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

Optimal. Leaf size=419 \[ \frac{2 c^2 \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{b-\sqrt{b^2-4 a c}} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )^{3/2}}+\frac{2 c^2 \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{\sqrt{b^2-4 a c}+b} \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )^{3/2}}-\frac{e x \left (a c e+b^2 (-e)+b c d\right )}{a^2 d \sqrt{d+e x^2} \left (e (a e-b d)+c d^2\right )}+\frac{2 e x (4 a e+3 b d)}{3 a^2 d^3 \sqrt{d+e x^2}}+\frac{4 a e+3 b d}{3 a^2 d^2 x \sqrt{d+e x^2}}-\frac{1}{3 a d x^3 \sqrt{d+e x^2}} \]

[Out]

-1/(3*a*d*x^3*Sqrt[d + e*x^2]) + (3*b*d + 4*a*e)/(3*a^2*d^2*x*Sqrt[d + e*x^2]) +
(2*e*(3*b*d + 4*a*e)*x)/(3*a^2*d^3*Sqrt[d + e*x^2]) - (e*(b*c*d - b^2*e + a*c*e
)*x)/(a^2*d*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[d + e*x^2]) + (2*c^2*(b + (b^2 - 2*a
*c)/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])])/(a^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(2*c*
d - (b - Sqrt[b^2 - 4*a*c])*e)^(3/2)) + (2*c^2*(b - (b^2 - 2*a*c)/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])])/(a^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(2*c*d - (b + Sqrt[b^2
- 4*a*c])*e)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 12.8139, antiderivative size = 647, normalized size of antiderivative = 1.54, number of steps used = 15, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276 \[ \frac{\sqrt{d+e x^2} \left (a c e+b^2 (-e)+b c d\right )}{a^2 d x \left (a e^2-b d e+c d^2\right )}+\frac{c \left (\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac{c \left (-\frac{3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt{b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a^2 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac{2 e \sqrt{d+e x^2} (c d-b e)}{3 a d^2 x \left (a e^2-b d e+c d^2\right )}-\frac{e^2}{3 d x^3 \sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{d+e x^2} (c d-b e)}{3 a d x^3 \left (a e^2-b d e+c d^2\right )}+\frac{4 e^3}{3 d^2 x \sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )}+\frac{8 e^4 x}{3 d^3 \sqrt{d+e x^2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-e^2/(3*d*(c*d^2 - b*d*e + a*e^2)*x^3*Sqrt[d + e*x^2]) + (4*e^3)/(3*d^2*(c*d^2 -
b*d*e + a*e^2)*x*Sqrt[d + e*x^2]) + (8*e^4*x)/(3*d^3*(c*d^2 - b*d*e + a*e^2)*Sq
rt[d + e*x^2]) - ((c*d - b*e)*Sqrt[d + e*x^2])/(3*a*d*(c*d^2 - b*d*e + a*e^2)*x^
3) + (2*e*(c*d - b*e)*Sqrt[d + e*x^2])/(3*a*d^2*(c*d^2 - b*d*e + a*e^2)*x) + ((b
*c*d - b^2*e + a*c*e)*Sqrt[d + e*x^2])/(a^2*d*(c*d^2 - b*d*e + a*e^2)*x) + (c*(b
*c*d - b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*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])])/(a^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b
^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) + (c*(b*c*d - b^2*e + a*c*e - (b^2*c*d
- 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(a^2*Sqrt
[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e +
a*e^2))

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 1.36056, size = 0, normalized size = 0. \[ \int \frac{1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.052, size = 541, normalized size = 1.3 \[ -{\frac{1}{3\,ad{x}^{3}}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+{\frac{4\,e}{3\,a{d}^{2}x}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+{\frac{8\,{e}^{2}x}{3\,a{d}^{3}}{\frac{1}{\sqrt{e{x}^{2}+d}}}}-8\,{\frac{{e}^{3/2}c}{a \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}+8\,{\frac{{e}^{3/2}{b}^{2}}{{a}^{2} \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}-8\,{\frac{\sqrt{e}bcd}{{a}^{2} \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ) \left ( 2\,e{x}^{2}-2\,\sqrt{e}\sqrt{e{x}^{2}+d}x+2\,d \right ) }}-2\,{\frac{\sqrt{e}}{{a}^{2} \left ( 4\,a{e}^{2}-4\,bde+4\,c{d}^{2} \right ) }\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{ \left ( c \left ( ace-{b}^{2}e+bcd \right ){{\it \_R}}^{2}+2\, \left ( 4\,ab{e}^{2}c-3\,a{c}^{2}de-2\,{b}^{3}{e}^{2}+3\,{b}^{2}dec-b{c}^{2}{d}^{2} \right ){\it \_R}+a{c}^{2}{d}^{2}e-{b}^{2}c{d}^{2}e+b{c}^{2}{d}^{3} \right ) \ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}}}+{\frac{b}{{a}^{2}dx}{\frac{1}{\sqrt{e{x}^{2}+d}}}}+2\,{\frac{bex}{{a}^{2}{d}^{2}\sqrt{e{x}^{2}+d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/a/d/x^3/(e*x^2+d)^(1/2)+4/3/a/d^2*e/x/(e*x^2+d)^(1/2)+8/3/a/d^3*e^2*x/(e*x^
2+d)^(1/2)-8/a*e^(3/2)/(4*a*e^2-4*b*d*e+4*c*d^2)/(2*e*x^2-2*e^(1/2)*(e*x^2+d)^(1
/2)*x+2*d)*c+8/a^2*e^(3/2)/(4*a*e^2-4*b*d*e+4*c*d^2)/(2*e*x^2-2*e^(1/2)*(e*x^2+d
)^(1/2)*x+2*d)*b^2-8/a^2*e^(1/2)/(4*a*e^2-4*b*d*e+4*c*d^2)/(2*e*x^2-2*e^(1/2)*(e
*x^2+d)^(1/2)*x+2*d)*b*c*d-2/a^2*e^(1/2)/(4*a*e^2-4*b*d*e+4*c*d^2)*sum((c*(a*c*e
-b^2*e+b*c*d)*_R^2+2*(4*a*b*c*e^2-3*a*c^2*d*e-2*b^3*e^2+3*b^2*c*d*e-b*c^2*d^2)*_
R+a*c^2*d^2*e-b^2*c*d^2*e+b*c^2*d^3)/(_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=RootO
f(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))+b/a^2/d/x/(e*x^2+d)^(1/2)+2*b/a^2*e/d^2*x/(e*x^2+d)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Timed out