3.3.28 \(\int \frac {(a+b x^3)^2}{x^6} \, dx\) [228]

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 = {276} \begin {gather*} -\frac {a^2}{5 x^5}-\frac {a b}{x^2}+b^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^6,x]

[Out]

-1/5*a^2/x^5 - (a*b)/x^2 + b^2*x

Rule 276

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 \begin {gather*} -\frac {a^2}{5 x^5}-\frac {a b}{x^2}+b^2 x \end {gather*}

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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Maple [A]
time = 0.12, size = 22, normalized size = 0.96

method	result	size
default	\(-\frac {a^{2}}{5 x^{5}}-\frac {a b}{x^{2}}+b^{2} x\)	\(22\)
risch	\(b^{2} x +\frac {-a b \,x^{3}-\frac {1}{5} a^{2}}{x^{5}}\)	\(24\)
gosper	\(-\frac {-5 b^{2} x^{6}+5 a b \,x^{3}+a^{2}}{5 x^{5}}\)	\(25\)
norman	\(\frac {b^{2} x^{6}-a b \,x^{3}-\frac {1}{5} a^{2}}{x^{5}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 22, normalized size = 0.96 \begin {gather*} b^{2} x - \frac {5 \, a b x^{3} + a^{2}}{5 \, x^{5}} \end {gather*}

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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Fricas [A]
time = 0.34, size = 26, normalized size = 1.13 \begin {gather*} \frac {5 \, b^{2} x^{6} - 5 \, a b x^{3} - a^{2}}{5 \, x^{5}} \end {gather*}

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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Sympy [A]
time = 0.06, size = 22, normalized size = 0.96 \begin {gather*} b^{2} x + \frac {- a^{2} - 5 a b x^{3}}{5 x^{5}} \end {gather*}

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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Giac [A]
time = 1.16, size = 22, normalized size = 0.96 \begin {gather*} b^{2} x - \frac {5 \, a b x^{3} + a^{2}}{5 \, x^{5}} \end {gather*}

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

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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