3.34 \(\int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx\)

Optimal. Leaf size=134 \[ \frac {e^{3/2} \log \left (\sqrt {e} x \sqrt {2 c d-b e}+\sqrt {c} d+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}-\frac {e^{3/2} \log \left (-\sqrt {e} x \sqrt {2 c d-b e}+\sqrt {c} d+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}} \]

[Out]

-1/2*e^(3/2)*ln(d*c^(1/2)+e*x^2*c^(1/2)-x*e^(1/2)*(-b*e+2*c*d)^(1/2))/c^(1/2)/(-b*e+2*c*d)^(1/2)+1/2*e^(3/2)*l
n(d*c^(1/2)+e*x^2*c^(1/2)+x*e^(1/2)*(-b*e+2*c*d)^(1/2))/c^(1/2)/(-b*e+2*c*d)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1164, 628} \[ \frac {e^{3/2} \log \left (\sqrt {e} x \sqrt {2 c d-b e}+\sqrt {c} d+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}-\frac {e^{3/2} \log \left (-\sqrt {e} x \sqrt {2 c d-b e}+\sqrt {c} d+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}} \]

Antiderivative was successfully verified.

[In]

Int[(d - e*x^2)/((c*d^2)/e^2 + b*x^2 + c*x^4),x]

[Out]

-(e^(3/2)*Log[Sqrt[c]*d - Sqrt[e]*Sqrt[2*c*d - b*e]*x + Sqrt[c]*e*x^2])/(2*Sqrt[c]*Sqrt[2*c*d - b*e]) + (e^(3/
2)*Log[Sqrt[c]*d + Sqrt[e]*Sqrt[2*c*d - b*e]*x + Sqrt[c]*e*x^2])/(2*Sqrt[c]*Sqrt[2*c*d - b*e])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1164

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x
- x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] &&  !GtQ[b^2
- 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {d-e x^2}{\frac {c d^2}{e^2}+b x^2+c x^4} \, dx &=-\frac {e^{3/2} \int \frac {\frac {\sqrt {2 c d-b e}}{\sqrt {c} \sqrt {e}}+2 x}{-\frac {d}{e}-\frac {\sqrt {2 c d-b e} x}{\sqrt {c} \sqrt {e}}-x^2} \, dx}{2 \sqrt {c} \sqrt {2 c d-b e}}-\frac {e^{3/2} \int \frac {\frac {\sqrt {2 c d-b e}}{\sqrt {c} \sqrt {e}}-2 x}{-\frac {d}{e}+\frac {\sqrt {2 c d-b e} x}{\sqrt {c} \sqrt {e}}-x^2} \, dx}{2 \sqrt {c} \sqrt {2 c d-b e}}\\ &=-\frac {e^{3/2} \log \left (\sqrt {c} d-\sqrt {e} \sqrt {2 c d-b e} x+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}+\frac {e^{3/2} \log \left (\sqrt {c} d+\sqrt {e} \sqrt {2 c d-b e} x+\sqrt {c} e x^2\right )}{2 \sqrt {c} \sqrt {2 c d-b e}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 250, normalized size = 1.87 \[ \frac {e^{3/2} \left (-\frac {\left (\sqrt {b^2 e^2-4 c^2 d^2}-b e-2 c d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {e} x}{\sqrt {b e-\sqrt {b^2 e^2-4 c^2 d^2}}}\right )}{\sqrt {b e-\sqrt {b^2 e^2-4 c^2 d^2}}}-\frac {\left (\sqrt {b^2 e^2-4 c^2 d^2}+b e+2 c d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {e} x}{\sqrt {\sqrt {b^2 e^2-4 c^2 d^2}+b e}}\right )}{\sqrt {\sqrt {b^2 e^2-4 c^2 d^2}+b e}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2 e^2-4 c^2 d^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d - e*x^2)/((c*d^2)/e^2 + b*x^2 + c*x^4),x]

[Out]

(e^(3/2)*(-(((-2*c*d - b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e - Sqrt[-4
*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b^2*e^2]]) - ((2*c*d + b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*A
rcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^
2]]))/(Sqrt[2]*Sqrt[c]*Sqrt[-4*c^2*d^2 + b^2*e^2])

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fricas [A]  time = 0.42, size = 244, normalized size = 1.82 \[ \left [\frac {1}{2} \, e \sqrt {\frac {e}{2 \, c^{2} d - b c e}} \log \left (\frac {c e^{2} x^{4} + c d^{2} + {\left (4 \, c d e - b e^{2}\right )} x^{2} + 2 \, {\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} x^{3} + {\left (2 \, c^{2} d^{2} - b c d e\right )} x\right )} \sqrt {\frac {e}{2 \, c^{2} d - b c e}}}{c e^{2} x^{4} + b e^{2} x^{2} + c d^{2}}\right ), -e \sqrt {-\frac {e}{2 \, c^{2} d - b c e}} \arctan \left (c x \sqrt {-\frac {e}{2 \, c^{2} d - b c e}}\right ) + e \sqrt {-\frac {e}{2 \, c^{2} d - b c e}} \arctan \left (\frac {{\left (c e x^{3} - {\left (c d - b e\right )} x\right )} \sqrt {-\frac {e}{2 \, c^{2} d - b c e}}}{d}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),x, algorithm="fricas")

