3.3.49 \(\int \frac {(a+b x)^2}{x^{10}} \, dx\) [249]

Optimal. Leaf size=30 \[ -\frac {a^2}{9 x^9}-\frac {a b}{4 x^8}-\frac {b^2}{7 x^7} \]

[Out]

-1/9*a^2/x^9-1/4*a*b/x^8-1/7*b^2/x^7

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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}{9 x^9}-\frac {a b}{4 x^8}-\frac {b^2}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^10,x]

[Out]

-1/9*a^2/x^9 - (a*b)/(4*x^8) - b^2/(7*x^7)

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^{10}} \, dx &=\int \left (\frac {a^2}{x^{10}}+\frac {2 a b}{x^9}+\frac {b^2}{x^8}\right ) \, dx\\ &=-\frac {a^2}{9 x^9}-\frac {a b}{4 x^8}-\frac {b^2}{7 x^7}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} -\frac {a^2}{9 x^9}-\frac {a b}{4 x^8}-\frac {b^2}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^10,x]

[Out]

-1/9*a^2/x^9 - (a*b)/(4*x^8) - b^2/(7*x^7)

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Mathics [A]
time = 1.73, size = 23, normalized size = 0.77 \begin {gather*} \frac {-\frac {a^2}{9}-\frac {a b x}{4}-\frac {b^2 x^2}{7}}{x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^2/x^10,x]')

[Out]

(-a ^ 2 / 9 - a b x / 4 - b ^ 2 x ^ 2 / 7) / x ^ 9

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Maple [A]
time = 0.08, size = 25, normalized size = 0.83

method	result	size
norman	\(\frac {-\frac {1}{7} x^{2} b^{2}-\frac {1}{4} a b x -\frac {1}{9} a^{2}}{x^{9}}\)	\(24\)
risch	\(\frac {-\frac {1}{7} x^{2} b^{2}-\frac {1}{4} a b x -\frac {1}{9} a^{2}}{x^{9}}\)	\(24\)
gosper	\(-\frac {36 x^{2} b^{2}+63 a b x +28 a^{2}}{252 x^{9}}\)	\(25\)
default	\(-\frac {a^{2}}{9 x^{9}}-\frac {a b}{4 x^{8}}-\frac {b^{2}}{7 x^{7}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/x^10,x,method=_RETURNVERBOSE)

[Out]

-1/9*a^2/x^9-1/4*a*b/x^8-1/7*b^2/x^7

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Maxima [A]
time = 0.27, size = 24, normalized size = 0.80 \begin {gather*} -\frac {36 \, b^{2} x^{2} + 63 \, a b x + 28 \, a^{2}}{252 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^10,x, algorithm="maxima")

[Out]

-1/252*(36*b^2*x^2 + 63*a*b*x + 28*a^2)/x^9

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Fricas [A]
time = 0.30, size = 24, normalized size = 0.80 \begin {gather*} -\frac {36 \, b^{2} x^{2} + 63 \, a b x + 28 \, a^{2}}{252 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^10,x, algorithm="fricas")

[Out]

-1/252*(36*b^2*x^2 + 63*a*b*x + 28*a^2)/x^9

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Sympy [A]
time = 0.12, size = 26, normalized size = 0.87 \begin {gather*} \frac {- 28 a^{2} - 63 a b x - 36 b^{2} x^{2}}{252 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/x**10,x)

[Out]

(-28*a**2 - 63*a*b*x - 36*b**2*x**2)/(252*x**9)

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Giac [A]
time = 0.00, size = 29, normalized size = 0.97 \begin {gather*} \frac {-36 x^{2} b^{2}-63 x b a-28 a^{2}}{252 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^10,x)

[Out]

-1/252*(36*b^2*x^2 + 63*a*b*x + 28*a^2)/x^9

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Mupad [B]
time = 0.04, size = 24, normalized size = 0.80 \begin {gather*} -\frac {\frac {a^2}{9}+\frac {a\,b\,x}{4}+\frac {b^2\,x^2}{7}}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/x^10,x)

[Out]

-(a^2/9 + (b^2*x^2)/7 + (a*b*x)/4)/x^9

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