3.55 \(\int \frac {-b-\sqrt {b^2-4 a c}+2 c x^2}{\sqrt {1+\frac {2 c x^2}{-b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}}} \, dx\)
Optimal. Leaf size=113 \[ -\frac {\sqrt {b-\sqrt {b^2-4 a c}} \left (\sqrt {b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \]
[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 = 0.20, 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, 424} \[ -\frac {\sqrt {b-\sqrt {b^2-4 a c}} \left (\sqrt {b^2-4 a c}+b\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {c}} \]
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 424
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), 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] time = 0.44, size = 104, normalized size = 0.92 \[ -2 i \sqrt {2} a \sqrt {\frac {c}{\sqrt {b^2-4 a c}-b}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{\sqrt {b^2-4 a c}-b}} x\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right ) \]
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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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (2 \, a c x^{2} - a b - \sqrt {b^{2} - 4 \, a c} a\right )} \sqrt {-\frac {b x^{2} + \sqrt {b^{2} - 4 \, a c} x^{2} - 2 \, a}{a}} \sqrt {-\frac {b x^{2} - \sqrt {b^{2} - 4 \, a c} x^{2} - 2 \, a}{a}}}{2 \, {\left (c x^{4} - b x^{2} + a\right )}}, x\right ) \]
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]
integral(1/2*(2*a*c*x^2 - a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(-(b*x^2 + sqrt(b^2 - 4*a*c)*x^2 - 2*a)/a)*sqrt(-(b*x
^2 - sqrt(b^2 - 4*a*c)*x^2 - 2*a)/a)/(c*x^4 - b*x^2 + a), x)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, choosing root of [1,0,%%%{8,[1,0,
1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[
1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%%%}] at parameters values [18.6420984049,-49,-86]Warning, choosing
root of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%
%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%%%}] at parameters values [35.2935628123,
0,0]Warning, choosing root of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%
}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%%%}] at parameters v
alues [73.519035968,-9,-13]Warning, choosing root of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%
%%},0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,
0]%%%}] at parameters values [62.4600259969,0,0]Warning, choosing root of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0
]%%%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1
,2,0]%%%}+%%%{1,[0,4,0]%%%}] at parameters values [85.089694743,0,0]Warning, choosing root of [1,0,%%%{8,[1,1,
0]%%%}+%%%{-4,[1,0,0]%%%}+%%%{-2,[0,0,2]%%%},0,%%%{16,[2,2,0]%%%}+%%%{16,[2,1,0]%%%}+%%%{4,[2,0,0]%%%}+%%%{-8,
[1,1,2]%%%}+%%%{-4,[1,0,2]%%%}+%%%{1,[0,0,4]%%%}] at parameters values [-33,-70,8]Warning, choosing root of [1
,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0
]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%%%}] at parameters values [82.7280518371,-80,-23]War
ning, choosing root of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%%%{8
,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%%%}] at parameters values [
8.05231268331,0,0]Warning, choosing root of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%},0,%%%
{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%%%}] a
t parameters values [39.9828299829,91,31]Warning, choosing root of [1,0,%%%{8,[1,0,1]%%%}+%%%{-2,[1,0,0]%%%}+%
%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]%%%}+%%%{-2,[1,2,0]%%
%}+%%%{1,[0,4,0]%%%}] at parameters values [94.1262030317,0,0]Warning, choosing root of [1,0,%%%{8,[1,0,1]%%%}
+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,2,0]%%%},0,%%%{16,[2,0,2]%%%}+%%%{8,[2,0,1]%%%}+%%%{1,[2,0,0]%%%}+%%%{-8,[1,2,1]
%%%}+%%%{-2,[1,2,0]%%%}+%%%{1,[0,4,0]%%%}] at parameters values [91.6686590291,0,0]Warning, choosing root of [
1,0,%%%{8,[1,1,0]%%%}+%%%{-4,[1,0,0]%%%}+%%%{-2,[0,0,2]%%%},0,%%%{16,[2,2,0]%%%}+%%%{16,[2,1,0]%%%}+%%%{4,[2,0
,0]%%%}+%%%{-8,[1,1,2]%%%}+%%%{-4,[1,0,2]%%%}+%%%{1,[0,0,4]%%%}] at parameters values [-88,9,6]Evaluation time
: 22.75
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {2 c \,x^{2}-b -\sqrt {-4 a c +b^{2}}}{\sqrt {\frac {2 c \,x^{2}}{-b -\sqrt {-4 a c +b^{2}}}+1}\, \sqrt {\frac {2 c \,x^{2}}{-b +\sqrt {-4 a c +b^{2}}}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-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]
int((-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)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, c x^{2} - b - \sqrt {b^{2} - 4 \, a c}}{\sqrt {-\frac {2 \, c x^{2}}{b + \sqrt {b^{2} - 4 \, a c}} + 1} \sqrt {-\frac {2 \, c x^{2}}{b - \sqrt {b^{2} - 4 \, a c}} + 1}}\,{d x} \]
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {b-2\,c\,x^2+\sqrt {b^2-4\,a\,c}}{\sqrt {1-\frac {2\,c\,x^2}{b-\sqrt {b^2-4\,a\,c}}}\,\sqrt {1-\frac {2\,c\,x^2}{b+\sqrt {b^2-4\,a\,c}}}} \,d x \]
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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {- b + 2 c x^{2} - \sqrt {- 4 a c + b^{2}}}{\sqrt {\frac {- b + 2 c x^{2} - \sqrt {- 4 a c + b^{2}}}{- b - \sqrt {- 4 a c + b^{2}}}} \sqrt {\frac {- b + 2 c x^{2} + \sqrt {- 4 a c + b^{2}}}{- b + \sqrt {- 4 a c + b^{2}}}}}\, dx \]
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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