[Out]

[1/2*e*sqrt(e/(2*c^2*d - b*c*e))*log((c*e^2*x^4 + c*d^2 + (4*c*d*e - b*e^2)*x^2 + 2*((2*c^2*d*e - b*c*e^2)*x^3
+ (2*c^2*d^2 - b*c*d*e)*x)*sqrt(e/(2*c^2*d - b*c*e)))/(c*e^2*x^4 + b*e^2*x^2 + c*d^2)), -e*sqrt(-e/(2*c^2*d -
b*c*e))*arctan(c*x*sqrt(-e/(2*c^2*d - b*c*e))) + e*sqrt(-e/(2*c^2*d - b*c*e))*arctan((c*e*x^3 - (c*d - b*e)*x
)*sqrt(-e/(2*c^2*d - b*c*e))/d)]

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giac [B]  time = 1.37, size = 2202, normalized size = 16.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),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 + b^2*e^4)*c*e^2)*c^4*d^4*e^2 - 16*b^2*c^
3*d^2*e^6 + 8*b*c^4*d^2*e^6 + 8*sqrt(2)*sqrt(b*c*e^4 + sqrt(-4*c^2*d^2*e^2 + 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 + 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 + b^2*e^4)*c*e^2)*c^4*d^2*e^4 - 4*sqrt(2)*sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(-4
*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b*c^2*d^2*e^2 - 8*(4*c^2*d^2*e^2 - b^2*e^4)*c^3*d^2*e^2 + 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 + 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 + 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 + b^2*e^4)*c*e^2)*b^2*
c^2*e^6 + sqrt(2)*sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b^3*e^4
- 2*sqrt(2)*sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b^2*c*e^4 + sq
rt(2)*sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b*c^2*e^4 + 2*(4*c^2
*d^2*e^2 - b^2*e^4)*b^2*c*e^4 - 2*(4*c^2*d^2*e^2 - b^2*e^4)*b*c^2*e^4 + 2*(8*c^5*d^3*e^4 - 4*sqrt(2)*sqrt(-4*c
^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*c^3*d^3 - 2*b^2*c^3*d*e^6 + sqrt(2)
*sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b^2*c*d*e^2 - 2*sqrt(2)*s
qrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b*c^2*d*e^2 + sqrt(2)*sqrt(
-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*c^3*d*e^2 - 2*(4*c^2*d^2*e^2 -
b^2*e^4)*c^3*d*e^2)*e)*arctan(2*sqrt(1/2)*x/sqrt((b + sqrt(-4*c^2*d^2*e^(-2) + b^2))/c))/((16*c^5*d^5*e^2 - 8*
b^2*c^3*d^3*e^4 + 8*b*c^4*d^3*e^4 - 4*c^5*d^3*e^4 + b^4*c*d*e^6 - 2*b^3*c^2*d*e^6 + b^2*c^3*d*e^6)*abs(c)) + 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 + b^2*e^4)*c*e^2)*c^4*d^4*e^2 - 16*b^2*c^3*
d^2*e^6 + 8*b*c^4*d^2*e^6 - 8*sqrt(2)*sqrt(b*c*e^4 - sqrt(-4*c^2*d^2*e^2 + 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 + 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 + b^2*e^4)*c*e^2)*c^4*d^2*e^4 - 4*sqrt(2)*sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(-4*c
^2*d^2*e^2 + b^2*e^4)*c*e^2)*b*c^2*d^2*e^2 - 8*(4*c^2*d^2*e^2 - b^2*e^4)*c^3*d^2*e^2 + 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 + 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 + 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 + b^2*e^4)*c*e^2)*b^2*c^
2*e^6 + sqrt(2)*sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b^3*e^4 -
2*sqrt(2)*sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b^2*c*e^4 + sqrt
(2)*sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b*c^2*e^4 + 2*(4*c^2*d
^2*e^2 - b^2*e^4)*b^2*c*e^4 - 2*(4*c^2*d^2*e^2 - b^2*e^4)*b*c^2*e^4 + 2*(8*c^5*d^3*e^4 - 4*sqrt(2)*sqrt(-4*c^2
*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*c^3*d^3 - 2*b^2*c^3*d*e^6 + sqrt(2)*s
qrt(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b^2*c*d*e^2 - 2*sqrt(2)*sqr
t(-4*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*b*c^2*d*e^2 + sqrt(2)*sqrt(-4
*c^2*d^2*e^2 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*c*e^2)*c^3*d*e^2 - 2*(4*c^2*d^2*e^2 - b^
2*e^4)*c^3*d*e^2)*e)*arctan(2*sqrt(1/2)*x/sqrt((b - sqrt(-4*c^2*d^2*e^(-2) + b^2))/c))/((16*c^5*d^5*e^2 - 8*b^
2*c^3*d^3*e^4 + 8*b*c^4*d^3*e^4 - 4*c^5*d^3*e^4 + b^4*c*d*e^6 - 2*b^3*c^2*d*e^6 + b^2*c^3*d*e^6)*abs(c))

