*(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,ind
ex=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^
4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2))+(-RootOf(_Z^4*a+_Z^3*b+
_Z^2*c+_Z*b+a,index=2)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))*EllipticPi(((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_
Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(R
ootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*
b+_Z^2*c+_Z*b+a,index=2)))^(1/2),(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b
+a,index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)),((RootO
f(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3))*(-RootOf(_Z^4*a+_Z^3*b+_Z^
2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3)
+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3
*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)))-b/a*(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^
2*c+_Z*b+a,index=1))*((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)
)*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_
Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)*(x-RootOf(_Z^4*a+_Z^3*b+_
Z^2*c+_Z*b+a,index=2))^2*((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,inde
x=1))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3)-RootOf(_Z^4
*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)*((RootOf(_Z^4*a+_Z^3*
b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,in
dex=4))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_Z
^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2)/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_
Z^2*c+_Z*b+a,index=2))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1
))/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)*(EllipticF(((RootOf(_Z^4*a+_Z^3
*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2))*(x-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,i
ndex=1))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(x-RootOf(_
Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2),((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b
+_Z^2*c+_Z*b+a,index=3))*(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,inde
x=1))/(-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=3)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/(-RootOf(_Z^4
*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)+RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=2)))^(1/2))+(RootOf(_Z^4*a+_Z^3*b+_Z
^2*c+_Z*b+a,index=2)-RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1))/RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=1)*
EllipticPi(((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a...
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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((-x^4+1)/x^2/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
[Out]
-integrate((x^4 - 1)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*x^2), x)
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Fricas [A]
time = 0.97, size = 216, normalized size = 2.14 \begin {gather*} \left [\frac {\sqrt {a} b x \log \left (\frac {8 \, a^{2} x^{4} + 8 \, a b x^{3} + 8 \, a b x + {\left (8 \, a^{2} + b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (2 \, a x^{2} + b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} a}{4 \, a^{2} x}, -\frac {\sqrt {-a} b x \arctan \left (\frac {2 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} \sqrt {-a}}{2 \, a x^{2} + b x + 2 \, a}\right ) + 2 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} a}{2 \, a^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-x^4+1)/x^2/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
[Out]
[1/4*(sqrt(a)*b*x*log((8*a^2*x^4 + 8*a*b*x^3 + 8*a*b*x + (8*a^2 + b^2 + 4*a*c)*x^2 + 4*sqrt(a*x^4 + b*x^3 + c*
x^2 + b*x + a)*(2*a*x^2 + b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*a)/(a^2*x
), -1/2*(sqrt(-a)*b*x*arctan(2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*sqrt(-a)/(2*a*x^2 + b*x + 2*a)) + 2*sqrt(
a*x^4 + b*x^3 + c*x^2 + b*x + a)*a)/(a^2*x)]
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {1}{x^{2} \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\right )\, dx - \int \frac {x^{2}}{\sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-x**4+1)/x**2/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2),x)
[Out]
-Integral(-1/(x**2*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2)), x) - Integral(x**2/sqrt(a*x**4 + a + b*x**3 + b*
x + c*x**2), 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((-x^4+1)/x^2/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="giac")
[Out]
integrate(-(x^4 - 1)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*x^2), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^4-1}{x^2\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(x^4 - 1)/(x^2*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)
[Out]
-int((x^4 - 1)/(x^2*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)), x)
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