3.1.55 \(\int \frac {-b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{-b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}}} \, dx\) [55]
Optimal. Leaf size=113 \[ -\frac {\sqrt {b-\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \]
[Out]
-1/2*EllipticE(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2),((b-(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^
(1/2))*(b+(-4*a*c+b^2)^(1/2))*(b-(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2)/c^(1/2)
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Rubi [A]
time = 0.15, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 87, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {21, 435}
\begin {gather*} -\frac {\sqrt {b-\sqrt {b^2-4 a c}} \left (\sqrt {b^2-4 a c}+b\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(-b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(Sqrt[1 + (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(-b +
Sqrt[b^2 - 4*a*c])]),x]
[Out]
-((Sqrt[b - Sqrt[b^2 - 4*a*c]]*(b + Sqrt[b^2 - 4*a*c])*EllipticE[ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2
- 4*a*c]]], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])])/(Sqrt[2]*Sqrt[c]))
Rule 21
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])
Rule 435
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]
Rubi steps
\begin {align*} \int \frac {-b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{-b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}}} \, dx &=\left (-b-\sqrt {b^2-4 a c}\right ) \int \frac {\sqrt {1+\frac {2 c x^2}{-b-\sqrt {b^2-4 a c}}}}{\sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}}} \, dx\\ &=-\frac {\sqrt {b-\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.56, size = 104, normalized size = 0.92 \begin {gather*} -2 i \sqrt {2} a \sqrt {\frac {c}{-b+\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{-b+\sqrt {b^2-4 a c}}} x\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(Sqrt[1 + (2*c*x^2)/(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/
(-b + Sqrt[b^2 - 4*a*c])]),x]
[Out]
(-2*I)*Sqrt[2]*a*Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(-b + Sqrt[b^2 - 4*a*c])]
*x], (b - Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c])]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2537\) vs.
\(2(92)=184\).
time = 0.47, size = 2538, normalized size = 22.46
| | |
| method |
result |
size |
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| elliptic |
\(\text {Expression too large to display}\) |
\(2538\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*c*x^2-(-4*a*c+b^2)^(1/2)-b)/(1+2*c*x^2/(-b-(-4*a*c+b^2)^(1/2)))^(1/2)/(1+2*c/(-b+(-4*a*c+b^2)^(1/2))*x^
2)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/2*(-2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*((-2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(4*a*c-b^2
)/a/c)^(1/2)*(-(2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(-2*c*x^2+(-4*a*c+b^2)^(1/2)+b)/a/c)^(1/2)/((-2*c*x^2+(-4*a*c+b^
2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/((2*c*x^2+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))^(1/2)/(-2*(
