3.227 \(\int \frac {(a+b x^3)^2}{x^5} \, dx\)

Optimal. Leaf size=28 \[ -\frac {a^2}{4 x^4}-\frac {2 a b}{x}+\frac {b^2 x^2}{2} \]

[Out]

-1/4*a^2/x^4-2*a*b/x+1/2*b^2*x^2

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Rubi [A]  time = 0.01, antiderivative size = 28, 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} \[ -\frac {a^2}{4 x^4}-\frac {2 a b}{x}+\frac {b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^5,x]

[Out]

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

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^5} \, dx &=\int \left (\frac {a^2}{x^5}+\frac {2 a b}{x^2}+b^2 x\right ) \, dx\\ &=-\frac {a^2}{4 x^4}-\frac {2 a b}{x}+\frac {b^2 x^2}{2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \[ -\frac {a^2}{4 x^4}-\frac {2 a b}{x}+\frac {b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^5,x]

[Out]

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

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fricas [A]  time = 0.69, size = 26, normalized size = 0.93 \[ \frac {2 \, b^{2} x^{6} - 8 \, a b x^{3} - a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^5,x, algorithm="fricas")

[Out]

1/4*(2*b^2*x^6 - 8*a*b*x^3 - a^2)/x^4

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giac [A]  time = 0.15, size = 25, normalized size = 0.89 \[ \frac {1}{2} \, b^{2} x^{2} - \frac {8 \, a b x^{3} + a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^5,x, algorithm="giac")

[Out]

1/2*b^2*x^2 - 1/4*(8*a*b*x^3 + a^2)/x^4

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maple [A]  time = 0.00, size = 25, normalized size = 0.89 \[ \frac {b^{2} x^{2}}{2}-\frac {2 a b}{x}-\frac {a^{2}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^5,x)

[Out]

-1/4*a^2/x^4-2*a*b/x+1/2*b^2*x^2

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maxima [A]  time = 1.37, size = 25, normalized size = 0.89 \[ \frac {1}{2} \, b^{2} x^{2} - \frac {8 \, a b x^{3} + a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^5,x, algorithm="maxima")

[Out]

1/2*b^2*x^2 - 1/4*(8*a*b*x^3 + a^2)/x^4

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mupad [B]  time = 0.03, size = 27, normalized size = 0.96 \[ \frac {b^2\,x^2}{2}-\frac {\frac {a^2}{4}+2\,b\,a\,x^3}{x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^2/x^5,x)

[Out]

(b^2*x^2)/2 - (a^2/4 + 2*a*b*x^3)/x^4

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sympy [A]  time = 0.18, size = 26, normalized size = 0.93 \[ \frac {b^{2} x^{2}}{2} + \frac {- a^{2} - 8 a b x^{3}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**5,x)

[Out]

b**2*x**2/2 + (-a**2 - 8*a*b*x**3)/(4*x**4)

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