3.338 \(\int \frac{1}{x \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=266 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 a^{3/2} d}+\frac{e \left (2 a c e+b^2 (-e)+c x^2 (2 c d-b e)+b c d\right )}{d \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}+\frac{-2 a c+b^2+b c x^2}{a d \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{e^3 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

(b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*d*Sqrt[a + b*x^2 + c*x^4]) + (e*(b*c*d
- b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x^2))/((b^2 - 4*a*c)*d*(c*d^2 - b*d*e + a*e^
2)*Sqrt[a + b*x^2 + c*x^4]) - ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 +
c*x^4])]/(2*a^(3/2)*d) - (e^3*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*(c*d^2 - b*d*e + a*e^2)^(
3/2))

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Rubi [A]  time = 0.923201, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{\tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 a^{3/2} d}+\frac{e \left (2 a c e+b^2 (-e)+c x^2 (2 c d-b e)+b c d\right )}{d \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}+\frac{-2 a c+b^2+b c x^2}{a d \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{e^3 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]

[Out]

(b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*d*Sqrt[a + b*x^2 + c*x^4]) + (e*(b*c*d
- b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x^2))/((b^2 - 4*a*c)*d*(c*d^2 - b*d*e + a*e^
2)*Sqrt[a + b*x^2 + c*x^4]) - ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 +
c*x^4])]/(2*a^(3/2)*d) - (e^3*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*(c*d^2 - b*d*e + a*e^2)^(
3/2))

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Rubi in Sympy [A]  time = 111.14, size = 238, normalized size = 0.89 \[ \frac{e^{3} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 d \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} - \frac{e \left (- 2 a c e + b^{2} e - b c d + c x^{2} \left (b e - 2 c d\right )\right )}{d \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}} \left (a e^{2} - b d e + c d^{2}\right )} + \frac{- 2 a c + b^{2} + b c x^{2}}{a d \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}} - \frac{\operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 a^{\frac{3}{2}} d} \]

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)**(3/2),x)

[Out]

e**3*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*(a*e**2 - b*d*e + c*d**2)**(3/2)) - e*(-2*a*c*e
+ b**2*e - b*c*d + c*x**2*(b*e - 2*c*d))/(d*(-4*a*c + b**2)*sqrt(a + b*x**2 + c*
x**4)*(a*e**2 - b*d*e + c*d**2)) + (-2*a*c + b**2 + b*c*x**2)/(a*d*(-4*a*c + b**
2)*sqrt(a + b*x**2 + c*x**4)) - atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2
+ c*x**4)))/(2*a**(3/2)*d)

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Mathematica [A]  time = 3.09718, size = 378, normalized size = 1.42 \[ \frac{\frac{-\frac{a^{3/2} e^3 \log \left (d+e x^2\right )}{\sqrt{a e^2-b d e+c d^2}}+\frac{a^{3/2} e^3 \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )}{\sqrt{a e^2-b d e+c d^2}}+\log \left (x^2\right ) \left (e (a e-b d)+c d^2\right )-c d^2 \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )+b d e \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )-a e^2 \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )}{d}+\frac{2 \sqrt{a} \left (b c \left (3 a e+c d x^2\right )-2 a c^2 \left (d-e x^2\right )+b^3 (-e)+b^2 c \left (d-e x^2\right )\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}}{2 a^{3/2} \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]

[Out]

((2*Sqrt[a]*(-(b^3*e) + b*c*(3*a*e + c*d*x^2) + b^2*c*(d - e*x^2) - 2*a*c^2*(d -
e*x^2)))/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) + ((c*d^2 + e*(-(b*d) + a*e))*
Log[x^2] - (a^(3/2)*e^3*Log[d + e*x^2])/Sqrt[c*d^2 - b*d*e + a*e^2] - c*d^2*Log[
2*a + b*x^2 + 2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4]] + b*d*e*Log[2*a + b*x^2 + 2*Sqr
t[a]*Sqrt[a + b*x^2 + c*x^4]] - a*e^2*Log[2*a + b*x^2 + 2*Sqrt[a]*Sqrt[a + b*x^2
+ c*x^4]] + (a^(3/2)*e^3*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]])/Sqrt[c*d^2 - b*d*e + a*e^2])/d)/(2*
a^(3/2)*(c*d^2 + e*(-(b*d) + a*e)))

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Maple [B]  time = 0.018, size = 612, normalized size = 2.3 \[{\frac{1}{2\,ad}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{bc{x}^{2}}{ad \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{{b}^{2}}{2\,ad \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{1}{2\,d}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}+2\,{\frac{ce}{d \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( -4\,ac+{b}^{2} \right ) }\sqrt{ \left ({x}^{2}-1/2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{2}c+\sqrt{-4\,ac+{b}^{2}} \left ({x}^{2}-1/2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) } \left ({x}^{2}+1/2\,{\frac{b}{c}}-1/2\,{\frac{\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{-1}}-2\,{\frac{ce}{d \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( -4\,ac+{b}^{2} \right ) }\sqrt{ \left ({x}^{2}+1/2\,{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{2}c-\sqrt{-4\,ac+{b}^{2}} \left ({x}^{2}+1/2\,{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) } \left ({x}^{2}+1/2\,{\frac{\sqrt{-4\,ac+{b}^{2}}}{c}}+1/2\,{\frac{b}{c}} \right ) ^{-1}}-2\,{\frac{{e}^{2}c}{d \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({x}^{2}+{\frac{d}{e}} \right ) ^{2}c+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}} \]

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)^(3/2),x)

[Out]

1/2/d/a/(c*x^4+b*x^2+a)^(1/2)-1/d*b/a/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)*x^2*c-1/
2/d*b^2/a/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)-1/2/d/a^(3/2)*ln((2*a+b*x^2+2*a^(1/2
)*(c*x^4+b*x^2+a)^(1/2))/x^2)+2/d*e*c/(e*(-4*a*c+b^2)^(1/2)-b*e+2*c*d)/(-4*a*c+b
^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))*((x^2-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))^
2*c+(-4*a*c+b^2)^(1/2)*(x^2-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)-2/d*e*c/(e*(-4
*a*c+b^2)^(1/2)+b*e-2*c*d)/(-4*a*c+b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)*(
(x^2+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2*c-(-4*a*c+b^2)^(1/2)*(x^2+1/2*(b+(-4*a*c+b^
2)^(1/2))/c))^(1/2)-2/d*e^2*c/(e*(-4*a*c+b^2)^(1/2)-b*e+2*c*d)/(e*(-4*a*c+b^2)^(
1/2)+b*e-2*c*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. \[ \int \frac{1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}{\left (e x^{2} + d\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)*x),x, algorithm="maxima")

[Out]

integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)*x), x)

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Fricas [A]  time = 3.51356, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)*x),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (d + e x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral(1/(x*(d + e*x**2)*(a + b*x**2 + c*x**4)**(3/2)), 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/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)*x),x, algorithm="giac")

[Out]

Exception raised: RuntimeError