3.630 \(\int \frac {(a+b x^4)^2}{x^4} \, dx\)

Optimal. Leaf size=26 \[ -\frac {a^2}{3 x^3}+2 a b x+\frac {b^2 x^5}{5} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 26, 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} \[ -\frac {a^2}{3 x^3}+2 a b x+\frac {b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2/x^4,x]

[Out]

-a^2/(3*x^3) + 2*a*b*x + (b^2*x^5)/5

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

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Mathematica [A]  time = 0.00, size = 26, normalized size = 1.00 \[ -\frac {a^2}{3 x^3}+2 a b x+\frac {b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^2/x^4,x]

[Out]

-1/3*a^2/x^3 + 2*a*b*x + (b^2*x^5)/5

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fricas [A]  time = 0.72, size = 26, normalized size = 1.00 \[ \frac {3 \, b^{2} x^{8} + 30 \, a b x^{4} - 5 \, a^{2}}{15 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*b^2*x^8 + 30*a*b*x^4 - 5*a^2)/x^3

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giac [A]  time = 0.15, size = 22, normalized size = 0.85 \[ \frac {1}{5} \, b^{2} x^{5} + 2 \, a b x - \frac {a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*x^5 + 2*a*b*x - 1/3*a^2/x^3

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maple [A]  time = 0.00, size = 23, normalized size = 0.88 \[ \frac {b^{2} x^{5}}{5}+2 a b x -\frac {a^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 22, normalized size = 0.85 \[ \frac {1}{5} \, b^{2} x^{5} + 2 \, a b x - \frac {a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*x^5 + 2*a*b*x - 1/3*a^2/x^3

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mupad [B]  time = 0.03, size = 22, normalized size = 0.85 \[ \frac {b^2\,x^5}{5}-\frac {a^2}{3\,x^3}+2\,a\,b\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^4)^2/x^4,x)

[Out]

(b^2*x^5)/5 - a^2/(3*x^3) + 2*a*b*x

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sympy [A]  time = 0.19, size = 22, normalized size = 0.85 \[ - \frac {a^{2}}{3 x^{3}} + 2 a b x + \frac {b^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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