3.49 \(\int \frac {(a+b x^2)^3}{x^8} \, dx\)

Optimal. Leaf size=39 \[ -\frac {a^3}{7 x^7}-\frac {3 a^2 b}{5 x^5}-\frac {a b^2}{x^3}-\frac {b^3}{x} \]

[Out]

-1/7*a^3/x^7-3/5*a^2*b/x^5-a*b^2/x^3-b^3/x

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Rubi [A]  time = 0.01, antiderivative size = 39, 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} \[ -\frac {3 a^2 b}{5 x^5}-\frac {a^3}{7 x^7}-\frac {a b^2}{x^3}-\frac {b^3}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^8,x]

[Out]

-a^3/(7*x^7) - (3*a^2*b)/(5*x^5) - (a*b^2)/x^3 - b^3/x

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^2\right )^3}{x^8} \, dx &=\int \left (\frac {a^3}{x^8}+\frac {3 a^2 b}{x^6}+\frac {3 a b^2}{x^4}+\frac {b^3}{x^2}\right ) \, dx\\ &=-\frac {a^3}{7 x^7}-\frac {3 a^2 b}{5 x^5}-\frac {a b^2}{x^3}-\frac {b^3}{x}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 39, normalized size = 1.00 \[ -\frac {a^3}{7 x^7}-\frac {3 a^2 b}{5 x^5}-\frac {a b^2}{x^3}-\frac {b^3}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^8,x]

[Out]

-1/7*a^3/x^7 - (3*a^2*b)/(5*x^5) - (a*b^2)/x^3 - b^3/x

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fricas [A]  time = 0.82, size = 37, normalized size = 0.95 \[ -\frac {35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^8,x, algorithm="fricas")

[Out]

-1/35*(35*b^3*x^6 + 35*a*b^2*x^4 + 21*a^2*b*x^2 + 5*a^3)/x^7

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giac [A]  time = 1.09, size = 37, normalized size = 0.95 \[ -\frac {35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^8,x, algorithm="giac")

[Out]

-1/35*(35*b^3*x^6 + 35*a*b^2*x^4 + 21*a^2*b*x^2 + 5*a^3)/x^7

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maple [A]  time = 0.00, size = 36, normalized size = 0.92 \[ -\frac {b^{3}}{x}-\frac {a \,b^{2}}{x^{3}}-\frac {3 a^{2} b}{5 x^{5}}-\frac {a^{3}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^8,x)

[Out]

-1/7*a^3/x^7-3/5*a^2*b/x^5-a*b^2/x^3-b^3/x

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maxima [A]  time = 1.25, size = 37, normalized size = 0.95 \[ -\frac {35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^8,x, algorithm="maxima")

[Out]

-1/35*(35*b^3*x^6 + 35*a*b^2*x^4 + 21*a^2*b*x^2 + 5*a^3)/x^7

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mupad [B]  time = 0.03, size = 35, normalized size = 0.90 \[ -\frac {\frac {a^3}{7}+\frac {3\,a^2\,b\,x^2}{5}+a\,b^2\,x^4+b^3\,x^6}{x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^8,x)

[Out]

-(a^3/7 + b^3*x^6 + (3*a^2*b*x^2)/5 + a*b^2*x^4)/x^7

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sympy [A]  time = 0.26, size = 39, normalized size = 1.00 \[ \frac {- 5 a^{3} - 21 a^{2} b x^{2} - 35 a b^{2} x^{4} - 35 b^{3} x^{6}}{35 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**8,x)

[Out]

(-5*a**3 - 21*a**2*b*x**2 - 35*a*b**2*x**4 - 35*b**3*x**6)/(35*x**7)

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