3.326 \(\int \frac{x}{\left (d+e x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=86 \[ \frac{\tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \sqrt{a e^2-b d e+c d^2}} \]

[Out]

ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a
+ b*x^2 + c*x^4])]/(2*Sqrt[c*d^2 - b*d*e + a*e^2])

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Rubi [A]  time = 0.244218, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \sqrt{a e^2-b d e+c d^2}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a
+ b*x^2 + c*x^4])]/(2*Sqrt[c*d^2 - b*d*e + a*e^2])

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Rubi in Sympy [A]  time = 30.9928, size = 78, normalized size = 0.91 \[ - \frac{\operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 \sqrt{a e^{2} - b d e + c d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(e*x**2+d)/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

-atanh((2*a*e - b*d + x**2*(b*e - 2*c*d))/(2*sqrt(a + b*x**2 + c*x**4)*sqrt(a*e*
*2 - b*d*e + c*d**2)))/(2*sqrt(a*e**2 - b*d*e + c*d**2))

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Mathematica [A]  time = 0.0907648, size = 96, normalized size = 1.12 \[ \frac{\log \left (d+e x^2\right )-\log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )}{2 \sqrt{e (a e-b d)+c d^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Log[d + e*x^2] - Log[-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2 + 2*Sqrt[c*d^2 - b*d*
e + a*e^2]*Sqrt[a + b*x^2 + c*x^4]])/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)])

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Maple [B]  time = 0.008, size = 165, normalized size = 1.9 \[ -{\frac{1}{2\,e}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({x}^{2}+{\frac{d}{e}} \right ) ^{2}c+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(e*x^2+d)/(c*x^4+b*x^2+a)^(1/2),x)

[Out]

-1/2/e/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)
/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x^2+d/e)^2*c+(b*e-2*c*d)/e*(x^2
+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.359839, size = 1, normalized size = 0.01 \[ \left [\frac{\log \left (-\frac{4 \,{\left (b c d^{3} + 3 \, a b d e^{2} - 2 \, a^{2} e^{3} -{\left (b^{2} + 2 \, a c\right )} d^{2} e +{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} +{\left ({\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{4} - 8 \, a b d e + 8 \, a^{2} e^{2} +{\left (b^{2} + 4 \, a c\right )} d^{2} + 2 \,{\left (4 \, b c d^{2} + 4 \, a b e^{2} -{\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x^{2}\right )} \sqrt{c d^{2} - b d e + a e^{2}}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right )}{4 \, \sqrt{c d^{2} - b d e + a e^{2}}}, -\frac{\arctan \left (-\frac{\sqrt{-c d^{2} + b d e - a e^{2}}{\left ({\left (2 \, c d - b e\right )} x^{2} + b d - 2 \, a e\right )}}{2 \, \sqrt{c x^{4} + b x^{2} + a}{\left (c d^{2} - b d e + a e^{2}\right )}}\right )}{2 \, \sqrt{-c d^{2} + b d e - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="fricas")

[Out]

[1/4*log(-(4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d
^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x^2)*sqrt(c*x^4 + b*x^2 + a) +
((8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^4 - 8*a*b*d*e + 8*a^2*e^2 + (b^2
+ 4*a*c)*d^2 + 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x^2)*sqrt(c*d^2
- b*d*e + a*e^2))/(e^2*x^4 + 2*d*e*x^2 + d^2))/sqrt(c*d^2 - b*d*e + a*e^2), -1/2
*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*((2*c*d - b*e)*x^2 + b*d - 2*a*e)/(sqr
t(c*x^4 + b*x^2 + a)*(c*d^2 - b*d*e + a*e^2)))/sqrt(-c*d^2 + b*d*e - a*e^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (d + e x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(e*x**2+d)/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

Integral(x/((d + e*x**2)*sqrt(a + b*x**2 + c*x**4)), x)

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GIAC/XCAS [A]  time = 0.287555, size = 101, normalized size = 1.17 \[ \frac{\arctan \left (-\frac{{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e - a e^{2}}}\right )}{\sqrt{-c d^{2} + b d e - a e^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="giac")

[Out]

arctan(-((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*
d*e - a*e^2))/sqrt(-c*d^2 + b*d*e - a*e^2)