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

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*a*b*x^3)/3 + (b^2*x^7)/7

________________________________________________________________________________________

Rubi [A]  time = 0.0094924, antiderivative size = 28, normalized size of antiderivative = 1., 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} \[ -\frac{a^2}{x}+\frac{2}{3} a b x^3+\frac{b^2 x^7}{7} \]

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 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^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*}

Mathematica [A]  time = 0.0006762, size = 28, normalized size = 1. \[ -\frac{a^2}{x}+\frac{2}{3} a b x^3+\frac{b^2 x^7}{7} \]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 25, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{x}}+{\frac{2\,{x}^{3}ab}{3}}+{\frac{{b}^{2}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.962468, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{7} \, b^{2} x^{7} + \frac{2}{3} \, a b x^{3} - \frac{a^{2}}{x} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.3644, size = 57, normalized size = 2.04 \begin{align*} \frac{3 \, b^{2} x^{8} + 14 \, a b x^{4} - 21 \, a^{2}}{21 \, x} \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 0.31635, size = 22, normalized size = 0.79 \begin{align*} - \frac{a^{2}}{x} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{7}}{7} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.09199, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{7} \, b^{2} x^{7} + \frac{2}{3} \, a b x^{3} - \frac{a^{2}}{x} \end{align*}

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