3.385 \(\int \frac{1}{x^6 \sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=443 \[ -\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{a^3 d x}-\frac{c \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^3 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{c \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a^3 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{2 b e \sqrt{d+e x^2}}{3 a^2 d^2 x}+\frac{b \sqrt{d+e x^2}}{3 a^2 d x^3}-\frac{8 e^2 \sqrt{d+e x^2}}{15 a d^3 x}+\frac{4 e \sqrt{d+e x^2}}{15 a d^2 x^3}-\frac{\sqrt{d+e x^2}}{5 a d x^5} \]

[Out]

-Sqrt[d + e*x^2]/(5*a*d*x^5) + (b*Sqrt[d + e*x^2])/(3*a^2*d*x^3) + (4*e*Sqrt[d +
e*x^2])/(15*a*d^2*x^3) - ((b^2 - a*c)*Sqrt[d + e*x^2])/(a^3*d*x) - (2*b*e*Sqrt[
d + e*x^2])/(3*a^2*d^2*x) - (8*e^2*Sqrt[d + e*x^2])/(15*a*d^3*x) - (c*(b^2 - a*c
+ (b*(b^2 - 3*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^3*Sqrt[b - Sqrt[b^
2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (c*(b^2 - a*c - (b*(b^2 -
3*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^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]*
Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi [A]  time = 3.27137, antiderivative size = 443, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{\left (b^2-a c\right ) \sqrt{d+e x^2}}{a^3 d x}-\frac{c \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a^3 \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{c \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a^3 \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{2 b e \sqrt{d+e x^2}}{3 a^2 d^2 x}+\frac{b \sqrt{d+e x^2}}{3 a^2 d x^3}-\frac{8 e^2 \sqrt{d+e x^2}}{15 a d^3 x}+\frac{4 e \sqrt{d+e x^2}}{15 a d^2 x^3}-\frac{\sqrt{d+e x^2}}{5 a d x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

-Sqrt[d + e*x^2]/(5*a*d*x^5) + (b*Sqrt[d + e*x^2])/(3*a^2*d*x^3) + (4*e*Sqrt[d +
e*x^2])/(15*a*d^2*x^3) - ((b^2 - a*c)*Sqrt[d + e*x^2])/(a^3*d*x) - (2*b*e*Sqrt[
d + e*x^2])/(3*a^2*d^2*x) - (8*e^2*Sqrt[d + e*x^2])/(15*a*d^3*x) - (c*(b^2 - a*c
+ (b*(b^2 - 3*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^3*Sqrt[b - Sqrt[b^
2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (c*(b^2 - a*c - (b*(b^2 -
3*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^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]*
Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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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(1/x**6/(c*x**4+b*x**2+a)/(e*x**2+d)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.942758, size = 0, normalized size = 0. \[ \int \frac{1}{x^6 \sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/(x^6*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)),x]

[Out]

Integrate[1/(x^6*Sqrt[d + e*x^2]*(a + b*x^2 + c*x^4)), x]

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Maple [C]  time = 0.042, size = 350, normalized size = 0.8 \[ -{\frac{1}{5\,ad{x}^{5}}\sqrt{e{x}^{2}+d}}+{\frac{4\,e}{15\,a{d}^{2}{x}^{3}}\sqrt{e{x}^{2}+d}}-{\frac{8\,{e}^{2}}{15\,a{d}^{3}x}\sqrt{e{x}^{2}+d}}-{\frac{-ac+{b}^{2}}{{a}^{3}dx}\sqrt{e{x}^{2}+d}}-{\frac{1}{2\,{a}^{3}}\sqrt{e}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,bde+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{c \left ( ac-{b}^{2} \right ){{\it \_R}}^{2}+2\, \left ( 4\,abce-a{c}^{2}d-2\,{b}^{3}e+{b}^{2}cd \right ){\it \_R}+a{c}^{2}{d}^{2}-{b}^{2}c{d}^{2}}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-x\sqrt{e} \right ) ^{2}-{\it \_R} \right ) }}+{\frac{b}{3\,{a}^{2}d{x}^{3}}\sqrt{e{x}^{2}+d}}-{\frac{2\,be}{3\,{a}^{2}{d}^{2}x}\sqrt{e{x}^{2}+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(c*x^4+b*x^2+a)/(e*x^2+d)^(1/2),x)

