3.3.43 \(\int \frac {(a+b x)^8}{x^{10}} \, dx\) [243]

Optimal. Leaf size=17 \[ -\frac {(a+b x)^9}{9 a x^9} \]

[Out]

-1/9*(b*x+a)^9/a/x^9

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {37} \begin {gather*} -\frac {(a+b x)^9}{9 a x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^8/x^10,x]

[Out]

-1/9*(a + b*x)^9/(a*x^9)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin {align*} \int \frac {(a+b x)^8}{x^{10}} \, dx &=-\frac {(a+b x)^9}{9 a x^9}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(17)=34\).
time = 0.01, size = 96, normalized size = 5.65 \begin {gather*} -\frac {a^8}{9 x^9}-\frac {a^7 b}{x^8}-\frac {4 a^6 b^2}{x^7}-\frac {28 a^5 b^3}{3 x^6}-\frac {14 a^4 b^4}{x^5}-\frac {14 a^3 b^5}{x^4}-\frac {28 a^2 b^6}{3 x^3}-\frac {4 a b^7}{x^2}-\frac {b^8}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^8/x^10,x]

[Out]

-1/9*a^8/x^9 - (a^7*b)/x^8 - (4*a^6*b^2)/x^7 - (28*a^5*b^3)/(3*x^6) - (14*a^4*b^4)/x^5 - (14*a^3*b^5)/x^4 - (2
8*a^2*b^6)/(3*x^3) - (4*a*b^7)/x^2 - b^8/x

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(92\) vs. \(2(17)=34\).
time = 2.28, size = 90, normalized size = 5.29 \begin {gather*} \frac {-a^8-9 a^7 b x-36 a^6 b^2 x^2-84 a^5 b^3 x^3-126 a^4 b^4 x^4-126 a^3 b^5 x^5-84 a^2 b^6 x^6-36 a b^7 x^7-9 b^8 x^8}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^8/x^10,x]')

[Out]

(-a ^ 8 - 9 a ^ 7 b x - 36 a ^ 6 b ^ 2 x ^ 2 - 84 a ^ 5 b ^ 3 x ^ 3 - 126 a ^ 4 b ^ 4 x ^ 4 - 126 a ^ 3 b ^ 5
x ^ 5 - 84 a ^ 2 b ^ 6 x ^ 6 - 36 a b ^ 7 x ^ 7 - 9 b ^ 8 x ^ 8) / (9 x ^ 9)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(15)=30\).
time = 0.08, size = 91, normalized size = 5.35

method	result	size
gosper	\(-\frac {9 b^{8} x^{8}+36 a \,b^{7} x^{7}+84 a^{2} x^{6} b^{6}+126 a^{3} x^{5} b^{5}+126 a^{4} x^{4} b^{4}+84 a^{5} x^{3} b^{3}+36 a^{6} x^{2} b^{2}+9 a^{7} x b +a^{8}}{9 x^{9}}\)	\(89\)
norman	\(\frac {-b^{8} x^{8}-4 a \,b^{7} x^{7}-\frac {28}{3} a^{2} x^{6} b^{6}-14 a^{3} x^{5} b^{5}-14 a^{4} x^{4} b^{4}-\frac {28}{3} a^{5} x^{3} b^{3}-4 a^{6} x^{2} b^{2}-a^{7} x b -\frac {1}{9} a^{8}}{x^{9}}\)	\(90\)
risch	\(\frac {-b^{8} x^{8}-4 a \,b^{7} x^{7}-\frac {28}{3} a^{2} x^{6} b^{6}-14 a^{3} x^{5} b^{5}-14 a^{4} x^{4} b^{4}-\frac {28}{3} a^{5} x^{3} b^{3}-4 a^{6} x^{2} b^{2}-a^{7} x b -\frac {1}{9} a^{8}}{x^{9}}\)	\(90\)
default	\(-\frac {b^{8}}{x}-\frac {28 b^{6} a^{2}}{3 x^{3}}-\frac {14 a^{3} b^{5}}{x^{4}}-\frac {a^{8}}{9 x^{9}}-\frac {4 a \,b^{7}}{x^{2}}-\frac {a^{7} b}{x^{8}}-\frac {14 a^{4} b^{4}}{x^{5}}-\frac {28 a^{5} b^{3}}{3 x^{6}}-\frac {4 b^{2} a^{6}}{x^{7}}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^8/x^10,x,method=_RETURNVERBOSE)

[Out]

-b^8/x-28/3*b^6*a^2/x^3-14*a^3*b^5/x^4-1/9*a^8/x^9-4*a*b^7/x^2-a^7*b/x^8-14*a^4*b^4/x^5-28/3*a^5*b^3/x^6-4*b^2
*a^6/x^7

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (15) = 30\).
time = 0.27, size = 88, normalized size = 5.18 \begin {gather*} -\frac {9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8/x^10,x, algorithm="maxima")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*x^4 + 84*a^5*b^3*x^3 + 36*a^6*
b^2*x^2 + 9*a^7*b*x + a^8)/x^9

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (15) = 30\).
time = 0.30, size = 88, normalized size = 5.18 \begin {gather*} -\frac {9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8/x^10,x, algorithm="fricas")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*x^4 + 84*a^5*b^3*x^3 + 36*a^6*
b^2*x^2 + 9*a^7*b*x + a^8)/x^9

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (14) = 28\).
time = 0.33, size = 95, normalized size = 5.59 \begin {gather*} \frac {- a^{8} - 9 a^{7} b x - 36 a^{6} b^{2} x^{2} - 84 a^{5} b^{3} x^{3} - 126 a^{4} b^{4} x^{4} - 126 a^{3} b^{5} x^{5} - 84 a^{2} b^{6} x^{6} - 36 a b^{7} x^{7} - 9 b^{8} x^{8}}{9 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**8/x**10,x)

[Out]

(-a**8 - 9*a**7*b*x - 36*a**6*b**2*x**2 - 84*a**5*b**3*x**3 - 126*a**4*b**4*x**4 - 126*a**3*b**5*x**5 - 84*a**
2*b**6*x**6 - 36*a*b**7*x**7 - 9*b**8*x**8)/(9*x**9)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (15) = 30\).
time = 0.00, size = 99, normalized size = 5.82 \begin {gather*} \frac {-9 x^{8} b^{8}-36 x^{7} b^{7} a-84 x^{6} b^{6} a^{2}-126 x^{5} b^{5} a^{3}-126 x^{4} b^{4} a^{4}-84 x^{3} b^{3} a^{5}-36 x^{2} b^{2} a^{6}-9 x b a^{7}-a^{8}}{9 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8/x^10,x)

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*x^4 + 84*a^5*b^3*x^3 + 36*a^6*
b^2*x^2 + 9*a^7*b*x + a^8)/x^9

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Mupad [B]
time = 0.09, size = 88, normalized size = 5.18 \begin {gather*} -\frac {\frac {a^8}{9}+a^7\,b\,x+4\,a^6\,b^2\,x^2+\frac {28\,a^5\,b^3\,x^3}{3}+14\,a^4\,b^4\,x^4+14\,a^3\,b^5\,x^5+\frac {28\,a^2\,b^6\,x^6}{3}+4\,a\,b^7\,x^7+b^8\,x^8}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^8/x^10,x)

[Out]

-(a^8/9 + b^8*x^8 + 4*a*b^7*x^7 + 4*a^6*b^2*x^2 + (28*a^5*b^3*x^3)/3 + 14*a^4*b^4*x^4 + 14*a^3*b^5*x^5 + (28*a
^2*b^6*x^6)/3 + a^7*b*x)/x^9

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