3.988 \(\int \frac{1}{x^5 \left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=195 \[ -\frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 a^{7/2}}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a+b x^2+c x^4}}{8 a^3 x^2 \left (b^2-4 a c\right )}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a+b x^2+c x^4}}{4 a^2 x^4 \left (b^2-4 a c\right )}+\frac{-2 a c+b^2+b c x^2}{a x^4 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

[Out]

(b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*x^4*Sqrt[a + b*x^2 + c*x^4]) - ((5*b^2
- 12*a*c)*Sqrt[a + b*x^2 + c*x^4])/(4*a^2*(b^2 - 4*a*c)*x^4) + (b*(15*b^2 - 52*a
*c)*Sqrt[a + b*x^2 + c*x^4])/(8*a^3*(b^2 - 4*a*c)*x^2) - (3*(5*b^2 - 4*a*c)*ArcT
anh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(16*a^(7/2))

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Rubi [A]  time = 0.483718, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{16 a^{7/2}}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a+b x^2+c x^4}}{8 a^3 x^2 \left (b^2-4 a c\right )}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a+b x^2+c x^4}}{4 a^2 x^4 \left (b^2-4 a c\right )}+\frac{-2 a c+b^2+b c x^2}{a x^4 \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(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)*x^4*Sqrt[a + b*x^2 + c*x^4]) - ((5*b^2
- 12*a*c)*Sqrt[a + b*x^2 + c*x^4])/(4*a^2*(b^2 - 4*a*c)*x^4) + (b*(15*b^2 - 52*a
*c)*Sqrt[a + b*x^2 + c*x^4])/(8*a^3*(b^2 - 4*a*c)*x^2) - (3*(5*b^2 - 4*a*c)*ArcT
anh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(16*a^(7/2))

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Rubi in Sympy [A]  time = 46.7064, size = 180, normalized size = 0.92 \[ \frac{- 2 a c + b^{2} + b c x^{2}}{a x^{4} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}} - \frac{\left (- 12 a c + 5 b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{4 a^{2} x^{4} \left (- 4 a c + b^{2}\right )} + \frac{b \left (- 52 a c + 15 b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{8 a^{3} x^{2} \left (- 4 a c + b^{2}\right )} - \frac{3 \left (- 4 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{16 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

(-2*a*c + b**2 + b*c*x**2)/(a*x**4*(-4*a*c + b**2)*sqrt(a + b*x**2 + c*x**4)) -
(-12*a*c + 5*b**2)*sqrt(a + b*x**2 + c*x**4)/(4*a**2*x**4*(-4*a*c + b**2)) + b*(
-52*a*c + 15*b**2)*sqrt(a + b*x**2 + c*x**4)/(8*a**3*x**2*(-4*a*c + b**2)) - 3*(
-4*a*c + 5*b**2)*atanh((2*a + b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2 + c*x**4)))/(16
*a**(7/2))

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Mathematica [A]  time = 0.326525, size = 159, normalized size = 0.82 \[ \frac{3 \left (5 b^2-4 a c\right ) \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )\right )}{16 a^{7/2}}+\frac{\sqrt{a+b x^2+c x^4} \left (\frac{8 \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x^2+b^4+b^3 c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{2 a}{x^4}+\frac{7 b}{x^2}\right )}{8 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^2 + c*x^4)^(3/2)),x]

[Out]

(Sqrt[a + b*x^2 + c*x^4]*((-2*a)/x^4 + (7*b)/x^2 + (8*(b^4 - 4*a*b^2*c + 2*a^2*c
^2 + b^3*c*x^2 - 3*a*b*c^2*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4))))/(8*a^3) +
(3*(5*b^2 - 4*a*c)*(Log[x^2] - Log[2*a + b*x^2 + 2*Sqrt[a]*Sqrt[a + b*x^2 + c*x
^4]]))/(16*a^(7/2))

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Maple [A]  time = 0.023, size = 314, normalized size = 1.6 \[ -{\frac{1}{4\,a{x}^{4}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{5\,b}{8\,{a}^{2}{x}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{15\,{b}^{2}}{16\,{a}^{3}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{15\,{x}^{2}{b}^{3}c}{8\,{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{15\,{b}^{4}}{16\,{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{15\,{b}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}+{\frac{13\,b{c}^{2}{x}^{2}}{2\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{13\,{b}^{2}c}{4\,{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,c}{4\,{a}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{3\,c}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(c*x^4+b*x^2+a)^(3/2),x)

[Out]

-1/4/a/x^4/(c*x^4+b*x^2+a)^(1/2)+5/8*b/a^2/x^2/(c*x^4+b*x^2+a)^(1/2)+15/16*b^2/a
^3/(c*x^4+b*x^2+a)^(1/2)-15/8*b^3/a^3/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)*c*x^2-15
/16*b^4/a^3/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)-15/16*b^2/a^(7/2)*ln((2*a+b*x^2+2*
a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)+13/2*b/a^2*c^2/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(
1/2)*x^2+13/4*b^2/a^2*c/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)-3/4*c/a^2/(c*x^4+b*x^2
+a)^(1/2)+3/4*c/a^(5/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.354233, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} x^{6} +{\left (15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}\right )} x^{4} - 2 \, a^{2} b^{2} + 8 \, a^{3} c + 5 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{a} - 3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{8} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{6} +{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{4}\right )} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right )}{32 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{8} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{6} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{4}\right )} \sqrt{a}}, \frac{2 \,{\left ({\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} x^{6} +{\left (15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}\right )} x^{4} - 2 \, a^{2} b^{2} + 8 \, a^{3} c + 5 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a} - 3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{8} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{6} +{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{4}\right )} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right )}{16 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{8} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{6} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{4}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*x^5),x, algorithm="fricas")

[Out]

[1/32*(4*((15*b^3*c - 52*a*b*c^2)*x^6 + (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)*x^4 -
2*a^2*b^2 + 8*a^3*c + 5*(a*b^3 - 4*a^2*b*c)*x^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(a
) - 3*((5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*x^8 + (5*b^5 - 24*a*b^3*c + 16*a^2*
b*c^2)*x^6 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*x^4)*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^3*b^2*c - 4*a^4*c^2)*x^8 + (a^3*b^3 - 4*a^4*b*c)*x^6 + (a^4*b^2 - 4*a^5*
c)*x^4)*sqrt(a)), 1/16*(2*((15*b^3*c - 52*a*b*c^2)*x^6 + (15*b^4 - 62*a*b^2*c +
24*a^2*c^2)*x^4 - 2*a^2*b^2 + 8*a^3*c + 5*(a*b^3 - 4*a^2*b*c)*x^2)*sqrt(c*x^4 +
b*x^2 + a)*sqrt(-a) - 3*((5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*x^8 + (5*b^5 - 24
*a*b^3*c + 16*a^2*b*c^2)*x^6 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*x^4)*arctan
(1/2*(b*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)))/(((a^3*b^2*c - 4*a^4*c
^2)*x^8 + (a^3*b^3 - 4*a^4*b*c)*x^6 + (a^4*b^2 - 4*a^5*c)*x^4)*sqrt(-a))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{5} \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Integral(1/(x**5*(a + b*x**2 + c*x**4)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*x^5),x, algorithm="giac")

[Out]

integrate(1/((c*x^4 + b*x^2 + a)^(3/2)*x^5), x)