[Out]

-1/5*(e*x^2+d)^(1/2)/a/d/x^5+4/15*e*(e*x^2+d)^(1/2)/a/d^2/x^3-8/15*e^2*(e*x^2+d)
^(1/2)/a/d^3/x-(-a*c+b^2)*(e*x^2+d)^(1/2)/a^3/d/x-1/2/a^3*e^(1/2)*sum((c*(a*c-b^
2)*_R^2+2*(4*a*b*c*e-a*c^2*d-2*b^3*e+b^2*c*d)*_R+a*c^2*d^2-b^2*c*d^2)/(_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))+1/3*b*(e*x^2+d)^(1/2)/a^2/d/x^3-2/3*
b*e*(e*x^2+d)^(1/2)/a^2/d^2/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2} + a\right )} \sqrt{e x^{2} + d} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)*x^6),x, algorithm="maxima")

[Out]

integrate(1/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)*x^6), x)

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Fricas [A]  time = 94.5071, size = 13497, normalized size = 30.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)*x^6),x, algorithm="fricas")

[Out]

-1/60*(15*sqrt(1/2)*a^3*d^3*x^5*sqrt(-((b^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7
*a^3*b*c^4)*d - (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 2*a^4*c^4)*
e - ((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*
c)*e^2)*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a
^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*
b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e +
(b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a
^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3
*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a
^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)))/((a^7*b^2*c - 4*a^8*c
^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2))*log((((a^7*b^2*c
^5 - 4*a^8*c^6)*d^3 - (a^7*b^3*c^4 - 4*a^8*b*c^5)*d^2*e + (a^8*b^2*c^4 - 4*a^9*c
^5)*d*e^2)*x^2*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5
+ 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c - 11*a*b^11*c^2 +
46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)
*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4
- 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a
^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2
- 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)) + 2*(a*b^6*c^5
- 5*a^2*b^4*c^6 + 6*a^3*b^2*c^7 - a^4*c^8)*d^2 - 2*(a*b^7*c^4 - 6*a^2*b^5*c^5 +
10*a^3*b^3*c^6 - 4*a^4*b*c^7)*d*e - ((b^7*c^5 - 5*a*b^5*c^6 + 6*a^2*b^3*c^7 - a^
3*b*c^8)*d^2 - (b^8*c^4 - 2*a*b^6*c^5 - 10*a^2*b^4*c^6 + 20*a^3*b^2*c^7 - 4*a^4*
c^8)*d*e + 4*(a*b^7*c^4 - 6*a^2*b^5*c^5 + 10*a^3*b^3*c^6 - 4*a^4*b*c^7)*e^2)*x^2
+ 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((a^8*b^5*c^2 - 7*a^9*b^3*c^3 + 12*a^10*b*c^4)*d
^3 - (2*a^8*b^6*c - 15*a^9*b^4*c^2 + 30*a^10*b^2*c^3 - 8*a^11*c^4)*d^2*e + (a^8*
b^7 - 7*a^9*b^5*c + 11*a^10*b^3*c^2 + 4*a^11*b*c^3)*d*e^2 - (a^9*b^6 - 8*a^10*b^
4*c + 18*a^11*b^2*c^2 - 8*a^12*c^3)*e^3)*x*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*
a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 -
2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 3
4*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a
^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*
c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15
*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a
^17*c)*e^4)) + ((a*b^10*c^2 - 10*a^2*b^8*c^3 + 35*a^3*b^6*c^4 - 51*a^4*b^4*c^5 +
29*a^5*b^2*c^6 - 4*a^6*c^7)*d^2 - (2*a*b^11*c - 22*a^2*b^9*c^2 + 88*a^3*b^7*c^3
- 155*a^4*b^5*c^4 + 114*a^5*b^3*c^5 - 24*a^6*b*c^6)*d*e + (a*b^12 - 12*a^2*b^10
*c + 54*a^3*b^8*c^2 - 112*a^4*b^6*c^3 + 104*a^5*b^4*c^4 - 32*a^6*b^2*c^5)*e^2)*x
)*sqrt(-((b^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7*a^3*b*c^4)*d - (b^8 - 8*a*b^6
*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 2*a^4*c^4)*e - ((a^7*b^2*c - 4*a^8*c^2)*d
^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)*sqrt(((b^12*c^2 - 10*a
