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

Optimal. Leaf size=260 \[ -\frac{\left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )^{3/2} \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{3/2}}+\frac{\left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )^{3/2} \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{\sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{3/2}}-\frac{d \sqrt{d+e x^2}}{a x} \]

[Out]

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

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Rubi [A]  time = 2.16023, antiderivative size = 432, normalized size of antiderivative = 1.66, number of steps used = 16, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276 \[ -\frac{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{2 a \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{2 a \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )}{2 a}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right )}{2 a}-\frac{d \sqrt{d+e x^2}}{a x}+\frac{d \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.736161, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^{3/2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.043, size = 360, normalized size = 1.4 \[ -{\frac{1}{adx} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{ex}{ad} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{5\,ex}{4\,a}\sqrt{e{x}^{2}+d}}+{\frac{3\,d}{2\,a}\sqrt{e}\ln \left ( x\sqrt{e}+\sqrt{e{x}^{2}+d} \right ) }+{\frac{{x}^{2}}{4\,a}{e}^{{\frac{3}{2}}}}+{\frac{d}{8\,a}\sqrt{e}}-{\frac{1}{2\,a}\sqrt{e}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{ \left ( a{e}^{2}-c{d}^{2} \right ){{\it \_R}}^{2}+2\,d \left ( 3\,a{e}^{2}-2\,bde+c{d}^{2} \right ){\it \_R}+a{d}^{2}{e}^{2}-c{d}^{4}}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }}-{\frac{{d}^{2}}{8\,a}\sqrt{e} \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{-2}}+{\frac{3\,d}{2\,a}\sqrt{e}\ln \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a/d/x*(e*x^2+d)^(5/2)+1/a/d*e*x*(e*x^2+d)^(3/2)+5/4/a*e*x*(e*x^2+d)^(1/2)+3/2
/a*d*e^(1/2)*ln(x*e^(1/2)+(e*x^2+d)^(1/2))+1/4/a*e^(3/2)*x^2+1/8/a*e^(1/2)*d-1/2
/a*e^(1/2)*sum(((a*e^2-c*d^2)*_R^2+2*d*(3*a*e^2-2*b*d*e+c*d^2)*_R+a*d^2*e^2-c*d^
4)/(_R^3*c+3*_R^2*b*e-3*_R^2*c*d+8*_R*a*e^2-4*_R*b*d*e+3*_R*c*d^2+b*d^2*e-c*d^3)
*ln(((e*x^2+d)^(1/2)-x*e^(1/2))^2-_R),_R=RootOf(c*_Z^4+(4*b*e-4*c*d)*_Z^3+(16*a*
e^2-8*b*d*e+6*c*d^2)*_Z^2+(4*b*d^2*e-4*c*d^3)*_Z+c*d^4))-1/8/a*e^(1/2)*d^2/((e*x
^2+d)^(1/2)-x*e^(1/2))^2+3/2/a*e^(1/2)*d*ln((e*x^2+d)^(1/2)-x*e^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 12.3863, size = 5480, normalized size = 21.08 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(sqrt(1/2)*a*x*sqrt(-(3*a^2*b*d*e^2 - 2*a^3*e^3 + (b^3 - 3*a*b*c)*d^3 - 3*(
a*b^2 - 2*a^2*c)*d^2*e + (a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - 9*a^4*d^2
*e^4 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^
2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(-(12*a^3*b*
