3.248 \(\int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^{17}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{128 c^4 \left (b x^2+c x^4\right )^{5/2}}{15015 b^5 x^{10}}+\frac{64 c^3 \left (b x^2+c x^4\right )^{5/2}}{3003 b^4 x^{12}}-\frac{16 c^2 \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{14}}+\frac{8 c \left (b x^2+c x^4\right )^{5/2}}{143 b^2 x^{16}}-\frac{\left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}} \]

[Out]

-(b*x^2 + c*x^4)^(5/2)/(13*b*x^18) + (8*c*(b*x^2 + c*x^4)^(5/2))/(143*b^2*x^16)
- (16*c^2*(b*x^2 + c*x^4)^(5/2))/(429*b^3*x^14) + (64*c^3*(b*x^2 + c*x^4)^(5/2))
/(3003*b^4*x^12) - (128*c^4*(b*x^2 + c*x^4)^(5/2))/(15015*b^5*x^10)

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Rubi [A]  time = 0.385765, 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 \[ -\frac{128 c^4 \left (b x^2+c x^4\right )^{5/2}}{15015 b^5 x^{10}}+\frac{64 c^3 \left (b x^2+c x^4\right )^{5/2}}{3003 b^4 x^{12}}-\frac{16 c^2 \left (b x^2+c x^4\right )^{5/2}}{429 b^3 x^{14}}+\frac{8 c \left (b x^2+c x^4\right )^{5/2}}{143 b^2 x^{16}}-\frac{\left (b x^2+c x^4\right )^{5/2}}{13 b x^{18}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2 + c*x^4)^(3/2)/x^17,x]

[Out]

-(b*x^2 + c*x^4)^(5/2)/(13*b*x^18) + (8*c*(b*x^2 + c*x^4)^(5/2))/(143*b^2*x^16)
- (16*c^2*(b*x^2 + c*x^4)^(5/2))/(429*b^3*x^14) + (64*c^3*(b*x^2 + c*x^4)^(5/2))
/(3003*b^4*x^12) - (128*c^4*(b*x^2 + c*x^4)^(5/2))/(15015*b^5*x^10)

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Rubi in Sympy [A]  time = 37.577, size = 126, normalized size = 0.93 \[ - \frac{\left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{13 b x^{18}} + \frac{8 c \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{143 b^{2} x^{16}} - \frac{16 c^{2} \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{429 b^{3} x^{14}} + \frac{64 c^{3} \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{3003 b^{4} x^{12}} - \frac{128 c^{4} \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{15015 b^{5} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(3/2)/x**17,x)

[Out]

-(b*x**2 + c*x**4)**(5/2)/(13*b*x**18) + 8*c*(b*x**2 + c*x**4)**(5/2)/(143*b**2*
x**16) - 16*c**2*(b*x**2 + c*x**4)**(5/2)/(429*b**3*x**14) + 64*c**3*(b*x**2 + c
*x**4)**(5/2)/(3003*b**4*x**12) - 128*c**4*(b*x**2 + c*x**4)**(5/2)/(15015*b**5*
x**10)

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Mathematica [A]  time = 0.0515323, size = 68, normalized size = 0.5 \[ -\frac{\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (1155 b^4-840 b^3 c x^2+560 b^2 c^2 x^4-320 b c^3 x^6+128 c^4 x^8\right )}{15015 b^5 x^{18}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2 + c*x^4)^(3/2)/x^17,x]

[Out]

-((x^2*(b + c*x^2))^(5/2)*(1155*b^4 - 840*b^3*c*x^2 + 560*b^2*c^2*x^4 - 320*b*c^
3*x^6 + 128*c^4*x^8))/(15015*b^5*x^18)

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Maple [A]  time = 0.009, size = 72, normalized size = 0.5 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 128\,{c}^{4}{x}^{8}-320\,{c}^{3}{x}^{6}b+560\,{c}^{2}{x}^{4}{b}^{2}-840\,c{x}^{2}{b}^{3}+1155\,{b}^{4} \right ) }{15015\,{x}^{16}{b}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(3/2)/x^17,x)

[Out]

-1/15015*(c*x^2+b)*(128*c^4*x^8-320*b*c^3*x^6+560*b^2*c^2*x^4-840*b^3*c*x^2+1155
*b^4)*(c*x^4+b*x^2)^(3/2)/x^16/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((c*x^4 + b*x^2)^(3/2)/x^17,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.412493, size = 116, normalized size = 0.85 \[ -\frac{{\left (128 \, c^{6} x^{12} - 64 \, b c^{5} x^{10} + 48 \, b^{2} c^{4} x^{8} - 40 \, b^{3} c^{3} x^{6} + 35 \, b^{4} c^{2} x^{4} + 1470 \, b^{5} c x^{2} + 1155 \, b^{6}\right )} \sqrt{c x^{4} + b x^{2}}}{15015 \, b^{5} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^17,x, algorithm="fricas")

[Out]

-1/15015*(128*c^6*x^12 - 64*b*c^5*x^10 + 48*b^2*c^4*x^8 - 40*b^3*c^3*x^6 + 35*b^
4*c^2*x^4 + 1470*b^5*c*x^2 + 1155*b^6)*sqrt(c*x^4 + b*x^2)/(b^5*x^14)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{17}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2)**(3/2)/x**17,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)/x**17, x)

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GIAC/XCAS [A]  time = 0.315218, size = 356, normalized size = 2.62 \[ \frac{256 \,{\left (6006 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{16} c^{\frac{13}{2}}{\rm sign}\left (x\right ) + 12012 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{14} b c^{\frac{13}{2}}{\rm sign}\left (x\right ) + 13728 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{12} b^{2} c^{\frac{13}{2}}{\rm sign}\left (x\right ) + 4719 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} b^{3} c^{\frac{13}{2}}{\rm sign}\left (x\right ) + 715 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} b^{4} c^{\frac{13}{2}}{\rm sign}\left (x\right ) - 286 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} b^{5} c^{\frac{13}{2}}{\rm sign}\left (x\right ) + 78 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} b^{6} c^{\frac{13}{2}}{\rm sign}\left (x\right ) - 13 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} b^{7} c^{\frac{13}{2}}{\rm sign}\left (x\right ) + b^{8} c^{\frac{13}{2}}{\rm sign}\left (x\right )\right )}}{15015 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^17,x, algorithm="giac")

[Out]

256/15015*(6006*(sqrt(c)*x - sqrt(c*x^2 + b))^16*c^(13/2)*sign(x) + 12012*(sqrt(
c)*x - sqrt(c*x^2 + b))^14*b*c^(13/2)*sign(x) + 13728*(sqrt(c)*x - sqrt(c*x^2 +
b))^12*b^2*c^(13/2)*sign(x) + 4719*(sqrt(c)*x - sqrt(c*x^2 + b))^10*b^3*c^(13/2)
*sign(x) + 715*(sqrt(c)*x - sqrt(c*x^2 + b))^8*b^4*c^(13/2)*sign(x) - 286*(sqrt(
c)*x - sqrt(c*x^2 + b))^6*b^5*c^(13/2)*sign(x) + 78*(sqrt(c)*x - sqrt(c*x^2 + b)
)^4*b^6*c^(13/2)*sign(x) - 13*(sqrt(c)*x - sqrt(c*x^2 + b))^2*b^7*c^(13/2)*sign(
x) + b^8*c^(13/2)*sign(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^13