3.3.42 \(\int \frac {(a+b x)^9}{x^{10}} \, dx\) [242]

Optimal. Leaf size=109 \[ -\frac {a^9}{9 x^9}-\frac {9 a^8 b}{8 x^8}-\frac {36 a^7 b^2}{7 x^7}-\frac {14 a^6 b^3}{x^6}-\frac {126 a^5 b^4}{5 x^5}-\frac {63 a^4 b^5}{2 x^4}-\frac {28 a^3 b^6}{x^3}-\frac {18 a^2 b^7}{x^2}-\frac {9 a b^8}{x}+b^9 \log (x) \]

[Out]

-1/9*a^9/x^9-9/8*a^8*b/x^8-36/7*a^7*b^2/x^7-14*a^6*b^3/x^6-126/5*a^5*b^4/x^5-63/2*a^4*b^5/x^4-28*a^3*b^6/x^3-1
8*a^2*b^7/x^2-9*a*b^8/x+b^9*ln(x)

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Rubi [A]
time = 0.03, antiderivative size = 109, 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^9}{9 x^9}-\frac {9 a^8 b}{8 x^8}-\frac {36 a^7 b^2}{7 x^7}-\frac {14 a^6 b^3}{x^6}-\frac {126 a^5 b^4}{5 x^5}-\frac {63 a^4 b^5}{2 x^4}-\frac {28 a^3 b^6}{x^3}-\frac {18 a^2 b^7}{x^2}-\frac {9 a b^8}{x}+b^9 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^9/x^10,x]

[Out]

-1/9*a^9/x^9 - (9*a^8*b)/(8*x^8) - (36*a^7*b^2)/(7*x^7) - (14*a^6*b^3)/x^6 - (126*a^5*b^4)/(5*x^5) - (63*a^4*b
^5)/(2*x^4) - (28*a^3*b^6)/x^3 - (18*a^2*b^7)/x^2 - (9*a*b^8)/x + b^9*Log[x]

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)^9}{x^{10}} \, dx &=\int \left (\frac {a^9}{x^{10}}+\frac {9 a^8 b}{x^9}+\frac {36 a^7 b^2}{x^8}+\frac {84 a^6 b^3}{x^7}+\frac {126 a^5 b^4}{x^6}+\frac {126 a^4 b^5}{x^5}+\frac {84 a^3 b^6}{x^4}+\frac {36 a^2 b^7}{x^3}+\frac {9 a b^8}{x^2}+\frac {b^9}{x}\right ) \, dx\\ &=-\frac {a^9}{9 x^9}-\frac {9 a^8 b}{8 x^8}-\frac {36 a^7 b^2}{7 x^7}-\frac {14 a^6 b^3}{x^6}-\frac {126 a^5 b^4}{5 x^5}-\frac {63 a^4 b^5}{2 x^4}-\frac {28 a^3 b^6}{x^3}-\frac {18 a^2 b^7}{x^2}-\frac {9 a b^8}{x}+b^9 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 109, normalized size = 1.00 \begin {gather*} -\frac {a^9}{9 x^9}-\frac {9 a^8 b}{8 x^8}-\frac {36 a^7 b^2}{7 x^7}-\frac {14 a^6 b^3}{x^6}-\frac {126 a^5 b^4}{5 x^5}-\frac {63 a^4 b^5}{2 x^4}-\frac {28 a^3 b^6}{x^3}-\frac {18 a^2 b^7}{x^2}-\frac {9 a b^8}{x}+b^9 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^9/x^10,x]

[Out]

-1/9*a^9/x^9 - (9*a^8*b)/(8*x^8) - (36*a^7*b^2)/(7*x^7) - (14*a^6*b^3)/x^6 - (126*a^5*b^4)/(5*x^5) - (63*a^4*b
^5)/(2*x^4) - (28*a^3*b^6)/x^3 - (18*a^2*b^7)/x^2 - (9*a*b^8)/x + b^9*Log[x]

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Mathics [A]
time = 2.65, size = 102, normalized size = 0.94 \begin {gather*} \frac {-\frac {a \left (280 a^8+2835 a^7 b x+12960 a^6 b^2 x^2+35280 a^5 b^3 x^3+63504 a^4 b^4 x^4+79380 a^3 b^5 x^5+70560 a^2 b^6 x^6+45360 a b^7 x^7+22680 b^8 x^8\right )}{2520}+b^9 x^9 \text {Log}\left [x\right ]}{x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^9/x^10,x]')

[Out]

(-a (280 a ^ 8 + 2835 a ^ 7 b x + 12960 a ^ 6 b ^ 2 x ^ 2 + 35280 a ^ 5 b ^ 3 x ^ 3 + 63504 a ^ 4 b ^ 4 x ^ 4
+ 79380 a ^ 3 b ^ 5 x ^ 5 + 70560 a ^ 2 b ^ 6 x ^ 6 + 45360 a b ^ 7 x ^ 7 + 22680 b ^ 8 x ^ 8) / 2520 + b ^ 9
x ^ 9 Log[x]) / x ^ 9

