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

Optimal. Leaf size=145 \[ \frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{7/2}}-\frac{\left (15 b^2-16 a c\right ) \sqrt{a+b x^2+c x^4}}{48 a^3 x^2}+\frac{5 b \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}-\frac{\sqrt{a+b x^2+c x^4}}{6 a x^6} \]

[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)
*ArcTanh[(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 = 0.388971, antiderivative size = 145, 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 \[ \frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{7/2}}-\frac{\left (15 b^2-16 a c\right ) \sqrt{a+b x^2+c x^4}}{48 a^3 x^2}+\frac{5 b \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}-\frac{\sqrt{a+b x^2+c x^4}}{6 a x^6} \]

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)
*ArcTanh[(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 = 35.4144, size = 133, normalized size = 0.92 \[ - \frac{\sqrt{a + b x^{2} + c x^{4}}}{6 a x^{6}} + \frac{5 b \sqrt{a + b x^{2} + c x^{4}}}{24 a^{2} x^{4}} - \frac{\left (- 16 a c + 15 b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{48 a^{3} x^{2}} + \frac{b \left (- 12 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{32 a^{\frac{7}{2}}} \]

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)*atanh((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 = 0.439272, size = 117, normalized size = 0.81 \[ \frac{b \left (12 a c-5 b^2\right ) \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )\right )}{32 a^{7/2}}-\frac{\sqrt{a+b x^2+c x^4} \left (8 a^2-2 a \left (5 b x^2+8 c x^4\right )+15 b^2 x^4\right )}{48 a^3 x^6} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x^2 + c*x^4]*(8*a^2 + 15*b^2*x^4 - 2*a*(5*b*x^2 + 8*c*x^4)))/(48*a^
3*x^6) + (b*(-5*b^2 + 12*a*c)*(Log[x^2] - Log[2*a + b*x^2 + 2*Sqrt[a]*Sqrt[a + b
*x^2 + c*x^4]]))/(32*a^(7/2))

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Maple [A]  time = 0.02, size = 176, normalized size = 1.2 \[ -{\frac{1}{6\,a{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,b}{24\,{a}^{2}{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,{b}^{2}}{16\,{a}^{3}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,{b}^{3}}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{3\,bc}{8}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{c}{3\,{a}^{2}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}} \]

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^(7/2)*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^(5/2)*c*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 = 0.309992, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, b^{3} - 12 \, a b c\right )} x^{6} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) + 4 \,{\left ({\left (15 \, b^{2} - 16 \, a c\right )} x^{4} - 10 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{a}}{192 \, a^{\frac{7}{2}} x^{6}}, \frac{3 \,{\left (5 \, b^{3} - 12 \, a b c\right )} x^{6} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) - 2 \,{\left ({\left (15 \, b^{2} - 16 \, a c\right )} x^{4} - 10 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}{96 \, \sqrt{-a} a^{3} x^{6}}\right ] \]

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))/(a^(7/2)*x^6), 1/96
*(3*(5*b^3 - 12*a*b*c)*x^6*arctan(1/2*(b*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2
+ a)*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))/(sqrt(-a)*a^3*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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{4} + b x^{2} + a} x^{7}}\,{d x} \]

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]

integrate(1/(sqrt(c*x^4 + b*x^2 + a)*x^7), x)