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

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

[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 = 0.0231058, 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, 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 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]]

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])

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*}

Mathematica [A]  time = 0.0054169, size = 40, normalized size = 1. \[ \frac{3 a^2 b}{4 x^4}-\frac{a^3}{6 x^6}-\frac{3 a b^2}{2 x^2}-b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

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 = 0.007, size = 35, normalized size = 0.9 \begin{align*} -{\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 ) \end{align*}

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 = 0.930659, size = 53, normalized size = 1.32 \begin{align*} -\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}} \end{align*}

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 = 1.73601, size = 92, normalized size = 2.3 \begin{align*} -\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}} \end{align*}

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 = 0.377801, size = 37, normalized size = 0.92 \begin{align*} - 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}} \end{align*}

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 [A]  time = 1.05544, size = 63, normalized size = 1.58 \begin{align*} -\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}} \end{align*}

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