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

Optimal. Leaf size=186 \[ -\frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d e}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 d}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 e} \]

[Out]

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

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Rubi [A]  time = 0.614251, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{\sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d e}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 d}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 72.8193, size = 162, normalized size = 0.87 \[ - \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 d} + \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 e} + \frac{\sqrt{a e^{2} - b d e + c d^{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 d e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.793336, size = 214, normalized size = 1.15 \[ \frac{-\log \left (d+e x^2\right ) \sqrt{a e^2-b d e+c d^2}+\sqrt{a e^2-b d e+c d^2} \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )+\sqrt{c} d \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )-\sqrt{a} e \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )+\sqrt{a} e \log \left (x^2\right )}{2 d e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.016, size = 851, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/d*(c*x^4+b*x^2+a)^(1/2)+1/4/d*b*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/
2))/c^(1/2)-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*((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)-1/4/d*
ln((1/2*(b*e-2*c*d)/e+c*(x^2+d/e))/c^(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))/c^(1/2)*b+1/2/e*ln((1/2*(b*e-2*c*d)/e+c*(x^2+d
/e))/c^(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))*c^(1/2)+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))*a-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))*b+1/2*d/e^2/((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))*c

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 73.2405, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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