3.1.63 \(\int \frac {(a+b x)^2}{x^8} \, dx\) [63]

Optimal. Leaf size=30 \[ -\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \]

[Out]

-1/7*a^2/x^7-1/3*a*b/x^6-1/5*b^2/x^5

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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \begin {gather*} -\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^8,x]

[Out]

-1/7*a^2/x^7 - (a*b)/(3*x^6) - b^2/(5*x^5)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} -\frac {a^2}{7 x^7}-\frac {a b}{3 x^6}-\frac {b^2}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^8,x]

[Out]

-1/7*a^2/x^7 - (a*b)/(3*x^6) - b^2/(5*x^5)

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Mathics [A]
time = 1.78, size = 23, normalized size = 0.77 \begin {gather*} \frac {-\frac {a^2}{7}-\frac {a b x}{3}-\frac {b^2 x^2}{5}}{x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^2/x^8,x]')

[Out]

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

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Maple [A]
time = 0.09, size = 25, normalized size = 0.83

method	result	size
norman	\(\frac {-\frac {1}{5} x^{2} b^{2}-\frac {1}{3} a b x -\frac {1}{7} a^{2}}{x^{7}}\)	\(24\)
risch	\(\frac {-\frac {1}{5} x^{2} b^{2}-\frac {1}{3} a b x -\frac {1}{7} a^{2}}{x^{7}}\)	\(24\)
gosper	\(-\frac {21 x^{2} b^{2}+35 a b x +15 a^{2}}{105 x^{7}}\)	\(25\)
default	\(-\frac {a^{2}}{7 x^{7}}-\frac {a b}{3 x^{6}}-\frac {b^{2}}{5 x^{5}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a^2/x^7-1/3*a*b/x^6-1/5*b^2/x^5

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Maxima [A]
time = 0.25, size = 24, normalized size = 0.80 \begin {gather*} -\frac {21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(21*b^2*x^2 + 35*a*b*x + 15*a^2)/x^7

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Fricas [A]
time = 0.30, size = 24, normalized size = 0.80 \begin {gather*} -\frac {21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(21*b^2*x^2 + 35*a*b*x + 15*a^2)/x^7

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Sympy [A]
time = 0.11, size = 26, normalized size = 0.87 \begin {gather*} \frac {- 15 a^{2} - 35 a b x - 21 b^{2} x^{2}}{105 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-15*a**2 - 35*a*b*x - 21*b**2*x**2)/(105*x**7)

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Giac [A]
time = 0.00, size = 29, normalized size = 0.97 \begin {gather*} \frac {-21 x^{2} b^{2}-35 x b a-15 a^{2}}{105 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(21*b^2*x^2 + 35*a*b*x + 15*a^2)/x^7

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Mupad [B]
time = 0.04, size = 24, normalized size = 0.80 \begin {gather*} -\frac {\frac {a^2}{7}+\frac {a\,b\,x}{3}+\frac {b^2\,x^2}{5}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/x^8,x)

[Out]

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

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