3.167 \(\int \frac{\left (b x^2+c x^4\right )^3}{x^{11}} \, dx\)

Optimal. Leaf size=40 \[ -\frac{b^3}{4 x^4}-\frac{3 b^2 c}{2 x^2}+3 b c^2 \log (x)+\frac{c^3 x^2}{2} \]

[Out]

-b^3/(4*x^4) - (3*b^2*c)/(2*x^2) + (c^3*x^2)/2 + 3*b*c^2*Log[x]

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Rubi [A]  time = 0.060033, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{b^3}{4 x^4}-\frac{3 b^2 c}{2 x^2}+3 b c^2 \log (x)+\frac{c^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2 + c*x^4)^3/x^11,x]

[Out]

-b^3/(4*x^4) - (3*b^2*c)/(2*x^2) + (c^3*x^2)/2 + 3*b*c^2*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b^{3}}{4 x^{4}} - \frac{3 b^{2} c}{2 x^{2}} + \frac{3 b c^{2} \log{\left (x^{2} \right )}}{2} + \frac{\int ^{x^{2}} c^{3}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**3/x**11,x)

[Out]

-b**3/(4*x**4) - 3*b**2*c/(2*x**2) + 3*b*c**2*log(x**2)/2 + Integral(c**3, (x, x
**2))/2

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Mathematica [A]  time = 0.00723674, size = 40, normalized size = 1. \[ -\frac{b^3}{4 x^4}-\frac{3 b^2 c}{2 x^2}+3 b c^2 \log (x)+\frac{c^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2 + c*x^4)^3/x^11,x]

[Out]

-b^3/(4*x^4) - (3*b^2*c)/(2*x^2) + (c^3*x^2)/2 + 3*b*c^2*Log[x]

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Maple [A]  time = 0.009, size = 35, normalized size = 0.9 \[ -{\frac{{b}^{3}}{4\,{x}^{4}}}-{\frac{3\,{b}^{2}c}{2\,{x}^{2}}}+{\frac{{c}^{3}{x}^{2}}{2}}+3\,b{c}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^3/x^11,x)

[Out]

-1/4*b^3/x^4-3/2*b^2*c/x^2+1/2*c^3*x^2+3*b*c^2*ln(x)

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Maxima [A]  time = 0.687281, size = 50, normalized size = 1.25 \[ \frac{1}{2} \, c^{3} x^{2} + \frac{3}{2} \, b c^{2} \log \left (x^{2}\right ) - \frac{6 \, b^{2} c x^{2} + b^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3/x^11,x, algorithm="maxima")

[Out]

1/2*c^3*x^2 + 3/2*b*c^2*log(x^2) - 1/4*(6*b^2*c*x^2 + b^3)/x^4

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Fricas [A]  time = 0.254501, size = 53, normalized size = 1.32 \[ \frac{2 \, c^{3} x^{6} + 12 \, b c^{2} x^{4} \log \left (x\right ) - 6 \, b^{2} c x^{2} - b^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3/x^11,x, algorithm="fricas")

[Out]

1/4*(2*c^3*x^6 + 12*b*c^2*x^4*log(x) - 6*b^2*c*x^2 - b^3)/x^4

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Sympy [A]  time = 1.35278, size = 36, normalized size = 0.9 \[ 3 b c^{2} \log{\left (x \right )} + \frac{c^{3} x^{2}}{2} - \frac{b^{3} + 6 b^{2} c x^{2}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2)**3/x**11,x)

[Out]

3*b*c**2*log(x) + c**3*x**2/2 - (b**3 + 6*b**2*c*x**2)/(4*x**4)

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GIAC/XCAS [A]  time = 0.268968, size = 62, normalized size = 1.55 \[ \frac{1}{2} \, c^{3} x^{2} + \frac{3}{2} \, b c^{2}{\rm ln}\left (x^{2}\right ) - \frac{9 \, b c^{2} x^{4} + 6 \, b^{2} c x^{2} + b^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^3/x^11,x, algorithm="giac")

[Out]

1/2*c^3*x^2 + 3/2*b*c^2*ln(x^2) - 1/4*(9*b*c^2*x^4 + 6*b^2*c*x^2 + b^3)/x^4