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

Optimal. Leaf size=108 \[ \frac{16 c^3 \sqrt{b x^2+c x^4}}{35 b^4 x^2}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{35 b^3 x^4}+\frac{6 c \sqrt{b x^2+c x^4}}{35 b^2 x^6}-\frac{\sqrt{b x^2+c x^4}}{7 b x^8} \]

[Out]

-Sqrt[b*x^2 + c*x^4]/(7*b*x^8) + (6*c*Sqrt[b*x^2 + c*x^4])/(35*b^2*x^6) - (8*c^2
*Sqrt[b*x^2 + c*x^4])/(35*b^3*x^4) + (16*c^3*Sqrt[b*x^2 + c*x^4])/(35*b^4*x^2)

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Rubi [A]  time = 0.294481, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{16 c^3 \sqrt{b x^2+c x^4}}{35 b^4 x^2}-\frac{8 c^2 \sqrt{b x^2+c x^4}}{35 b^3 x^4}+\frac{6 c \sqrt{b x^2+c x^4}}{35 b^2 x^6}-\frac{\sqrt{b x^2+c x^4}}{7 b x^8} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[b*x^2 + c*x^4]/(7*b*x^8) + (6*c*Sqrt[b*x^2 + c*x^4])/(35*b^2*x^6) - (8*c^2
*Sqrt[b*x^2 + c*x^4])/(35*b^3*x^4) + (16*c^3*Sqrt[b*x^2 + c*x^4])/(35*b^4*x^2)

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Rubi in Sympy [A]  time = 29.2785, size = 99, normalized size = 0.92 \[ - \frac{\sqrt{b x^{2} + c x^{4}}}{7 b x^{8}} + \frac{6 c \sqrt{b x^{2} + c x^{4}}}{35 b^{2} x^{6}} - \frac{8 c^{2} \sqrt{b x^{2} + c x^{4}}}{35 b^{3} x^{4}} + \frac{16 c^{3} \sqrt{b x^{2} + c x^{4}}}{35 b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(b*x**2 + c*x**4)/(7*b*x**8) + 6*c*sqrt(b*x**2 + c*x**4)/(35*b**2*x**6) - 8
*c**2*sqrt(b*x**2 + c*x**4)/(35*b**3*x**4) + 16*c**3*sqrt(b*x**2 + c*x**4)/(35*b
**4*x**2)

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Mathematica [A]  time = 0.0414759, size = 57, normalized size = 0.53 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (-5 b^3+6 b^2 c x^2-8 b c^2 x^4+16 c^3 x^6\right )}{35 b^4 x^8} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x^2*(b + c*x^2)]*(-5*b^3 + 6*b^2*c*x^2 - 8*b*c^2*x^4 + 16*c^3*x^6))/(35*b^
4*x^8)

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Maple [A]  time = 0.008, size = 61, normalized size = 0.6 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -16\,{c}^{3}{x}^{6}+8\,b{c}^{2}{x}^{4}-6\,{b}^{2}c{x}^{2}+5\,{b}^{3} \right ) }{35\,{b}^{4}{x}^{6}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(c*x^2+b)*(-16*c^3*x^6+8*b*c^2*x^4-6*b^2*c*x^2+5*b^3)/x^6/b^4/(c*x^4+b*x^2
)^(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)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272968, size = 72, normalized size = 0.67 \[ \frac{{\left (16 \, c^{3} x^{6} - 8 \, b c^{2} x^{4} + 6 \, b^{2} c x^{2} - 5 \, b^{3}\right )} \sqrt{c x^{4} + b x^{2}}}{35 \, b^{4} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35*(16*c^3*x^6 - 8*b*c^2*x^4 + 6*b^2*c*x^2 - 5*b^3)*sqrt(c*x^4 + b*x^2)/(b^4*x
^8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{7} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.278944, size = 77, normalized size = 0.71 \[ -\frac{5 \,{\left (c + \frac{b}{x^{2}}\right )}^{\frac{7}{2}} - 21 \,{\left (c + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} c + 35 \,{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} c^{2} - 35 \, \sqrt{c + \frac{b}{x^{2}}} c^{3}}{35 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(5*(c + b/x^2)^(7/2) - 21*(c + b/x^2)^(5/2)*c + 35*(c + b/x^2)^(3/2)*c^2 -
35*sqrt(c + b/x^2)*c^3)/b^4