3.3.30 \(\int \frac {(a+b x^3)^2}{x^8} \, dx\) [230]

Optimal. Leaf size=28 \[ -\frac {a^2}{7 x^7}-\frac {a b}{2 x^4}-\frac {b^2}{x} \]

[Out]

-1/7*a^2/x^7-1/2*a*b/x^4-b^2/x

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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}{7 x^7}-\frac {a b}{2 x^4}-\frac {b^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^8,x]

[Out]

-1/7*a^2/x^7 - (a*b)/(2*x^4) - b^2/x

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

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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -\frac {a^2}{7 x^7}-\frac {a b}{2 x^4}-\frac {b^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^8,x]

[Out]

-1/7*a^2/x^7 - (a*b)/(2*x^4) - b^2/x

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Maple [A]
time = 0.12, size = 25, normalized size = 0.89

method	result	size
default	\(-\frac {a^{2}}{7 x^{7}}-\frac {a b}{2 x^{4}}-\frac {b^{2}}{x}\)	\(25\)
norman	\(\frac {-b^{2} x^{6}-\frac {1}{2} a b \,x^{3}-\frac {1}{7} a^{2}}{x^{7}}\)	\(26\)
risch	\(\frac {-b^{2} x^{6}-\frac {1}{2} a b \,x^{3}-\frac {1}{7} a^{2}}{x^{7}}\)	\(26\)
gosper	\(-\frac {14 b^{2} x^{6}+7 a b \,x^{3}+2 a^{2}}{14 x^{7}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^2/x^7-1/2*a*b/x^4-b^2/x

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Maxima [A]
time = 0.30, size = 26, normalized size = 0.93 \begin {gather*} -\frac {14 \, b^{2} x^{6} + 7 \, a b x^{3} + 2 \, a^{2}}{14 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14*(14*b^2*x^6 + 7*a*b*x^3 + 2*a^2)/x^7

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Fricas [A]
time = 0.34, size = 26, normalized size = 0.93 \begin {gather*} -\frac {14 \, b^{2} x^{6} + 7 \, a b x^{3} + 2 \, a^{2}}{14 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14*(14*b^2*x^6 + 7*a*b*x^3 + 2*a^2)/x^7

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Sympy [A]
time = 0.08, size = 27, normalized size = 0.96 \begin {gather*} \frac {- 2 a^{2} - 7 a b x^{3} - 14 b^{2} x^{6}}{14 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*a**2 - 7*a*b*x**3 - 14*b**2*x**6)/(14*x**7)

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Giac [A]
time = 1.54, size = 26, normalized size = 0.93 \begin {gather*} -\frac {14 \, b^{2} x^{6} + 7 \, a b x^{3} + 2 \, a^{2}}{14 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14*(14*b^2*x^6 + 7*a*b*x^3 + 2*a^2)/x^7

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Mupad [B]
time = 0.03, size = 25, normalized size = 0.89 \begin {gather*} -\frac {\frac {a^2}{7}+\frac {a\,b\,x^3}{2}+b^2\,x^6}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^2/x^8,x)

[Out]

-(a^2/7 + b^2*x^6 + (a*b*x^3)/2)/x^7

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