3.230 \(\int \frac{\sqrt{b x^2+c x^4}}{x^{13}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{128 c^4 \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6}+\frac{64 c^3 \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac{16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac{8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac{\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}} \]

[Out]

-(b*x^2 + c*x^4)^(3/2)/(11*b*x^14) + (8*c*(b*x^2 + c*x^4)^(3/2))/(99*b^2*x^12) -
(16*c^2*(b*x^2 + c*x^4)^(3/2))/(231*b^3*x^10) + (64*c^3*(b*x^2 + c*x^4)^(3/2))/
(1155*b^4*x^8) - (128*c^4*(b*x^2 + c*x^4)^(3/2))/(3465*b^5*x^6)

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Rubi [A]  time = 0.364311, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{128 c^4 \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6}+\frac{64 c^3 \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}-\frac{16 c^2 \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}+\frac{8 c \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}-\frac{\left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x^2 + c*x^4]/x^13,x]

[Out]

-(b*x^2 + c*x^4)^(3/2)/(11*b*x^14) + (8*c*(b*x^2 + c*x^4)^(3/2))/(99*b^2*x^12) -
(16*c^2*(b*x^2 + c*x^4)^(3/2))/(231*b^3*x^10) + (64*c^3*(b*x^2 + c*x^4)^(3/2))/
(1155*b^4*x^8) - (128*c^4*(b*x^2 + c*x^4)^(3/2))/(3465*b^5*x^6)

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Rubi in Sympy [A]  time = 36.9205, size = 126, normalized size = 0.93 \[ - \frac{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{11 b x^{14}} + \frac{8 c \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{99 b^{2} x^{12}} - \frac{16 c^{2} \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{231 b^{3} x^{10}} + \frac{64 c^{3} \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{1155 b^{4} x^{8}} - \frac{128 c^{4} \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{3465 b^{5} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(1/2)/x**13,x)

[Out]

-(b*x**2 + c*x**4)**(3/2)/(11*b*x**14) + 8*c*(b*x**2 + c*x**4)**(3/2)/(99*b**2*x
**12) - 16*c**2*(b*x**2 + c*x**4)**(3/2)/(231*b**3*x**10) + 64*c**3*(b*x**2 + c*
x**4)**(3/2)/(1155*b**4*x**8) - 128*c**4*(b*x**2 + c*x**4)**(3/2)/(3465*b**5*x**
6)

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Mathematica [A]  time = 0.0368774, size = 79, normalized size = 0.58 \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )} \left (315 b^5+35 b^4 c x^2-40 b^3 c^2 x^4+48 b^2 c^3 x^6-64 b c^4 x^8+128 c^5 x^{10}\right )}{3465 b^5 x^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x^2 + c*x^4]/x^13,x]

[Out]

-(Sqrt[x^2*(b + c*x^2)]*(315*b^5 + 35*b^4*c*x^2 - 40*b^3*c^2*x^4 + 48*b^2*c^3*x^
6 - 64*b*c^4*x^8 + 128*c^5*x^10))/(3465*b^5*x^12)

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Maple [A]  time = 0.009, size = 72, normalized size = 0.5 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 128\,{c}^{4}{x}^{8}-192\,{c}^{3}{x}^{6}b+240\,{c}^{2}{x}^{4}{b}^{2}-280\,c{x}^{2}{b}^{3}+315\,{b}^{4} \right ) }{3465\,{x}^{12}{b}^{5}}\sqrt{c{x}^{4}+b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(1/2)/x^13,x)

[Out]

-1/3465*(c*x^2+b)*(128*c^4*x^8-192*b*c^3*x^6+240*b^2*c^2*x^4-280*b^3*c*x^2+315*b
^4)*(c*x^4+b*x^2)^(1/2)/x^12/b^5

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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^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.332517, size = 101, normalized size = 0.74 \[ -\frac{{\left (128 \, c^{5} x^{10} - 64 \, b c^{4} x^{8} + 48 \, b^{2} c^{3} x^{6} - 40 \, b^{3} c^{2} x^{4} + 35 \, b^{4} c x^{2} + 315 \, b^{5}\right )} \sqrt{c x^{4} + b x^{2}}}{3465 \, b^{5} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)/x^13,x, algorithm="fricas")

[Out]

-1/3465*(128*c^5*x^10 - 64*b*c^4*x^8 + 48*b^2*c^3*x^6 - 40*b^3*c^2*x^4 + 35*b^4*
c*x^2 + 315*b^5)*sqrt(c*x^4 + b*x^2)/(b^5*x^12)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )}}{x^{13}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2)**(1/2)/x**13,x)

[Out]

Integral(sqrt(x**2*(b + c*x**2))/x**13, x)

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GIAC/XCAS [A]  time = 0.306955, size = 278, normalized size = 2.04 \[ \frac{256 \,{\left (1386 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} c^{\frac{11}{2}}{\rm sign}\left (x\right ) + 924 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} b c^{\frac{11}{2}}{\rm sign}\left (x\right ) + 330 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} b^{2} c^{\frac{11}{2}}{\rm sign}\left (x\right ) - 165 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} b^{3} c^{\frac{11}{2}}{\rm sign}\left (x\right ) + 55 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} b^{4} c^{\frac{11}{2}}{\rm sign}\left (x\right ) - 11 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} b^{5} c^{\frac{11}{2}}{\rm sign}\left (x\right ) + b^{6} c^{\frac{11}{2}}{\rm sign}\left (x\right )\right )}}{3465 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)/x^13,x, algorithm="giac")

[Out]

256/3465*(1386*(sqrt(c)*x - sqrt(c*x^2 + b))^12*c^(11/2)*sign(x) + 924*(sqrt(c)*
x - sqrt(c*x^2 + b))^10*b*c^(11/2)*sign(x) + 330*(sqrt(c)*x - sqrt(c*x^2 + b))^8
*b^2*c^(11/2)*sign(x) - 165*(sqrt(c)*x - sqrt(c*x^2 + b))^6*b^3*c^(11/2)*sign(x)
+ 55*(sqrt(c)*x - sqrt(c*x^2 + b))^4*b^4*c^(11/2)*sign(x) - 11*(sqrt(c)*x - sqr
t(c*x^2 + b))^2*b^5*c^(11/2)*sign(x) + b^6*c^(11/2)*sign(x))/((sqrt(c)*x - sqrt(
c*x^2 + b))^2 - b)^11