3.7.29 \(\int \frac {(a+b x^4)^2}{x^3} \, dx\) [629]

Optimal. Leaf size=27 \[ -\frac {a^2}{2 x^2}+a b x^2+\frac {b^2 x^6}{6} \]

[Out]

-1/2*a^2/x^2+a*b*x^2+1/6*b^2*x^6

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Rubi [A]
time = 0.01, antiderivative size = 27, 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}{2 x^2}+a b x^2+\frac {b^2 x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2/x^3,x]

[Out]

-1/2*a^2/x^2 + a*b*x^2 + (b^2*x^6)/6

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

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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {a^2}{2 x^2}+a b x^2+\frac {b^2 x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^2/x^3,x]

[Out]

-1/2*a^2/x^2 + a*b*x^2 + (b^2*x^6)/6

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Maple [A]
time = 0.13, size = 24, normalized size = 0.89

method	result	size
default	\(-\frac {a^{2}}{2 x^{2}}+a b \,x^{2}+\frac {b^{2} x^{6}}{6}\)	\(24\)
risch	\(-\frac {a^{2}}{2 x^{2}}+a b \,x^{2}+\frac {b^{2} x^{6}}{6}\)	\(24\)
norman	\(\frac {\frac {1}{6} b^{2} x^{8}+a b \,x^{4}-\frac {1}{2} a^{2}}{x^{2}}\)	\(25\)
gosper	\(-\frac {-b^{2} x^{8}-6 a b \,x^{4}+3 a^{2}}{6 x^{2}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^4+a)^2/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*a^2/x^2+a*b*x^2+1/6*b^2*x^6

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Maxima [A]
time = 0.30, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + a b x^{2} - \frac {a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^3,x, algorithm="maxima")

[Out]

1/6*b^2*x^6 + a*b*x^2 - 1/2*a^2/x^2

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Fricas [A]
time = 0.34, size = 25, normalized size = 0.93 \begin {gather*} \frac {b^{2} x^{8} + 6 \, a b x^{4} - 3 \, a^{2}}{6 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^3,x, algorithm="fricas")

[Out]

1/6*(b^2*x^8 + 6*a*b*x^4 - 3*a^2)/x^2

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Sympy [A]
time = 0.03, size = 22, normalized size = 0.81 \begin {gather*} - \frac {a^{2}}{2 x^{2}} + a b x^{2} + \frac {b^{2} x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**4+a)**2/x**3,x)

[Out]

-a**2/(2*x**2) + a*b*x**2 + b**2*x**6/6

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Giac [A]
time = 0.58, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + a b x^{2} - \frac {a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^4+a)^2/x^3,x, algorithm="giac")

[Out]

1/6*b^2*x^6 + a*b*x^2 - 1/2*a^2/x^2

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Mupad [B]
time = 0.03, size = 25, normalized size = 0.93 \begin {gather*} \frac {-3\,a^2+6\,a\,b\,x^4+b^2\,x^8}{6\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^2/x^3,x)

[Out]

(b^2*x^8 - 3*a^2 + 6*a*b*x^4)/(6*x^2)

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