3.1.48 \(\int \frac {(a+b x^2)^3}{x^6} \, dx\) [48]

Optimal. Leaf size=34 \[ -\frac {a^3}{5 x^5}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{x}+b^3 x \]

[Out]

-1/5*a^3/x^5-a^2*b/x^3-3*a*b^2/x+b^3*x

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Rubi [A]
time = 0.01, antiderivative size = 34, 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^3}{5 x^5}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{x}+b^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^6,x]

[Out]

-1/5*a^3/x^5 - (a^2*b)/x^3 - (3*a*b^2)/x + b^3*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^2\right )^3}{x^6} \, dx &=\int \left (b^3+\frac {a^3}{x^6}+\frac {3 a^2 b}{x^4}+\frac {3 a b^2}{x^2}\right ) \, dx\\ &=-\frac {a^3}{5 x^5}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{x}+b^3 x\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 34, normalized size = 1.00 \begin {gather*} -\frac {a^3}{5 x^5}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{x}+b^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^6,x]

[Out]

-1/5*a^3/x^5 - (a^2*b)/x^3 - (3*a*b^2)/x + b^3*x

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Maple [A]
time = 0.01, size = 33, normalized size = 0.97

method	result	size
default	\(-\frac {a^{3}}{5 x^{5}}-\frac {a^{2} b}{x^{3}}-\frac {3 a \,b^{2}}{x}+b^{3} x\)	\(33\)
risch	\(b^{3} x +\frac {-3 a \,b^{2} x^{4}-a^{2} b \,x^{2}-\frac {1}{5} a^{3}}{x^{5}}\)	\(35\)
gosper	\(-\frac {-5 b^{3} x^{6}+15 a \,b^{2} x^{4}+5 a^{2} b \,x^{2}+a^{3}}{5 x^{5}}\)	\(36\)
norman	\(\frac {b^{3} x^{6}-3 a \,b^{2} x^{4}-a^{2} b \,x^{2}-\frac {1}{5} a^{3}}{x^{5}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*a^3/x^5-a^2*b/x^3-3*a*b^2/x+b^3*x

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Maxima [A]
time = 0.29, size = 33, normalized size = 0.97 \begin {gather*} b^{3} x - \frac {15 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*x - 1/5*(15*a*b^2*x^4 + 5*a^2*b*x^2 + a^3)/x^5

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Fricas [A]
time = 0.79, size = 37, normalized size = 1.09 \begin {gather*} \frac {5 \, b^{3} x^{6} - 15 \, a b^{2} x^{4} - 5 \, a^{2} b x^{2} - a^{3}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(5*b^3*x^6 - 15*a*b^2*x^4 - 5*a^2*b*x^2 - a^3)/x^5

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Sympy [A]
time = 0.07, size = 34, normalized size = 1.00 \begin {gather*} b^{3} x + \frac {- a^{3} - 5 a^{2} b x^{2} - 15 a b^{2} x^{4}}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*x + (-a**3 - 5*a**2*b*x**2 - 15*a*b**2*x**4)/(5*x**5)

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Giac [A]
time = 0.63, size = 33, normalized size = 0.97 \begin {gather*} b^{3} x - \frac {15 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*x - 1/5*(15*a*b^2*x^4 + 5*a^2*b*x^2 + a^3)/x^5

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Mupad [B]
time = 0.03, size = 34, normalized size = 1.00 \begin {gather*} b^3\,x-\frac {\frac {a^3}{5}+a^2\,b\,x^2+3\,a\,b^2\,x^4}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^6,x)

[Out]

b^3*x - (a^3/5 + a^2*b*x^2 + 3*a*b^2*x^4)/x^5

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