(-2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(4*a*c-b^2)/a/c)^(1/2)*c*x^2-4*(-(2*c*x^2+(-4*a
*c+b^2)^(1/2)-b)*(-2*c*x^2+(-4*a*c+b^2)^(1/2)+b)/a/c)^(1/2)*a*c+(-(2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(-2*c*x^2+(-4
*a*c+b^2)^(1/2)+b)/a/c)^(1/2)*b^2+((-2*c*x^2+(-4*a*c+b^2)^(1/2)+b)*(2*c*x^2+(-4*a*c+b^2)^(1/2)-b)*(4*a*c-b^2)/
a/c)^(1/2)*b)*(1/2*(4*a*c-b^2)/(-2*((-4*a*c+b^2)^(5/2)-(-4*a*c+b^2)^(3/2)*b^2+16*a^2*b*c^2-4*a*b^3*c)/(-b+(-4*
a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2))/a/(4*a*c-b^2))^(1/2)*(4+2*((-4*a*c+b^2)^(5/2)-(-4*a*c+b^2)^(3/2)*b^2+16
*a^2*b*c^2-4*a*b^3*c)/(-b+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2))/a/(4*a*c-b^2)*x^2)^(1/2)*(4-2*((-4*a*c+b^
2)^(5/2)-(-4*a*c+b^2)^(3/2)*b^2-16*a^2*b*c^2+4*a*b^3*c)/(-b+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2))/a/(4*a*
c-b^2)*x^2)^(1/2)/(-4*a*c+b^2-8*c^2/(-b+(-4*a*c+b^2)^(1/2))*x^2*a+2*c/(-b+(-4*a*c+b^2)^(1/2))*x^2*b^2-8*c^2*x^
2/(-b-(-4*a*c+b^2)^(1/2))*a+2*c*x^2/(-b-(-4*a*c+b^2)^(1/2))*b^2-16*c^3*x^4/(-b-(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c
+b^2)^(1/2))*a+4*c^2*x^4/(-b-(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2))*b^2)^(1/2)*EllipticF(1/2*x*(-2*((-4*a
*c+b^2)^(5/2)-(-4*a*c+b^2)^(3/2)*b^2+16*a^2*b*c^2-4*a*b^3*c)/(-b+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2))/a/
(4*a*c-b^2))^(1/2),1/2*(-4-2*(-8*c^2/(-b+(-4*a*c+b^2)^(1/2))*a+2*c/(-b+(-4*a*c+b^2)^(1/2))*b^2-8*c^2/(-b-(-4*a
*c+b^2)^(1/2))*a+2*c/(-b-(-4*a*c+b^2)^(1/2))*b^2)*((-4*a*c+b^2)^(5/2)-(-4*a*c+b^2)^(3/2)*b^2-16*a^2*b*c^2+4*a*
b^3*c)/(-b+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2))/a/(4*a*c-b^2)/(-16*c^3/(-b-(-4*a*c+b^2)^(1/2))/(-b+(-4*a
*c+b^2)^(1/2))*a+4*c^2/(-b-(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2))*b^2))^(1/2))-1/2*b/(-2*((-4*a*c+b^2)^(3
/2)-(-4*a*c+b^2)^(1/2)*b^2+4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4+2*((-4*a*c+b^2)
^(3/2)-(-4*a*c+b^2)^(1/2)*b^2+4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4-2*((-4*a
*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2-4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(1+2
*c/(-b+(-4*a*c+b^2)^(1/2))*x^2+2*c*x^2/(-b-(-4*a*c+b^2)^(1/2))+4*c^2*x^4/(-b-(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b
^2)^(1/2)))^(1/2)*EllipticF(1/2*x*(-2*((-4*a*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2+4*a*b*c)/(b+(-4*a*c+b^2)^(1/2
))/(-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/4*(-16-2*(2*c/(-b+(-4*a*c+b^2)^(1/2))+2*c/(-b-(-4*a*c+b^2)^(1/2)))*((-4*
a*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2-4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/a/c^2*(-b-(-4*a*c+b^2)^(1/2)))^(1/2))-2*
c/(-2*((-4*a*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2+4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2))/a)^(1
/2)*(4+2*((-4*a*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2+4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2))/a*
x^2)^(1/2)*(4-2*((-4*a*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2-4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1