*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 +
a^6*c^8)*d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*
a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b
^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e
^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^
14*b^4 - 2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 +
(a^16*b^2 - 4*a^17*c)*e^4)))/((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c
)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)))/x^2) - 15*sqrt(1/2)*a^3*d^3*x^5*sqrt(-((b^7*c
- 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7*a^3*b*c^4)*d - (b^8 - 8*a*b^6*c + 20*a^2*b^4
*c^2 - 16*a^3*b^2*c^3 + 2*a^4*c^4)*e - ((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 -
4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*
a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 -
2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 3
4*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a
^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*
c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15
*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a
^17*c)*e^4)))/((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^
2 - 4*a^9*c)*e^2))*log((((a^7*b^2*c^5 - 4*a^8*c^6)*d^3 - (a^7*b^3*c^4 - 4*a^8*b*
c^5)*d^2*e + (a^8*b^2*c^4 - 4*a^9*c^5)*d*e^2)*x^2*sqrt(((b^12*c^2 - 10*a*b^10*c^
3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)
*d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*
c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2
- 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^
14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 -
2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^
2 - 4*a^17*c)*e^4)) + 2*(a*b^6*c^5 - 5*a^2*b^4*c^6 + 6*a^3*b^2*c^7 - a^4*c^8)*d^
2 - 2*(a*b^7*c^4 - 6*a^2*b^5*c^5 + 10*a^3*b^3*c^6 - 4*a^4*b*c^7)*d*e - ((b^7*c^5
- 5*a*b^5*c^6 + 6*a^2*b^3*c^7 - a^3*b*c^8)*d^2 - (b^8*c^4 - 2*a*b^6*c^5 - 10*a^
2*b^4*c^6 + 20*a^3*b^2*c^7 - 4*a^4*c^8)*d*e + 4*(a*b^7*c^4 - 6*a^2*b^5*c^5 + 10*
a^3*b^3*c^6 - 4*a^4*b*c^7)*e^2)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((a^8*b^5*c^2
- 7*a^9*b^3*c^3 + 12*a^10*b*c^4)*d^3 - (2*a^8*b^6*c - 15*a^9*b^4*c^2 + 30*a^10*
b^2*c^3 - 8*a^11*c^4)*d^2*e + (a^8*b^7 - 7*a^9*b^5*c + 11*a^10*b^3*c^2 + 4*a^11*
b*c^3)*d*e^2 - (a^9*b^6 - 8*a^10*b^4*c + 18*a^11*b^2*c^2 - 8*a^12*c^3)*e^3)*x*sq
rt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6
- 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 -
91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12
*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5
+ 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^1
5*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 -
4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)) + ((a*b^10*c^2 - 10*a^2*b^8*c^3
+ 35*a^3*b^6*c^4 - 51*a^4*b^4*c^5 + 29*a^5*b^2*c^6 - 4*a^6*c^7)*d^2 - (2*a*b^11*
c - 22*a^2*b^9*c^2 + 88*a^3*b^7*c^3 - 155*a^4*b^5*c^4 + 114*a^5*b^3*c^5 - 24*a^6
*b*c^6)*d*e + (a*b^12 - 12*a^2*b^10*c + 54*a^3*b^8*c^2 - 112*a^4*b^6*c^3 + 104*a
^5*b^4*c^4 - 32*a^6*b^2*c^5)*e^2)*x)*sqrt(-((b^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^
3 - 7*a^3*b*c^4)*d - (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 2*a^4*
c^4)*e - ((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4
*a^9*c)*e^2)*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 +
46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46
*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d
*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 -
80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^1
4*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 -
