3.640 \(\int \frac{(a+b x^4)^3}{x^2} \, dx\)

Optimal. Leaf size=38 \[ a^2 b x^3-\frac{a^3}{x}+\frac{3}{7} a b^2 x^7+\frac{b^3 x^{11}}{11} \]

[Out]

-(a^3/x) + a^2*b*x^3 + (3*a*b^2*x^7)/7 + (b^3*x^11)/11

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Rubi [A]  time = 0.0129084, antiderivative size = 38, 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} \[ a^2 b x^3-\frac{a^3}{x}+\frac{3}{7} a b^2 x^7+\frac{b^3 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^3/x^2,x]

[Out]

-(a^3/x) + a^2*b*x^3 + (3*a*b^2*x^7)/7 + (b^3*x^11)/11

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 )^3}{x^2} \, dx &=\int \left (\frac{a^3}{x^2}+3 a^2 b x^2+3 a b^2 x^6+b^3 x^{10}\right ) \, dx\\ &=-\frac{a^3}{x}+a^2 b x^3+\frac{3}{7} a b^2 x^7+\frac{b^3 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.003639, size = 38, normalized size = 1. \[ a^2 b x^3-\frac{a^3}{x}+\frac{3}{7} a b^2 x^7+\frac{b^3 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^4)^3/x^2,x]

[Out]

-(a^3/x) + a^2*b*x^3 + (3*a*b^2*x^7)/7 + (b^3*x^11)/11

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Maple [A]  time = 0.003, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{x}}+{a}^{2}b{x}^{3}+{\frac{3\,a{b}^{2}{x}^{7}}{7}}+{\frac{{b}^{3}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^3/x+a^2*b*x^3+3/7*a*b^2*x^7+1/11*b^3*x^11

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Maxima [A]  time = 0.96531, size = 46, normalized size = 1.21 \begin{align*} \frac{1}{11} \, b^{3} x^{11} + \frac{3}{7} \, a b^{2} x^{7} + a^{2} b x^{3} - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^3*x^11 + 3/7*a*b^2*x^7 + a^2*b*x^3 - a^3/x

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Fricas [A]  time = 1.40876, size = 81, normalized size = 2.13 \begin{align*} \frac{7 \, b^{3} x^{12} + 33 \, a b^{2} x^{8} + 77 \, a^{2} b x^{4} - 77 \, a^{3}}{77 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/77*(7*b^3*x^12 + 33*a*b^2*x^8 + 77*a^2*b*x^4 - 77*a^3)/x

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Sympy [A]  time = 0.334861, size = 32, normalized size = 0.84 \begin{align*} - \frac{a^{3}}{x} + a^{2} b x^{3} + \frac{3 a b^{2} x^{7}}{7} + \frac{b^{3} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**3/x + a**2*b*x**3 + 3*a*b**2*x**7/7 + b**3*x**11/11

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Giac [A]  time = 1.1217, size = 46, normalized size = 1.21 \begin{align*} \frac{1}{11} \, b^{3} x^{11} + \frac{3}{7} \, a b^{2} x^{7} + a^{2} b x^{3} - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^3*x^11 + 3/7*a*b^2*x^7 + a^2*b*x^3 - a^3/x