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maple [B]  time = 0.08, size = 582, normalized size = 4.34 \[ -\frac {\sqrt {2}\, b \,e^{4} \arctanh \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{2 \sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\, \sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, b \,e^{4} \arctan \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{2 \sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\, \sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, c d \,e^{3} \arctanh \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\, \sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, c d \,e^{3} \arctan \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\, \sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}+\frac {\sqrt {2}\, e^{2} \arctanh \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, e^{2} \arctan \left (\frac {\sqrt {2}\, c e x}{\sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}}\right )}{2 \sqrt {\left (b \,e^{2}+\sqrt {\left (b e -2 c d \right ) \left (b e +2 c d \right ) e^{2}}\right ) c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),x)

[Out]

-1/2*e^4/(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2)*2^(1/2)/((-e^2*b+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2)*ar
ctanh(c*e*x*2^(1/2)/((-e^2*b+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2))*b-e^3*c/(e^2*(b*e-2*c*d)*(b*e+2*c*
d))^(1/2)*2^(1/2)/((-e^2*b+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2)*arctanh(c*e*x*2^(1/2)/((-e^2*b+(e^2*(
b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2))*d+1/2*e^2*2^(1/2)/((-e^2*b+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1
/2)*arctanh(c*e*x*2^(1/2)/((-e^2*b+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2))-1/2*e^4/(e^2*(b*e-2*c*d)*(b*
e+2*c*d))^(1/2)*2^(1/2)/((e^2*b+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2)*arctan(c*e*x*2^(1/2)/((e^2*b+(e^
2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2))*b-e^3*c/(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2)*2^(1/2)/((e^2*b+(e^2*(
b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2)*arctan(c*e*x*2^(1/2)/((e^2*b+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(
1/2))*d-1/2*e^2*2^(1/2)/((e^2*b+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2)*arctan(c*e*x*2^(1/2)/((e^2*b+(e^
2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {e x^{2} - d}{c x^{4} + b x^{2} + \frac {c d^{2}}{e^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),x, algorithm="maxima")

[Out]

-integrate((e*x^2 - d)/(c*x^4 + b*x^2 + c*d^2/e^2), x)

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mupad [B]  time = 0.18, size = 129, normalized size = 0.96 \[ -\frac {e^{3/2}\,\left (\mathrm {atan}\left (\frac {\sqrt {e}\,x\,\sqrt {b\,c\,e-2\,c^2\,d}}{b\,e-2\,c\,d}\right )+\mathrm {atan}\left (\frac {c\,e^{3/2}\,x^3\,\sqrt {b\,c\,e-2\,c^2\,d}+b\,e^{3/2}\,x\,\sqrt {b\,c\,e-2\,c^2\,d}-c\,d\,\sqrt {e}\,x\,\sqrt {b\,c\,e-2\,c^2\,d}}{d\,\left (2\,c^2\,d-b\,c\,e\right )}\right )\right )}{\sqrt {b\,c\,e-2\,c^2\,d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d - e*x^2)/(b*x^2 + c*x^4 + (c*d^2)/e^2),x)

[Out]

-(e^(3/2)*(atan((e^(1/2)*x*(b*c*e - 2*c^2*d)^(1/2))/(b*e - 2*c*d)) + atan((c*e^(3/2)*x^3*(b*c*e - 2*c^2*d)^(1/
2) + b*e^(3/2)*x*(b*c*e - 2*c^2*d)^(1/2) - c*d*e^(1/2)*x*(b*c*e - 2*c^2*d)^(1/2))/(d*(2*c^2*d - b*c*e)))))/(b*
c*e - 2*c^2*d)^(1/2)

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sympy [A]  time = 0.86, size = 158, normalized size = 1.18 \[ \frac {\sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} \log {\left (\frac {d}{e} + x^{2} + \frac {x \left (- b e \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} + 2 c d \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}}\right )}{e^{2}} \right )}}{2} - \frac {\sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} \log {\left (\frac {d}{e} + x^{2} + \frac {x \left (b e \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}} - 2 c d \sqrt {- \frac {e^{3}}{c \left (b e - 2 c d\right )}}\right )}{e^{2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-e*x**2+d)/(c*d**2/e**2+b*x**2+c*x**4),x)

[Out]

sqrt(-e**3/(c*(b*e - 2*c*d)))*log(d/e + x**2 + x*(-b*e*sqrt(-e**3/(c*(b*e - 2*c*d))) + 2*c*d*sqrt(-e**3/(c*(b*
e - 2*c*d))))/e**2)/2 - sqrt(-e**3/(c*(b*e - 2*c*d)))*log(d/e + x**2 + x*(b*e*sqrt(-e**3/(c*(b*e - 2*c*d))) -
2*c*d*sqrt(-e**3/(c*(b*e - 2*c*d))))/e**2)/2

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