3.975 \(\int \frac{1}{x^7 \sqrt{-a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=154 \[ -\frac{b \left (12 a c+5 b^2\right ) \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac{\left (16 a c+15 b^2\right ) \sqrt{-a+b x^2+c x^4}}{48 a^3 x^2}+\frac{5 b \sqrt{-a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\sqrt{-a+b x^2+c x^4}}{6 a x^6} \]

[Out]

Sqrt[-a + b*x^2 + c*x^4]/(6*a*x^6) + (5*b*Sqrt[-a + b*x^2 + c*x^4])/(24*a^2*x^4)
+ ((15*b^2 + 16*a*c)*Sqrt[-a + b*x^2 + c*x^4])/(48*a^3*x^2) - (b*(5*b^2 + 12*a*
c)*ArcTan[(2*a - b*x^2)/(2*Sqrt[a]*Sqrt[-a + b*x^2 + c*x^4])])/(32*a^(7/2))

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Rubi [A]  time = 0.434747, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{b \left (12 a c+5 b^2\right ) \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{32 a^{7/2}}+\frac{\left (16 a c+15 b^2\right ) \sqrt{-a+b x^2+c x^4}}{48 a^3 x^2}+\frac{5 b \sqrt{-a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\sqrt{-a+b x^2+c x^4}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

Sqrt[-a + b*x^2 + c*x^4]/(6*a*x^6) + (5*b*Sqrt[-a + b*x^2 + c*x^4])/(24*a^2*x^4)
+ ((15*b^2 + 16*a*c)*Sqrt[-a + b*x^2 + c*x^4])/(48*a^3*x^2) - (b*(5*b^2 + 12*a*
c)*ArcTan[(2*a - b*x^2)/(2*Sqrt[a]*Sqrt[-a + b*x^2 + c*x^4])])/(32*a^(7/2))

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Rubi in Sympy [A]  time = 37.0042, size = 133, normalized size = 0.86 \[ \frac{\sqrt{- a + b x^{2} + c x^{4}}}{6 a x^{6}} + \frac{5 b \sqrt{- a + b x^{2} + c x^{4}}}{24 a^{2} x^{4}} + \frac{\left (16 a c + 15 b^{2}\right ) \sqrt{- a + b x^{2} + c x^{4}}}{48 a^{3} x^{2}} + \frac{b \left (12 a c + 5 b^{2}\right ) \operatorname{atan}{\left (\frac{- 2 a + b x^{2}}{2 \sqrt{a} \sqrt{- a + b x^{2} + c x^{4}}} \right )}}{32 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(c*x**4+b*x**2-a)**(1/2),x)

[Out]

sqrt(-a + b*x**2 + c*x**4)/(6*a*x**6) + 5*b*sqrt(-a + b*x**2 + c*x**4)/(24*a**2*
x**4) + (16*a*c + 15*b**2)*sqrt(-a + b*x**2 + c*x**4)/(48*a**3*x**2) + b*(12*a*c
+ 5*b**2)*atan((-2*a + b*x**2)/(2*sqrt(a)*sqrt(-a + b*x**2 + c*x**4)))/(32*a**(
7/2))

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Mathematica [A]  time = 0.277932, size = 139, normalized size = 0.9 \[ \frac{b \left (12 a c+5 b^2\right ) \left (\frac{\log (x)}{\sqrt{-a}}-\frac{\log \left (2 \sqrt{-a} \sqrt{-a+b x^2+c x^4}-2 a+b x^2\right )}{2 \sqrt{-a}}\right )}{16 a^3}+\sqrt{-a+b x^2+c x^4} \left (\frac{16 a c+15 b^2}{48 a^3 x^2}+\frac{5 b}{24 a^2 x^4}+\frac{1}{6 a x^6}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

(1/(6*a*x^6) + (5*b)/(24*a^2*x^4) + (15*b^2 + 16*a*c)/(48*a^3*x^2))*Sqrt[-a + b*
x^2 + c*x^4] + (b*(5*b^2 + 12*a*c)*(Log[x]/Sqrt[-a] - Log[-2*a + b*x^2 + 2*Sqrt[
-a]*Sqrt[-a + b*x^2 + c*x^4]]/(2*Sqrt[-a])))/(16*a^3)