d^3*e^3 - 6*a^4*d^2*e^4 - 2*(a*b^2*c - a^2*c^2)*d^6 + 2*(a*b^3 + 2*a^2*b*c)*d^5*
e - 4*(2*a^2*b^2 + a^3*c)*d^4*e^2 + ((a^3*b^2*c - 4*a^4*c^2)*d^3 - (a^3*b^3 - 4*
a^4*b*c)*d^2*e + (a^4*b^2 - 4*a^5*c)*d*e^2)*x^2*sqrt(-(18*a^3*b*d^3*e^3 - 9*a^4*
d^2*e^4 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2
*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)) + (27*a^3*b*d^2*e^4 - 12*a^4*d*e^5
+ (b^3*c - a*b*c^2)*d^6 - (b^4 + 6*a*b^2*c - 4*a^2*c^2)*d^5*e + 2*(4*a*b^3 + 5*
a^2*b*c)*d^4*e^2 - 2*(11*a^2*b^2 + 4*a^3*c)*d^3*e^3)*x^2 + 2*sqrt(1/2)*sqrt(e*x^
2 + d)*(((a^4*b^3 - 4*a^5*b*c)*d - 2*(a^5*b^2 - 4*a^6*c)*e)*x*sqrt(-(18*a^3*b*d^
3*e^3 - 9*a^4*d^2*e^4 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^
5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)) - ((a*b^4 - 5*a^2*b^
2*c + 4*a^3*c^2)*d^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d^3*e + 3*(a^3*b^2 - 4*a^4*c)*d^2
*e^2)*x)*sqrt(-(3*a^2*b*d*e^2 - 2*a^3*e^3 + (b^3 - 3*a*b*c)*d^3 - 3*(a*b^2 - 2*a
^2*c)*d^2*e + (a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - 9*a^4*d^2*e^4 - (b^4
- 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c
)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c)))/x^2) - sqrt(1/2)*a*x*sqrt
(-(3*a^2*b*d*e^2 - 2*a^3*e^3 + (b^3 - 3*a*b*c)*d^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e +
(a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - 9*a^4*d^2*e^4 - (b^4 - 2*a*b^2*c
+ a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a
^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(-(12*a^3*b*d^3*e^3 - 6*a^4*d^2*e^4
- 2*(a*b^2*c - a^2*c^2)*d^6 + 2*(a*b^3 + 2*a^2*b*c)*d^5*e - 4*(2*a^2*b^2 + a^3*c
)*d^4*e^2 + ((a^3*b^2*c - 4*a^4*c^2)*d^3 - (a^3*b^3 - 4*a^4*b*c)*d^2*e + (a^4*b^
2 - 4*a^5*c)*d*e^2)*x^2*sqrt(-(18*a^3*b*d^3*e^3 - 9*a^4*d^2*e^4 - (b^4 - 2*a*b^2
*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)
/(a^6*b^2 - 4*a^7*c)) + (27*a^3*b*d^2*e^4 - 12*a^4*d*e^5 + (b^3*c - a*b*c^2)*d^6
- (b^4 + 6*a*b^2*c - 4*a^2*c^2)*d^5*e + 2*(4*a*b^3 + 5*a^2*b*c)*d^4*e^2 - 2*(11
*a^2*b^2 + 4*a^3*c)*d^3*e^3)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((a^4*b^3 - 4*a^
5*b*c)*d - 2*(a^5*b^2 - 4*a^6*c)*e)*x*sqrt(-(18*a^3*b*d^3*e^3 - 9*a^4*d^2*e^4 -
(b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a
^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)) - ((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d^4 - 3
*(a^2*b^3 - 4*a^3*b*c)*d^3*e + 3*(a^3*b^2 - 4*a^4*c)*d^2*e^2)*x)*sqrt(-(3*a^2*b*
d*e^2 - 2*a^3*e^3 + (b^3 - 3*a*b*c)*d^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e + (a^3*b^2 -
4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - 9*a^4*d^2*e^4 - (b^4 - 2*a*b^2*c + a^2*c^2)*
d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*
a^7*c)))/(a^3*b^2 - 4*a^4*c)))/x^2) - sqrt(1/2)*a*x*sqrt(-(3*a^2*b*d*e^2 - 2*a^3
*e^3 + (b^3 - 3*a*b*c)*d^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e - (a^3*b^2 - 4*a^4*c)*sqr
t(-(18*a^3*b*d^3*e^3 - 9*a^4*d^2*e^4 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^
