3.206 \(\int \frac{\left (a-b x^2\right )^3}{x^7} \, dx\)

Optimal. Leaf size=40 \[ -\frac{a^3}{6 x^6}+\frac{3 a^2 b}{4 x^4}-\frac{3 a b^2}{2 x^2}-b^3 \log (x) \]

[Out]

-a^3/(6*x^6) + (3*a^2*b)/(4*x^4) - (3*a*b^2)/(2*x^2) - b^3*Log[x]

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Rubi [A]  time = 0.049783, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{a^3}{6 x^6}+\frac{3 a^2 b}{4 x^4}-\frac{3 a b^2}{2 x^2}-b^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^3/x^7,x]

[Out]

-a^3/(6*x^6) + (3*a^2*b)/(4*x^4) - (3*a*b^2)/(2*x^2) - b^3*Log[x]

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Rubi in Sympy [A]  time = 4.59869, size = 41, normalized size = 1.02 \[ - \frac{a^{3}}{6 x^{6}} + \frac{3 a^{2} b}{4 x^{4}} - \frac{3 a b^{2}}{2 x^{2}} - \frac{b^{3} \log{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a)**3/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.00776439, size = 40, normalized size = 1. \[ -\frac{a^3}{6 x^6}+\frac{3 a^2 b}{4 x^4}-\frac{3 a b^2}{2 x^2}-b^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^2)^3/x^7,x]

[Out]

-a^3/(6*x^6) + (3*a^2*b)/(4*x^4) - (3*a*b^2)/(2*x^2) - b^3*Log[x]

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Maple [A]  time = 0.01, size = 35, normalized size = 0.9 \[ -{\frac{{a}^{3}}{6\,{x}^{6}}}+{\frac{3\,{a}^{2}b}{4\,{x}^{4}}}-{\frac{3\,{b}^{2}a}{2\,{x}^{2}}}-{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^3/x^7,x)

[Out]

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

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Maxima [A]  time = 1.34949, size = 53, normalized size = 1.32 \[ -\frac{1}{2} \, b^{3} \log \left (x^{2}\right ) - \frac{18 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} + 2 \, a^{3}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x^2 - a)^3/x^7,x, algorithm="maxima")

[Out]

-1/2*b^3*log(x^2) - 1/12*(18*a*b^2*x^4 - 9*a^2*b*x^2 + 2*a^3)/x^6

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Fricas [A]  time = 0.217696, size = 53, normalized size = 1.32 \[ -\frac{12 \, b^{3} x^{6} \log \left (x\right ) + 18 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} + 2 \, a^{3}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x^2 - a)^3/x^7,x, algorithm="fricas")

[Out]

-1/12*(12*b^3*x^6*log(x) + 18*a*b^2*x^4 - 9*a^2*b*x^2 + 2*a^3)/x^6

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Sympy [A]  time = 0.740808, size = 37, normalized size = 0.92 \[ - b^{3} \log{\left (x \right )} - \frac{2 a^{3} - 9 a^{2} b x^{2} + 18 a b^{2} x^{4}}{12 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a)**3/x**7,x)

[Out]

-b**3*log(x) - (2*a**3 - 9*a**2*b*x**2 + 18*a*b**2*x**4)/(12*x**6)

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GIAC/XCAS [A]  time = 0.201529, size = 63, normalized size = 1.58 \[ -\frac{1}{2} \, b^{3}{\rm ln}\left (x^{2}\right ) + \frac{11 \, b^{3} x^{6} - 18 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x^2 - a)^3/x^7,x, algorithm="giac")

[Out]

-1/2*b^3*ln(x^2) + 1/12*(11*b^3*x^6 - 18*a*b^2*x^4 + 9*a^2*b*x^2 - 2*a^3)/x^6