3.2.59 \(\int \frac {d+e x^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx\)
Optimal. Leaf size=49 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {c} \sqrt {e} \sqrt {c d-b e}} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used =
{1149, 208} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {c} \sqrt {e} \sqrt {c d-b e}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x^2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
-(ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]]/(Sqrt[c]*Sqrt[e]*Sqrt[c*d - b*e]))
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 1149
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2
, 0] && IntegerQ[p]
Rubi steps
\begin {align*} \int \frac {d+e x^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx &=\int \frac {1}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {c} \sqrt {e} \sqrt {c d-b e}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 48, normalized size = 0.98 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {b e-c d}}\right )}{\sqrt {c} \sqrt {e} \sqrt {b e-c d}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x^2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]]/(Sqrt[c]*Sqrt[e]*Sqrt[-(c*d) + b*e])
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x^2}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(d + e*x^2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]
[Out]
IntegrateAlgebraic[(d + e*x^2)/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4), x]
________________________________________________________________________________________
fricas [A] time = 0.63, size = 134, normalized size = 2.73 \begin {gather*} \left [\frac {\log \left (\frac {c e x^{2} + c d - b e - 2 \, \sqrt {c^{2} d e - b c e^{2}} x}{c e x^{2} - c d + b e}\right )}{2 \, \sqrt {c^{2} d e - b c e^{2}}}, -\frac {\sqrt {-c^{2} d e + b c e^{2}} \arctan \left (-\frac {\sqrt {-c^{2} d e + b c e^{2}} x}{c d - b e}\right )}{c^{2} d e - b c e^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")
[Out]
[1/2*log((c*e*x^2 + c*d - b*e - 2*sqrt(c^2*d*e - b*c*e^2)*x)/(c*e*x^2 - c*d + b*e))/sqrt(c^2*d*e - b*c*e^2), -
sqrt(-c^2*d*e + b*c*e^2)*arctan(-sqrt(-c^2*d*e + b*c*e^2)*x/(c*d - b*e))/(c^2*d*e - b*c*e^2)]
________________________________________________________________________________________
giac [B] time = 6.09, size = 3276, normalized size = 66.86
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")
[Out]
1/4*(32*c^5*d^4*e^4 - 16*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^4*d^4*e^2
- 64*b*c^4*d^3*e^5 - 16*c^5*d^3*e^5 + 32*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c
*e^2)*b*c^3*d^3*e^3 + 8*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2
- 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^3*d^3*e + 48*b^2*c^3*d^2*e^6 + 24*b*c^4*d^2*e^6 - 24*sqrt(2)*sqrt(b*c*e^4 +
sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^2*d^2*e^4 + 8*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e
^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^3*d^2*e^4 - 4*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*c*e^2)*c^4*d^2*e^4 - 12*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^
2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^2*d^2*e^2 - 16*b^3*c^2*d*e^7 - 12*b^2*c^3*d*e^7 + 8*sqrt(2)*sqrt
(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c*d*e^5 - 8*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^
2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^2*d*e^5 + 4*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*
d*e^3 + b^2*e^4)*c*e^2)*b*c^3*d*e^5 - 8*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^3*d^2*e^2 + 6*sqrt(2)*sqrt(4
*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c*
d*e^3 - 4*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3
+ b^2*e^4)*c*e^2)*b*c^2*d*e^3 + 2*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^
2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^3*d*e^3 + 2*b^4*c*e^8 + 2*b^3*c^2*e^8 - sqrt(2)*sqrt(b*c*e^4 + sqr
t(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*e^6 + 2*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*
d*e^3 + b^2*e^4)*c*e^2)*b^3*c*e^6 - sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*
b^2*c^2*e^6 + 8*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^2*d*e^3 + 4*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4
)*c^3*d*e^3 - sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*
e^3 + b^2*e^4)*c*e^2)*b^3*e^4 + 2*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^
2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c*e^4 - sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt
(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^2*e^4 - 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2