3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/(a^3
*b^2 - 4*a^4*c))*log(-(12*a^3*b*d^3*e^3 - 6*a^4*d^2*e^4 - 2*(a*b^2*c - a^2*c^2)*
d^6 + 2*(a*b^3 + 2*a^2*b*c)*d^5*e - 4*(2*a^2*b^2 + a^3*c)*d^4*e^2 - ((a^3*b^2*c
- 4*a^4*c^2)*d^3 - (a^3*b^3 - 4*a^4*b*c)*d^2*e + (a^4*b^2 - 4*a^5*c)*d*e^2)*x^2*
sqrt(-(18*a^3*b*d^3*e^3 - 9*a^4*d^2*e^4 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a
*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)) +
(27*a^3*b*d^2*e^4 - 12*a^4*d*e^5 + (b^3*c - a*b*c^2)*d^6 - (b^4 + 6*a*b^2*c - 4*
a^2*c^2)*d^5*e + 2*(4*a*b^3 + 5*a^2*b*c)*d^4*e^2 - 2*(11*a^2*b^2 + 4*a^3*c)*d^3*
e^3)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((a^4*b^3 - 4*a^5*b*c)*d - 2*(a^5*b^2 -
4*a^6*c)*e)*x*sqrt(-(18*a^3*b*d^3*e^3 - 9*a^4*d^2*e^4 - (b^4 - 2*a*b^2*c + a^2*c
^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2
- 4*a^7*c)) + ((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d
^3*e + 3*(a^3*b^2 - 4*a^4*c)*d^2*e^2)*x)*sqrt(-(3*a^2*b*d*e^2 - 2*a^3*e^3 + (b^3
- 3*a*b*c)*d^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e - (a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*
b*d^3*e^3 - 9*a^4*d^2*e^4 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c
)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^
4*c)))/x^2) + sqrt(1/2)*a*x*sqrt(-(3*a^2*b*d*e^2 - 2*a^3*e^3 + (b^3 - 3*a*b*c)*d
^3 - 3*(a*b^2 - 2*a^2*c)*d^2*e - (a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - 9
*a^4*d^2*e^4 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(
5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(-(1
2*a^3*b*d^3*e^3 - 6*a^4*d^2*e^4 - 2*(a*b^2*c - a^2*c^2)*d^6 + 2*(a*b^3 + 2*a^2*b
*c)*d^5*e - 4*(2*a^2*b^2 + a^3*c)*d^4*e^2 - ((a^3*b^2*c - 4*a^4*c^2)*d^3 - (a^3*
b^3 - 4*a^4*b*c)*d^2*e + (a^4*b^2 - 4*a^5*c)*d*e^2)*x^2*sqrt(-(18*a^3*b*d^3*e^3
- 9*a^4*d^2*e^4 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e -
3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)) + (27*a^3*b*d^2*e^4 - 12*a
^4*d*e^5 + (b^3*c - a*b*c^2)*d^6 - (b^4 + 6*a*b^2*c - 4*a^2*c^2)*d^5*e + 2*(4*a*
b^3 + 5*a^2*b*c)*d^4*e^2 - 2*(11*a^2*b^2 + 4*a^3*c)*d^3*e^3)*x^2 - 2*sqrt(1/2)*s
qrt(e*x^2 + d)*(((a^4*b^3 - 4*a^5*b*c)*d - 2*(a^5*b^2 - 4*a^6*c)*e)*x*sqrt(-(18*
a^3*b*d^3*e^3 - 9*a^4*d^2*e^4 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2
*b*c)*d^5*e - 3*(5*a^2*b^2 - 2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)) + ((a*b^4 -
5*a^2*b^2*c + 4*a^3*c^2)*d^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d^3*e + 3*(a^3*b^2 - 4*a^
4*c)*d^2*e^2)*x)*sqrt(-(3*a^2*b*d*e^2 - 2*a^3*e^3 + (b^3 - 3*a*b*c)*d^3 - 3*(a*b
^2 - 2*a^2*c)*d^2*e - (a^3*b^2 - 4*a^4*c)*sqrt(-(18*a^3*b*d^3*e^3 - 9*a^4*d^2*e^
4 - (b^4 - 2*a*b^2*c + a^2*c^2)*d^6 + 6*(a*b^3 - a^2*b*c)*d^5*e - 3*(5*a^2*b^2 -
2*a^3*c)*d^4*e^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c)))/x^2) + 4*sqrt(e*x
^2 + d)*d)/(a*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x**2)**(3/2)/(x**2*(a + b*x**2 + c*x**4)), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out