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

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

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {266, 43} \[ \frac {3 a^2 b}{4 x^4}-\frac {a^3}{6 x^6}-\frac {3 a b^2}{2 x^2}-b^3 \log (x) \]

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]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a-b x^2\right )^3}{x^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a-b x)^3}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^3}{x^4}-\frac {3 a^2 b}{x^3}+\frac {3 a b^2}{x^2}-\frac {b^3}{x}\right ) \, dx,x,x^2\right )\\ &=-\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)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 40, normalized size = 1.00 \[ -\frac {a^3}{6 x^6}+\frac {3 a^2 b}{4 x^4}-\frac {3 a b^2}{2 x^2}-b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^3/x^7,x]

[Out]

-1/6*a^3/x^6 + (3*a^2*b)/(4*x^4) - (3*a*b^2)/(2*x^2) - b^3*Log[x]

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IntegrateAlgebraic [A]  time = 0.01, size = 37, normalized size = 0.92 \[ b^3 (-\log (x))-\frac {a \left (2 a^2-9 a b x^2+18 b^2 x^4\right )}{12 x^6} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a - b*x^2)^3/x^7,x]

[Out]

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

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fricas [A]  time = 1.22, size = 39, normalized size = 0.98 \[ -\frac {12 \, b^{3} x^{6} \log \relax (x) + 18 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} + 2 \, a^{3}}{12 \, x^{6}} \]

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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giac [A]  time = 0.97, size = 47, normalized size = 1.18 \[ -\frac {1}{2} \, b^{3} \log \left (x^{2}\right ) + \frac {11 \, b^{3} x^{6} - 18 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, x^{6}} \]

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*log(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

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maple [A]  time = 0.26, size = 35, normalized size = 0.88

method	result	size
default	\(-\frac {a^{3}}{6 x^{6}}+\frac {3 a^{2} b}{4 x^{4}}-\frac {3 a \,b^{2}}{2 x^{2}}-b^{3} \ln \relax (x )\)	\(35\)
norman	\(\frac {-\frac {1}{6} a^{3}+\frac {3}{4} a^{2} b \,x^{2}-\frac {3}{2} b^{2} a \,x^{4}}{x^{6}}-b^{3} \ln \relax (x )\)	\(37\)
risch	\(\frac {-\frac {1}{6} a^{3}+\frac {3}{4} a^{2} b \,x^{2}-\frac {3}{2} b^{2} a \,x^{4}}{x^{6}}-b^{3} \ln \relax (x )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^2+a)^3/x^7,x,method=_RETURNVERBOSE)

[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 = 0.51, size = 39, normalized size = 0.98 \[ -\frac {1}{2} \, b^{3} \log \left (x^{2}\right ) - \frac {18 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} + 2 \, a^{3}}{12 \, x^{6}} \]

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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mupad [B]  time = 0.20, size = 37, normalized size = 0.92 \[ -b^3\,\ln \relax (x)-\frac {\frac {a^3}{6}-\frac {3\,a^2\,b\,x^2}{4}+\frac {3\,a\,b^2\,x^4}{2}}{x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x^2)^3/x^7,x)

[Out]

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

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sympy [A]  time = 0.25, size = 37, normalized size = 0.92 \[ - b^{3} \log {\relax (x )} - \frac {2 a^{3} - 9 a^{2} b x^{2} + 18 a b^{2} x^{4}}{12 x^{6}} \]

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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