/2))/a*x^2)^(1/2)/(1+2*c/(-b+(-4*a*c+b^2)^(1/2))*x^2+2*c*x^2/(-b-(-4*a*c+b^2)^(1/2))+4*c^2*x^4/(-b-(-4*a*c+b^2
)^(1/2))/(-b+(-4*a*c+b^2)^(1/2)))^(1/2)/(2*c/(-b+(-4*a*c+b^2)^(1/2))+2*c/(-b-(-4*a*c+b^2)^(1/2))-(-4*a*c+b^2)^
(1/2)/a)*(EllipticF(1/2*x*(-2*((-4*a*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2+4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(
-4*a*c+b^2)^(1/2))/a)^(1/2),1/4*(-16-2*(2*c/(-b+(-4*a*c+b^2)^(1/2))+2*c/(-b-(-4*a*c+b^2)^(1/2)))*((-4*a*c+b^2)
^(3/2)-(-4*a*c+b^2)^(1/2)*b^2-4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/a/c^2*(-b-(-4*a*c+b^2)^(1/2)))^(1/2))-EllipticE(
1/2*x*(-2*((-4*a*c+b^2)^(3/2)-(-4*a*c+b^2)^(1/2)*b^2+4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/(-b+(-4*a*c+b^2)^(1/2))/a
)^(1/2),1/4*(-16-2*(2*c/(-b+(-4*a*c+b^2)^(1/2))+2*c/(-b-(-4*a*c+b^2)^(1/2)))*((-4*a*c+b^2)^(3/2)-(-4*a*c+b^2)^
(1/2)*b^2-4*a*b*c)/(b+(-4*a*c+b^2)^(1/2))/a/c^2*(-b-(-4*a*c+b^2)^(1/2)))^(1/2))))
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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((-b+2*c*x^2-(-4*a*c+b^2)^(1/2))/(1+2*c*x^2/(-b-(-4*a*c+b^2)^(1/2)))^(1/2)/(1+2*c*x^2/(-b+(-4*a*c+b^2
)^(1/2)))^(1/2),x, algorithm="maxima")
[Out]
integrate((2*c*x^2 - b - sqrt(b^2 - 4*a*c))/(sqrt(-2*c*x^2/(b + sqrt(b^2 - 4*a*c)) + 1)*sqrt(-2*c*x^2/(b - sqr
t(b^2 - 4*a*c)) + 1)), x)
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-b+2*c*x^2-(-4*a*c+b^2)^(1/2))/(1+2*c*x^2/(-b-(-4*a*c+b^2)^(1/2)))^(1/2)/(1+2*c*x^2/(-b+(-4*a*c+b^2
)^(1/2)))^(1/2),x, algorithm="fricas")
[Out]
Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {- b + 2 c x^{2} - \sqrt {- 4 a c + b^{2}}}{\sqrt {\frac {- b + 2 c x^{2} - \sqrt {- 4 a c + b^{2}}}{- b - \sqrt {- 4 a c + b^{2}}}} \sqrt {\frac {- b + 2 c x^{2} + \sqrt {- 4 a c + b^{2}}}{- b + \sqrt {- 4 a c + b^{2}}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-b+2*c*x**2-(-4*a*c+b**2)**(1/2))/(1+2*c*x**2/(-b-(-4*a*c+b**2)**(1/2)))**(1/2)/(1+2*c*x**2/(-b+(-4
*a*c+b**2)**(1/2)))**(1/2),x)
[Out]
Integral((-b + 2*c*x**2 - sqrt(-4*a*c + b**2))/(sqrt((-b + 2*c*x**2 - sqrt(-4*a*c + b**2))/(-b - sqrt(-4*a*c +
b**2)))*sqrt((-b + 2*c*x**2 + sqrt(-4*a*c + b**2))/(-b + sqrt(-4*a*c + b**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((-b+2*c*x^2-(-4*a*c+b^2)^(1/2))/(1+2*c*x^2/(-b-(-4*a*c+b^2)^(1/2)))^(1/2)/(1+2*c*x^2/(-b+(-4*a*c+b^2
)^(1/2)))^(1/2),x, algorithm="giac")
[Out]
integrate((2*c*x^2 - b - sqrt(b^2 - 4*a*c))/(sqrt(-2*c*x^2/(b + sqrt(b^2 - 4*a*c)) + 1)*sqrt(-2*c*x^2/(b - sqr
t(b^2 - 4*a*c)) + 1)), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {b-2\,c\,x^2+\sqrt {b^2-4\,a\,c}}{\sqrt {1-\frac {2\,c\,x^2}{b-\sqrt {b^2-4\,a\,c}}}\,\sqrt {1-\frac {2\,c\,x^2}{b+\sqrt {b^2-4\,a\,c}}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(b - 2*c*x^2 + (b^2 - 4*a*c)^(1/2))/((1 - (2*c*x^2)/(b - (b^2 - 4*a*c)^(1/2)))^(1/2)*(1 - (2*c*x^2)/(b +
(b^2 - 4*a*c)^(1/2)))^(1/2)),x)
[Out]
int(-(b - 2*c*x^2 + (b^2 - 4*a*c)^(1/2))/((1 - (2*c*x^2)/(b - (b^2 - 4*a*c)^(1/2)))^(1/2)*(1 - (2*c*x^2)/(b +
(b^2 - 4*a*c)^(1/2)))^(1/2)), x)
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