3.229 \(\int \frac{\sqrt{b x^2+c x^4}}{x^{11}} \, dx\)

Optimal. Leaf size=108 \[ \frac{16 c^3 \left (b x^2+c x^4\right )^{3/2}}{315 b^4 x^6}-\frac{8 c^2 \left (b x^2+c x^4\right )^{3/2}}{105 b^3 x^8}+\frac{2 c \left (b x^2+c x^4\right )^{3/2}}{21 b^2 x^{10}}-\frac{\left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}} \]

[Out]

-(b*x^2 + c*x^4)^(3/2)/(9*b*x^12) + (2*c*(b*x^2 + c*x^4)^(3/2))/(21*b^2*x^10) -
(8*c^2*(b*x^2 + c*x^4)^(3/2))/(105*b^3*x^8) + (16*c^3*(b*x^2 + c*x^4)^(3/2))/(31
5*b^4*x^6)

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Rubi [A]  time = 0.285518, 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 \[ \frac{16 c^3 \left (b x^2+c x^4\right )^{3/2}}{315 b^4 x^6}-\frac{8 c^2 \left (b x^2+c x^4\right )^{3/2}}{105 b^3 x^8}+\frac{2 c \left (b x^2+c x^4\right )^{3/2}}{21 b^2 x^{10}}-\frac{\left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2 + c*x^4]/x^11,x]

[Out]

-(b*x^2 + c*x^4)^(3/2)/(9*b*x^12) + (2*c*(b*x^2 + c*x^4)^(3/2))/(21*b^2*x^10) -
(8*c^2*(b*x^2 + c*x^4)^(3/2))/(105*b^3*x^8) + (16*c^3*(b*x^2 + c*x^4)^(3/2))/(31
5*b^4*x^6)

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Rubi in Sympy [A]  time = 28.5699, size = 99, normalized size = 0.92 \[ - \frac{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{9 b x^{12}} + \frac{2 c \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{21 b^{2} x^{10}} - \frac{8 c^{2} \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{105 b^{3} x^{8}} + \frac{16 c^{3} \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{315 b^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(1/2)/x**11,x)

[Out]

-(b*x**2 + c*x**4)**(3/2)/(9*b*x**12) + 2*c*(b*x**2 + c*x**4)**(3/2)/(21*b**2*x*
*10) - 8*c**2*(b*x**2 + c*x**4)**(3/2)/(105*b**3*x**8) + 16*c**3*(b*x**2 + c*x**
4)**(3/2)/(315*b**4*x**6)

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Mathematica [A]  time = 0.0328914, size = 68, normalized size = 0.63 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (-35 b^4-5 b^3 c x^2+6 b^2 c^2 x^4-8 b c^3 x^6+16 c^4 x^8\right )}{315 b^4 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^2 + c*x^4]/x^11,x]

[Out]

(Sqrt[x^2*(b + c*x^2)]*(-35*b^4 - 5*b^3*c*x^2 + 6*b^2*c^2*x^4 - 8*b*c^3*x^6 + 16
*c^4*x^8))/(315*b^4*x^10)

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Maple [A]  time = 0.008, size = 61, normalized size = 0.6 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -16\,{c}^{3}{x}^{6}+24\,b{c}^{2}{x}^{4}-30\,{b}^{2}c{x}^{2}+35\,{b}^{3} \right ) }{315\,{x}^{10}{b}^{4}}\sqrt{c{x}^{4}+b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(1/2)/x^11,x)

[Out]

-1/315*(c*x^2+b)*(-16*c^3*x^6+24*b*c^2*x^4-30*b^2*c*x^2+35*b^3)*(c*x^4+b*x^2)^(1
/2)/x^10/b^4

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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(sqrt(c*x^4 + b*x^2)/x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294473, size = 86, normalized size = 0.8 \[ \frac{{\left (16 \, c^{4} x^{8} - 8 \, b c^{3} x^{6} + 6 \, b^{2} c^{2} x^{4} - 5 \, b^{3} c x^{2} - 35 \, b^{4}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, b^{4} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)/x^11,x, algorithm="fricas")

[Out]

1/315*(16*c^4*x^8 - 8*b*c^3*x^6 + 6*b^2*c^2*x^4 - 5*b^3*c*x^2 - 35*b^4)*sqrt(c*x
^4 + b*x^2)/(b^4*x^10)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )}}{x^{11}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2)**(1/2)/x**11,x)

[Out]

Integral(sqrt(x**2*(b + c*x**2))/x**11, x)

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GIAC/XCAS [A]  time = 0.302402, size = 240, normalized size = 2.22 \[ \frac{32 \,{\left (315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} c^{\frac{9}{2}}{\rm sign}\left (x\right ) + 189 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} b c^{\frac{9}{2}}{\rm sign}\left (x\right ) + 84 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} b^{2} c^{\frac{9}{2}}{\rm sign}\left (x\right ) - 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} b^{3} c^{\frac{9}{2}}{\rm sign}\left (x\right ) + 9 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} b^{4} c^{\frac{9}{2}}{\rm sign}\left (x\right ) - b^{5} c^{\frac{9}{2}}{\rm sign}\left (x\right )\right )}}{315 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)/x^11,x, algorithm="giac")

[Out]

32/315*(315*(sqrt(c)*x - sqrt(c*x^2 + b))^10*c^(9/2)*sign(x) + 189*(sqrt(c)*x -
sqrt(c*x^2 + b))^8*b*c^(9/2)*sign(x) + 84*(sqrt(c)*x - sqrt(c*x^2 + b))^6*b^2*c^
(9/2)*sign(x) - 36*(sqrt(c)*x - sqrt(c*x^2 + b))^4*b^3*c^(9/2)*sign(x) + 9*(sqrt
(c)*x - sqrt(c*x^2 + b))^2*b^4*c^(9/2)*sign(x) - b^5*c^(9/2)*sign(x))/((sqrt(c)*
x - sqrt(c*x^2 + b))^2 - b)^9