2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)))/((a^7*b^2*c - 4*
a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)))/x^2) + 15*
sqrt(1/2)*a^3*d^3*x^5*sqrt(-((b^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7*a^3*b*c^4
)*d - (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 2*a^4*c^4)*e + ((a^7*
b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)*sq
rt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6
- 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 -
91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12
*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5
+ 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^1
5*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 -
4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)))/((a^7*b^2*c - 4*a^8*c^2)*d^2 -
(a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2))*log(-(((a^7*b^2*c^5 - 4*a^
8*c^6)*d^3 - (a^7*b^3*c^4 - 4*a^8*b*c^5)*d^2*e + (a^8*b^2*c^4 - 4*a^9*c^5)*d*e^2
)*x^2*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4
*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^
9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b
^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5
*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c
- 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^1
5*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)) - 2*(a*b^6*c^5 - 5*a^2*b
^4*c^6 + 6*a^3*b^2*c^7 - a^4*c^8)*d^2 + 2*(a*b^7*c^4 - 6*a^2*b^5*c^5 + 10*a^3*b^
3*c^6 - 4*a^4*b*c^7)*d*e + ((b^7*c^5 - 5*a*b^5*c^6 + 6*a^2*b^3*c^7 - a^3*b*c^8)*
d^2 - (b^8*c^4 - 2*a*b^6*c^5 - 10*a^2*b^4*c^6 + 20*a^3*b^2*c^7 - 4*a^4*c^8)*d*e
+ 4*(a*b^7*c^4 - 6*a^2*b^5*c^5 + 10*a^3*b^3*c^6 - 4*a^4*b*c^7)*e^2)*x^2 + 2*sqrt
(1/2)*sqrt(e*x^2 + d)*(((a^8*b^5*c^2 - 7*a^9*b^3*c^3 + 12*a^10*b*c^4)*d^3 - (2*a
^8*b^6*c - 15*a^9*b^4*c^2 + 30*a^10*b^2*c^3 - 8*a^11*c^4)*d^2*e + (a^8*b^7 - 7*a
^9*b^5*c + 11*a^10*b^3*c^2 + 4*a^11*b*c^3)*d*e^2 - (a^9*b^6 - 8*a^10*b^4*c + 18*
a^11*b^2*c^2 - 8*a^12*c^3)*e^3)*x*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c
^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c
- 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3
*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^
3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a
^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c -
8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^
4)) - ((a*b^10*c^2 - 10*a^2*b^8*c^3 + 35*a^3*b^6*c^4 - 51*a^4*b^4*c^5 + 29*a^5*b
^2*c^6 - 4*a^6*c^7)*d^2 - (2*a*b^11*c - 22*a^2*b^9*c^2 + 88*a^3*b^7*c^3 - 155*a^
4*b^5*c^4 + 114*a^5*b^3*c^5 - 24*a^6*b*c^6)*d*e + (a*b^12 - 12*a^2*b^10*c + 54*a
^3*b^8*c^2 - 112*a^4*b^6*c^3 + 104*a^5*b^4*c^4 - 32*a^6*b^2*c^5)*e^2)*x)*sqrt(-(
(b^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7*a^3*b*c^4)*d - (b^8 - 8*a*b^6*c + 20*a
^2*b^4*c^2 - 16*a^3*b^2*c^3 + 2*a^4*c^4)*e + ((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7
*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)*sqrt(((b^12*c^2 - 10*a*b^10*c^3
+ 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*
d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c
^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 -
128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^1
4*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 -
2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2
- 4*a^17*c)*e^4)))/((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (
a^8*b^2 - 4*a^9*c)*e^2)))/x^2) - 15*sqrt(1/2)*a^3*d^3*x^5*sqrt(-((b^7*c - 7*a*b^
5*c^2 + 14*a^2*b^3*c^3 - 7*a^3*b*c^4)*d - (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16
*a^3*b^2*c^3 + 2*a^4*c^4)*e + ((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*