*e^4)*b^2*c*e^4 - 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^2*e^4)*arctan(2*sqrt(1/2)*x*e/sqrt((b*e^2 + sq
rt(b^2*e^4 + 4*(c*d^2 - b*d*e)*c*e^2))/c))/((16*c^5*d^5*e^4 - 48*b*c^4*d^4*e^5 + 56*b^2*c^3*d^3*e^6 - 8*b*c^4*
d^3*e^6 + 4*c^5*d^3*e^6 - 32*b^3*c^2*d^2*e^7 + 16*b^2*c^3*d^2*e^7 - 8*b*c^4*d^2*e^7 + 9*b^4*c*d*e^8 - 10*b^3*c
^2*d*e^8 + 5*b^2*c^3*d*e^8 - b^5*e^9 + 2*b^4*c*e^9 - b^3*c^2*e^9)*abs(c)) - 1/4*(32*c^5*d^4*e^4 + 16*sqrt(2)*s
qrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^4*d^4*e^2 - 64*b*c^4*d^3*e^5 - 16*c^5*d^3*e
^5 - 32*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^3*d^3*e^3 + 8*sqrt(2)*sq
rt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^
3*d^3*e + 48*b^2*c^3*d^2*e^6 + 24*b*c^4*d^2*e^6 + 24*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*c*e^2)*b^2*c^2*d^2*e^4 - 8*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)
*b*c^3*d^2*e^4 + 4*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^4*d^2*e^4 - 12*
sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)
*c*e^2)*b*c^2*d^2*e^2 - 16*b^3*c^2*d*e^7 - 12*b^2*c^3*d*e^7 - 8*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*
b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c*d*e^5 + 8*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)
*c*e^2)*b^2*c^2*d*e^5 - 4*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^3*d*e^
5 - 8*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^3*d^2*e^2 + 6*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e
^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c*d*e^3 - 4*sqrt(2)*sqrt(4*c^2*d^2*e
^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^2*d*e^3 + 2*
sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)
*c*e^2)*c^3*d*e^3 + 2*b^4*c*e^8 + 2*b^3*c^2*e^8 + sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^
2*e^4)*c*e^2)*b^4*e^6 - 2*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c*e^6
+ sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^2*e^6 + 8*(4*c^2*d^2*e^2 - 4
*b*c*d*e^3 + b^2*e^4)*b*c^2*d*e^3 + 4*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^3*d*e^3 - sqrt(2)*sqrt(4*c^2*d
^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*e^4 + 2*
sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)
*c*e^2)*b^2*c*e^4 - sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*
b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^2*e^4 - 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^2*c*e^4 - 2*(4*c^2*d^2*e^2
- 4*b*c*d*e^3 + b^2*e^4)*b*c^2*e^4)*arctan(2*sqrt(1/2)*x*e/sqrt((b*e^2 - sqrt(b^2*e^4 + 4*(c*d^2 - b*d*e)*c*e
^2))/c))/((16*c^5*d^5*e^4 - 48*b*c^4*d^4*e^5 + 56*b^2*c^3*d^3*e^6 - 8*b*c^4*d^3*e^6 + 4*c^5*d^3*e^6 - 32*b^3*c
^2*d^2*e^7 + 16*b^2*c^3*d^2*e^7 - 8*b*c^4*d^2*e^7 + 9*b^4*c*d*e^8 - 10*b^3*c^2*d*e^8 + 5*b^2*c^3*d*e^8 - b^5*e
^9 + 2*b^4*c*e^9 - b^3*c^2*e^9)*abs(c))
________________________________________________________________________________________
maple [A] time = 0.00, size = 33, normalized size = 0.67 \begin {gather*} \frac {\arctan \left (\frac {c e x}{\sqrt {\left (b e -c d \right ) c e}}\right )}{\sqrt {\left (b e -c d \right ) c e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)
[Out]
1/((b*e-c*d)*c*e)^(1/2)*arctan(1/((b*e-c*d)*c*e)^(1/2)*c*e*x)
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?
________________________________________________________________________________________
mupad [B] time = 4.49, size = 38, normalized size = 0.78 \begin {gather*} \frac {\mathrm {atan}\left (\frac {c\,e\,x}{\sqrt {b\,c\,e^2-c^2\,d\,e}}\right )}{\sqrt {b\,c\,e^2-c^2\,d\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x^2)/(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e),x)
[Out]
atan((c*e*x)/(b*c*e^2 - c^2*d*e)^(1/2))/(b*c*e^2 - c^2*d*e)^(1/2)
________________________________________________________________________________________
sympy [B] time = 0.32, size = 124, normalized size = 2.53 \begin {gather*} - \frac {\sqrt {- \frac {1}{c e \left (b e - c d\right )}} \log {\left (- b e \sqrt {- \frac {1}{c e \left (b e - c d\right )}} + c d \sqrt {- \frac {1}{c e \left (b e - c d\right )}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{c e \left (b e - c d\right )}} \log {\left (b e \sqrt {- \frac {1}{c e \left (b e - c d\right )}} - c d \sqrt {- \frac {1}{c e \left (b e - c d\right )}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x**2+d)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)
[Out]
-sqrt(-1/(c*e*(b*e - c*d)))*log(-b*e*sqrt(-1/(c*e*(b*e - c*d))) + c*d*sqrt(-1/(c*e*(b*e - c*d))) + x)/2 + sqrt
(-1/(c*e*(b*e - c*d)))*log(b*e*sqrt(-1/(c*e*(b*e - c*d))) - c*d*sqrt(-1/(c*e*(b*e - c*d))) + x)/2
________________________________________________________________________________________