3.7.28 \(\int \frac {(a+b x^4)^2}{x^2} \, dx\) [628]

Optimal. Leaf size=28 \[ -\frac {a^2}{x}+\frac {2}{3} a b x^3+\frac {b^2 x^7}{7} \]

[Out]

-a^2/x+2/3*a*b*x^3+1/7*b^2*x^7

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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 = {276} \begin {gather*} -\frac {a^2}{x}+\frac {2}{3} a b x^3+\frac {b^2 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2/x^2,x]

[Out]

-(a^2/x) + (2*a*b*x^3)/3 + (b^2*x^7)/7

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^2} \, dx &=\int \left (\frac {a^2}{x^2}+2 a b x^2+b^2 x^6\right ) \, dx\\ &=-\frac {a^2}{x}+\frac {2}{3} a b x^3+\frac {b^2 x^7}{7}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -\frac {a^2}{x}+\frac {2}{3} a b x^3+\frac {b^2 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^2/x^2,x]

[Out]

-(a^2/x) + (2*a*b*x^3)/3 + (b^2*x^7)/7

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Maple [A]
time = 0.13, size = 25, normalized size = 0.89

method	result	size
default	\(-\frac {a^{2}}{x}+\frac {2 a b \,x^{3}}{3}+\frac {b^{2} x^{7}}{7}\)	\(25\)
risch	\(-\frac {a^{2}}{x}+\frac {2 a b \,x^{3}}{3}+\frac {b^{2} x^{7}}{7}\)	\(25\)
norman	\(\frac {\frac {1}{7} b^{2} x^{8}+\frac {2}{3} a b \,x^{4}-a^{2}}{x}\)	\(26\)
gosper	\(-\frac {-3 b^{2} x^{8}-14 a b \,x^{4}+21 a^{2}}{21 x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2/x+2/3*a*b*x^3+1/7*b^2*x^7

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Maxima [A]
time = 0.29, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{7} \, b^{2} x^{7} + \frac {2}{3} \, a b x^{3} - \frac {a^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*x^7 + 2/3*a*b*x^3 - a^2/x

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Fricas [A]
time = 0.35, size = 26, normalized size = 0.93 \begin {gather*} \frac {3 \, b^{2} x^{8} + 14 \, a b x^{4} - 21 \, a^{2}}{21 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(3*b^2*x^8 + 14*a*b*x^4 - 21*a^2)/x

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Sympy [A]
time = 0.02, size = 22, normalized size = 0.79 \begin {gather*} - \frac {a^{2}}{x} + \frac {2 a b x^{3}}{3} + \frac {b^{2} x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2/x + 2*a*b*x**3/3 + b**2*x**7/7

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Giac [A]
time = 0.62, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{7} \, b^{2} x^{7} + \frac {2}{3} \, a b x^{3} - \frac {a^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*x^7 + 2/3*a*b*x^3 - a^2/x

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Mupad [B]
time = 0.03, size = 25, normalized size = 0.89 \begin {gather*} \frac {-a^2+\frac {2\,a\,b\,x^4}{3}+\frac {b^2\,x^8}{7}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^2/x^2,x)

[Out]

((b^2*x^8)/7 - a^2 + (2*a*b*x^4)/3)/x

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