c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c
^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c
- 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3
*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^
3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a
^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c -
8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^
4)))/((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9
*c)*e^2))*log(-(((a^7*b^2*c^5 - 4*a^8*c^6)*d^3 - (a^7*b^3*c^4 - 4*a^8*b*c^5)*d^2
*e + (a^8*b^2*c^4 - 4*a^9*c^5)*d*e^2)*x^2*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a
^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2
*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34
*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^
3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c
^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*
b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^
17*c)*e^4)) - 2*(a*b^6*c^5 - 5*a^2*b^4*c^6 + 6*a^3*b^2*c^7 - a^4*c^8)*d^2 + 2*(a
*b^7*c^4 - 6*a^2*b^5*c^5 + 10*a^3*b^3*c^6 - 4*a^4*b*c^7)*d*e + ((b^7*c^5 - 5*a*b
^5*c^6 + 6*a^2*b^3*c^7 - a^3*b*c^8)*d^2 - (b^8*c^4 - 2*a*b^6*c^5 - 10*a^2*b^4*c^
6 + 20*a^3*b^2*c^7 - 4*a^4*c^8)*d*e + 4*(a*b^7*c^4 - 6*a^2*b^5*c^5 + 10*a^3*b^3*
c^6 - 4*a^4*b*c^7)*e^2)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*(((a^8*b^5*c^2 - 7*a^9
*b^3*c^3 + 12*a^10*b*c^4)*d^3 - (2*a^8*b^6*c - 15*a^9*b^4*c^2 + 30*a^10*b^2*c^3
- 8*a^11*c^4)*d^2*e + (a^8*b^7 - 7*a^9*b^5*c + 11*a^10*b^3*c^2 + 4*a^11*b*c^3)*d
*e^2 - (a^9*b^6 - 8*a^10*b^4*c + 18*a^11*b^2*c^2 - 8*a^12*c^3)*e^3)*x*sqrt(((b^1
2*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*b^4*c^6 - 12*a^
5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9*c^3 - 91*a^3*b
^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^14 - 12*a*b^12*
c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*b^4*c^5 + 16*a^
6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c - 4*a^15*b*c^2)
*d^3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15*b^3 - 4*a^16*b
*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)) - ((a*b^10*c^2 - 10*a^2*b^8*c^3 + 35*a^3
*b^6*c^4 - 51*a^4*b^4*c^5 + 29*a^5*b^2*c^6 - 4*a^6*c^7)*d^2 - (2*a*b^11*c - 22*a
^2*b^9*c^2 + 88*a^3*b^7*c^3 - 155*a^4*b^5*c^4 + 114*a^5*b^3*c^5 - 24*a^6*b*c^6)*
d*e + (a*b^12 - 12*a^2*b^10*c + 54*a^3*b^8*c^2 - 112*a^4*b^6*c^3 + 104*a^5*b^4*c
^4 - 32*a^6*b^2*c^5)*e^2)*x)*sqrt(-((b^7*c - 7*a*b^5*c^2 + 14*a^2*b^3*c^3 - 7*a^
3*b*c^4)*d - (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 2*a^4*c^4)*e +
((a^7*b^2*c - 4*a^8*c^2)*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*
e^2)*sqrt(((b^12*c^2 - 10*a*b^10*c^3 + 37*a^2*b^8*c^4 - 62*a^3*b^6*c^5 + 46*a^4*
b^4*c^6 - 12*a^5*b^2*c^7 + a^6*c^8)*d^2 - 2*(b^13*c - 11*a*b^11*c^2 + 46*a^2*b^9
*c^3 - 91*a^3*b^7*c^4 + 86*a^4*b^5*c^5 - 34*a^5*b^3*c^6 + 4*a^6*b*c^7)*d*e + (b^
14 - 12*a*b^12*c + 56*a^2*b^10*c^2 - 128*a^3*b^8*c^3 + 148*a^4*b^6*c^4 - 80*a^5*
b^4*c^5 + 16*a^6*b^2*c^6)*e^2)/((a^14*b^2*c^2 - 4*a^15*c^3)*d^4 - 2*(a^14*b^3*c
- 4*a^15*b*c^2)*d^3*e + (a^14*b^4 - 2*a^15*b^2*c - 8*a^16*c^2)*d^2*e^2 - 2*(a^15
*b^3 - 4*a^16*b*c)*d*e^3 + (a^16*b^2 - 4*a^17*c)*e^4)))/((a^7*b^2*c - 4*a^8*c^2)
*d^2 - (a^7*b^3 - 4*a^8*b*c)*d*e + (a^8*b^2 - 4*a^9*c)*e^2)))/x^2) + 4*((10*a*b*
d*e + 8*a^2*e^2 + 15*(b^2 - a*c)*d^2)*x^4 + 3*a^2*d^2 - (5*a*b*d^2 + 4*a^2*d*e)*
x^2)*sqrt(e*x^2 + d))/(a^3*d^3*x^5)

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

[Out]

Timed out

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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(1/((c*x^4 + b*x^2 + a)*sqrt(e*x^2 + d)*x^6),x, algorithm="giac")

[Out]

Timed out