3.327 \(\int \frac{1}{x \left (d+e x^2\right ) \sqrt{a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=138 \[ -\frac{e \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d \sqrt{a e^2-b d e+c d^2}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a} d} \]

[Out]

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

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Rubi [A]  time = 0.520403, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\frac{e \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d \sqrt{a e^2-b d e+c d^2}}-\frac{\tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a} d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 53.6563, size = 121, normalized size = 0.88 \[ \frac{e \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 d \sqrt{a e^{2} - b d e + c d^{2}}} - \frac{\operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 \sqrt{a} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.942169, size = 174, normalized size = 1.26 \[ \frac{-\frac{e \log \left (d+e x^2\right )}{\sqrt{a e^2-b d e+c d^2}}+\frac{e \log \left (2 \sqrt{a+x^2 \left (b+c x^2\right )} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )}{\sqrt{a e^2-b d e+c d^2}}-\frac{\log \left (2 \sqrt{a} \sqrt{a+x^2 \left (b+c x^2\right )}+2 a+b x^2\right )}{\sqrt{a}}+\frac{\log \left (x^2\right )}{\sqrt{a}}}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 207, normalized size = 1.5 \[ -{\frac{1}{2\,d}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{1}{2\,d}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({x}^{2}+{\frac{d}{e}} \right ) ^{2}c+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/d/a^(1/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)+1/2/d/((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. \[ \int \frac{1}{\sqrt{c x^{4} + b x^{2} + a}{\left (e x^{2} + d\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)*x),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)*x), x)

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Fricas [A]  time = 0.486653, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)*x),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (d + e x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 + a)*(e*x^2 + d)*x),x, algorithm="giac")

[Out]

Exception raised: RuntimeError