3.14.5 \(\int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\)
Optimal. Leaf size=94 \[ -\frac {2 \sqrt {-2 a-2 b-c} \tan ^{-1}\left (\frac {x \sqrt {-2 a-2 b-c}}{-\sqrt {a x^4+a+b x^3+b x+c x^2}+\sqrt {a} x^2-2 \sqrt {a} x+\sqrt {a}}\right )}{2 a+2 b+c} \]
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Rubi [F] time = 0.50, 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}{(-1+x) \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)/((-1 + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]
[Out]
Defer[Int][1/Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4], x] + 2*Defer[Int][1/((-1 + x)*Sqrt[a + b*x + c*x^2 + b*x^3
+ a*x^4]), x]
Rubi steps
\begin {align*} \int \frac {1+x}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx &=\int \left (\frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {2}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\\ &=2 \int \frac {1}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\\ \end {align*}
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Mathematica [C] time = 6.40, size = 6023, normalized size = 64.07 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(1 + x)/((-1 + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]
[Out]
Result too large to show
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IntegrateAlgebraic [A] time = 0.76, size = 94, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {-2 a-2 b-c} \tan ^{-1}\left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 a+2 b+c} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(1 + x)/((-1 + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]
[Out]
(-2*Sqrt[-2*a - 2*b - c]*ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a] - 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x +
c*x^2 + b*x^3 + a*x^4])])/(2*a + 2*b + c)
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fricas [B] time = 0.84, size = 382, normalized size = 4.06 \begin {gather*} \left [\frac {\log \left (\frac {{\left (24 \, a^{2} + 16 \, a b + b^{2} + 4 \, a c\right )} x^{4} - 4 \, {\left (8 \, a^{2} - 4 \, a b - 3 \, b^{2} - 2 \, {\left (2 \, a + b\right )} c\right )} x^{3} + 2 \, {\left (24 \, a^{2} + 3 \, b^{2} - 4 \, {\left (a - 2 \, b\right )} c + 4 \, c^{2}\right )} x^{2} - 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left ({\left (4 \, a + b\right )} x^{2} - 2 \, {\left (2 \, a - b - c\right )} x + 4 \, a + b\right )} \sqrt {2 \, a + 2 \, b + c} + 24 \, a^{2} + 16 \, a b + b^{2} + 4 \, a c - 4 \, {\left (8 \, a^{2} - 4 \, a b - 3 \, b^{2} - 2 \, {\left (2 \, a + b\right )} c\right )} x}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )}{2 \, \sqrt {2 \, a + 2 \, b + c}}, \frac {\sqrt {-2 \, a - 2 \, b - c} \arctan \left (\frac {\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left ({\left (4 \, a + b\right )} x^{2} - 2 \, {\left (2 \, a - b - c\right )} x + 4 \, a + b\right )} \sqrt {-2 \, a - 2 \, b - c}}{2 \, {\left ({\left (2 \, a^{2} + 2 \, a b + a c\right )} x^{4} + {\left (2 \, a b + 2 \, b^{2} + b c\right )} x^{3} + {\left (2 \, {\left (a + b\right )} c + c^{2}\right )} x^{2} + 2 \, a^{2} + 2 \, a b + a c + {\left (2 \, a b + 2 \, b^{2} + b c\right )} x\right )}}\right )}{2 \, a + 2 \, b + c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+x)/(-1+x)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
[Out]
[1/2*log(((24*a^2 + 16*a*b + b^2 + 4*a*c)*x^4 - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x^3 + 2*(24*a^2 + 3*
b^2 - 4*(a - 2*b)*c + 4*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x
+ 4*a + b)*sqrt(2*a + 2*b + c) + 24*a^2 + 16*a*b + b^2 + 4*a*c - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x)/
(x^4 - 4*x^3 + 6*x^2 - 4*x + 1))/sqrt(2*a + 2*b + c), sqrt(-2*a - 2*b - c)*arctan(1/2*sqrt(a*x^4 + b*x^3 + c*x
^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(-2*a - 2*b - c)/((2*a^2 + 2*a*b + a*c)*x^4 +
(2*a*b + 2*b^2 + b*c)*x^3 + (2*(a + b)*c + c^2)*x^2 + 2*a^2 + 2*a*b + a*c + (2*a*b + 2*b^2 + b*c)*x))/(2*a + 2
*b + c)]
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+x)/(-1+x)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="giac")
[Out]
Timed out
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maple [C] time = 0.15, size = 2813, normalized size = 29.93
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(2813\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(2813\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((1+x)/(-1+x)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
[Out]
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))*(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-RootO
f(_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,index=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-R
ootOf(_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,index=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)*Ell
ipticF(((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,in
dex=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
))+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))*((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-Ro
otOf(_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,index=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,index=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)-1)*(EllipticF(((RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+a,index=4)-RootO
f(_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,inde
x=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))*(-Root
Of(_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))+(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)-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))/(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-Root
Of(_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)-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))/(RootOf(_Z^4*a+_Z^3*b+_Z^2*c+_Z*b+
a,index=1)-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)),((Root
Of(_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)))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((1+x)/(-1+x)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
[Out]
integrate((x + 1)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(x - 1)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x+1}{\left (x-1\right )\,\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 + 1)/((x - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)
[Out]
int((x + 1)/((x - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\left (x - 1\right ) \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((1+x)/(-1+x)/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2),x)
[Out]
Integral((x + 1)/((x - 1)*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2)), x)
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