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Maple [A]  time = 0.023, size = 202, normalized size = 1.3 \[{\frac{1}{6\,a{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}-a}}+{\frac{5\,b}{24\,{a}^{2}{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}-a}}+{\frac{5\,{b}^{2}}{16\,{a}^{3}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}-a}}-{\frac{5\,{b}^{3}}{32\,{a}^{3}}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,a+b{x}^{2}+2\,\sqrt{-a}\sqrt{c{x}^{4}+b{x}^{2}-a} \right ) } \right ){\frac{1}{\sqrt{-a}}}}-{\frac{3\,bc}{8\,{a}^{2}}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,a+b{x}^{2}+2\,\sqrt{-a}\sqrt{c{x}^{4}+b{x}^{2}-a} \right ) } \right ){\frac{1}{\sqrt{-a}}}}+{\frac{c}{3\,{a}^{2}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(c*x^4+b*x^2-a)^(1/2),x)

[Out]

1/6*(c*x^4+b*x^2-a)^(1/2)/a/x^6+5/24*b*(c*x^4+b*x^2-a)^(1/2)/a^2/x^4+5/16*b^2/a^
3/x^2*(c*x^4+b*x^2-a)^(1/2)-5/32*b^3/a^3/(-a)^(1/2)*ln((-2*a+b*x^2+2*(-a)^(1/2)*
(c*x^4+b*x^2-a)^(1/2))/x^2)-3/8*b/a^2*c/(-a)^(1/2)*ln((-2*a+b*x^2+2*(-a)^(1/2)*(
c*x^4+b*x^2-a)^(1/2))/x^2)+1/3*c/a^2/x^2*(c*x^4+b*x^2-a)^(1/2)

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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/(sqrt(c*x^4 + b*x^2 - a)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30754, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, b^{3} + 12 \, a b c\right )} x^{6} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} - a}{\left (a b x^{2} - 2 \, a^{2}\right )} +{\left ({\left (b^{2} - 4 \, a c\right )} x^{4} - 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{-a}}{x^{4}}\right ) + 4 \,{\left ({\left (15 \, b^{2} + 16 \, a c\right )} x^{4} + 10 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{c x^{4} + b x^{2} - a} \sqrt{-a}}{192 \, \sqrt{-a} a^{3} x^{6}}, \frac{3 \,{\left (5 \, b^{3} + 12 \, a b c\right )} x^{6} \arctan \left (\frac{b x^{2} - 2 \, a}{2 \, \sqrt{c x^{4} + b x^{2} - a} \sqrt{a}}\right ) + 2 \,{\left ({\left (15 \, b^{2} + 16 \, a c\right )} x^{4} + 10 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{c x^{4} + b x^{2} - a} \sqrt{a}}{96 \, a^{\frac{7}{2}} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 - a)*x^7),x, algorithm="fricas")

[Out]

[1/192*(3*(5*b^3 + 12*a*b*c)*x^6*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) + 4*((15*b^2 + 16*a*c
)*x^4 + 10*a*b*x^2 + 8*a^2)*sqrt(c*x^4 + b*x^2 - a)*sqrt(-a))/(sqrt(-a)*a^3*x^6)
, 1/96*(3*(5*b^3 + 12*a*b*c)*x^6*arctan(1/2*(b*x^2 - 2*a)/(sqrt(c*x^4 + b*x^2 -
a)*sqrt(a))) + 2*((15*b^2 + 16*a*c)*x^4 + 10*a*b*x^2 + 8*a^2)*sqrt(c*x^4 + b*x^2
- a)*sqrt(a))/(a^(7/2)*x^6)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{7} \sqrt{- a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(c*x**4+b*x**2-a)**(1/2),x)

[Out]

Integral(1/(x**7*sqrt(-a + b*x**2 + c*x**4)), x)

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GIAC/XCAS [A]  time = 0.521242, size = 158, normalized size = 1.03 \[ \frac{1}{48} \, \sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}}{\left (\frac{2 \,{\left (\frac{5 \, b}{a^{2}} + \frac{4}{a x^{2}}\right )}}{x^{2}} + \frac{15 \, a b^{2} + 16 \, a^{2} c}{a^{4}}\right )} + \frac{{\left (5 \, a b^{3} + 12 \, a^{2} b c\right )}{\rm ln}\left ({\left | -2 \, \sqrt{-a}{\left (\sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}} - \frac{\sqrt{-a}}{x^{2}}\right )} + b \right |}\right )}{32 \, \sqrt{-a} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^4 + b*x^2 - a)*x^7),x, algorithm="giac")

[Out]

1/48*sqrt(c + b/x^2 - a/x^4)*(2*(5*b/a^2 + 4/(a*x^2))/x^2 + (15*a*b^2 + 16*a^2*c
)/a^4) + 1/32*(5*a*b^3 + 12*a^2*b*c)*ln(abs(-2*sqrt(-a)*(sqrt(c + b/x^2 - a/x^4)
- sqrt(-a)/x^2) + b))/(sqrt(-a)*a^4)