3.228 \(\int \frac {(a+b x^3)^2}{x^6} \, dx\)

Optimal. Leaf size=23 \[ -\frac {a^2}{5 x^5}-\frac {a b}{x^2}+b^2 x \]

[Out]

-1/5*a^2/x^5-a*b/x^2+b^2*x

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ -\frac {a^2}{5 x^5}-\frac {a b}{x^2}+b^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^6,x]

[Out]

-a^2/(5*x^5) - (a*b)/x^2 + b^2*x

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^2}{x^6} \, dx &=\int \left (b^2+\frac {a^2}{x^6}+\frac {2 a b}{x^3}\right ) \, dx\\ &=-\frac {a^2}{5 x^5}-\frac {a b}{x^2}+b^2 x\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {a^2}{5 x^5}-\frac {a b}{x^2}+b^2 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^6,x]

[Out]

-1/5*a^2/x^5 - (a*b)/x^2 + b^2*x

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fricas [A]  time = 0.83, size = 26, normalized size = 1.13 \[ \frac {5 \, b^{2} x^{6} - 5 \, a b x^{3} - a^{2}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^6,x, algorithm="fricas")

[Out]

1/5*(5*b^2*x^6 - 5*a*b*x^3 - a^2)/x^5

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giac [A]  time = 0.16, size = 22, normalized size = 0.96 \[ b^{2} x - \frac {5 \, a b x^{3} + a^{2}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^6,x, algorithm="giac")

[Out]

b^2*x - 1/5*(5*a*b*x^3 + a^2)/x^5

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maple [A]  time = 0.01, size = 22, normalized size = 0.96 \[ b^{2} x -\frac {a b}{x^{2}}-\frac {a^{2}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^6,x)

[Out]

-1/5*a^2/x^5-a*b/x^2+b^2*x

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maxima [A]  time = 1.34, size = 22, normalized size = 0.96 \[ b^{2} x - \frac {5 \, a b x^{3} + a^{2}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^6,x, algorithm="maxima")

[Out]

b^2*x - 1/5*(5*a*b*x^3 + a^2)/x^5

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mupad [B]  time = 0.03, size = 23, normalized size = 1.00 \[ b^2\,x-\frac {\frac {a^2}{5}+b\,a\,x^3}{x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^2/x^6,x)

[Out]

b^2*x - (a^2/5 + a*b*x^3)/x^5

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sympy [A]  time = 0.18, size = 22, normalized size = 0.96 \[ b^{2} x + \frac {- a^{2} - 5 a b x^{3}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**6,x)

[Out]

b**2*x + (-a**2 - 5*a*b*x**3)/(5*x**5)

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