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Maple [A]
time = 0.08, size = 100, normalized size = 0.92

method	result	size
default	\(-\frac {a^{9}}{9 x^{9}}-\frac {9 a^{8} b}{8 x^{8}}-\frac {36 a^{7} b^{2}}{7 x^{7}}-\frac {14 a^{6} b^{3}}{x^{6}}-\frac {126 a^{5} b^{4}}{5 x^{5}}-\frac {63 a^{4} b^{5}}{2 x^{4}}-\frac {28 a^{3} b^{6}}{x^{3}}-\frac {18 a^{2} b^{7}}{x^{2}}-\frac {9 a \,b^{8}}{x}+b^{9} \ln \left (x \right )\)	\(100\)
norman	\(\frac {-\frac {1}{9} a^{9}-9 a \,b^{8} x^{8}-18 a^{2} b^{7} x^{7}-28 a^{3} b^{6} x^{6}-\frac {63}{2} a^{4} b^{5} x^{5}-\frac {126}{5} a^{5} b^{4} x^{4}-14 a^{6} b^{3} x^{3}-\frac {36}{7} a^{7} b^{2} x^{2}-\frac {9}{8} a^{8} b x}{x^{9}}+b^{9} \ln \left (x \right )\)	\(100\)
risch	\(\frac {-\frac {1}{9} a^{9}-9 a \,b^{8} x^{8}-18 a^{2} b^{7} x^{7}-28 a^{3} b^{6} x^{6}-\frac {63}{2} a^{4} b^{5} x^{5}-\frac {126}{5} a^{5} b^{4} x^{4}-14 a^{6} b^{3} x^{3}-\frac {36}{7} a^{7} b^{2} x^{2}-\frac {9}{8} a^{8} b x}{x^{9}}+b^{9} \ln \left (x \right )\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^9/x^10,x,method=_RETURNVERBOSE)

[Out]

-1/9*a^9/x^9-9/8*a^8*b/x^8-36/7*a^7*b^2/x^7-14*a^6*b^3/x^6-126/5*a^5*b^4/x^5-63/2*a^4*b^5/x^4-28*a^3*b^6/x^3-1
8*a^2*b^7/x^2-9*a*b^8/x+b^9*ln(x)

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Maxima [A]
time = 0.25, size = 100, normalized size = 0.92 \begin {gather*} b^{9} \log \left (x\right ) - \frac {22680 \, a b^{8} x^{8} + 45360 \, a^{2} b^{7} x^{7} + 70560 \, a^{3} b^{6} x^{6} + 79380 \, a^{4} b^{5} x^{5} + 63504 \, a^{5} b^{4} x^{4} + 35280 \, a^{6} b^{3} x^{3} + 12960 \, a^{7} b^{2} x^{2} + 2835 \, a^{8} b x + 280 \, a^{9}}{2520 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/x^10,x, algorithm="maxima")

[Out]

b^9*log(x) - 1/2520*(22680*a*b^8*x^8 + 45360*a^2*b^7*x^7 + 70560*a^3*b^6*x^6 + 79380*a^4*b^5*x^5 + 63504*a^5*b
^4*x^4 + 35280*a^6*b^3*x^3 + 12960*a^7*b^2*x^2 + 2835*a^8*b*x + 280*a^9)/x^9

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Fricas [A]
time = 0.31, size = 103, normalized size = 0.94 \begin {gather*} \frac {2520 \, b^{9} x^{9} \log \left (x\right ) - 22680 \, a b^{8} x^{8} - 45360 \, a^{2} b^{7} x^{7} - 70560 \, a^{3} b^{6} x^{6} - 79380 \, a^{4} b^{5} x^{5} - 63504 \, a^{5} b^{4} x^{4} - 35280 \, a^{6} b^{3} x^{3} - 12960 \, a^{7} b^{2} x^{2} - 2835 \, a^{8} b x - 280 \, a^{9}}{2520 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/x^10,x, algorithm="fricas")

[Out]

1/2520*(2520*b^9*x^9*log(x) - 22680*a*b^8*x^8 - 45360*a^2*b^7*x^7 - 70560*a^3*b^6*x^6 - 79380*a^4*b^5*x^5 - 63
504*a^5*b^4*x^4 - 35280*a^6*b^3*x^3 - 12960*a^7*b^2*x^2 - 2835*a^8*b*x - 280*a^9)/x^9

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Sympy [A]
time = 0.37, size = 107, normalized size = 0.98 \begin {gather*} b^{9} \log {\left (x \right )} + \frac {- 280 a^{9} - 2835 a^{8} b x - 12960 a^{7} b^{2} x^{2} - 35280 a^{6} b^{3} x^{3} - 63504 a^{5} b^{4} x^{4} - 79380 a^{4} b^{5} x^{5} - 70560 a^{3} b^{6} x^{6} - 45360 a^{2} b^{7} x^{7} - 22680 a b^{8} x^{8}}{2520 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**9/x**10,x)

[Out]

b**9*log(x) + (-280*a**9 - 2835*a**8*b*x - 12960*a**7*b**2*x**2 - 35280*a**6*b**3*x**3 - 63504*a**5*b**4*x**4
- 79380*a**4*b**5*x**5 - 70560*a**3*b**6*x**6 - 45360*a**2*b**7*x**7 - 22680*a*b**8*x**8)/(2520*x**9)

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Giac [A]
time = 0.00, size = 113, normalized size = 1.04 \begin {gather*} \frac {\frac {1}{2520} \left (-22680 b^{8} a x^{8}-45360 b^{7} a^{2} x^{7}-70560 b^{6} a^{3} x^{6}-79380 b^{5} a^{4} x^{5}-63504 b^{4} a^{5} x^{4}-35280 b^{3} a^{6} x^{3}-12960 b^{2} a^{7} x^{2}-2835 b a^{8} x-280 a^{9}\right )}{x^{9}}+b^{9} \ln \left |x\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^9/x^10,x)

[Out]

b^9*log(abs(x)) - 1/2520*(22680*a*b^8*x^8 + 45360*a^2*b^7*x^7 + 70560*a^3*b^6*x^6 + 79380*a^4*b^5*x^5 + 63504*
a^5*b^4*x^4 + 35280*a^6*b^3*x^3 + 12960*a^7*b^2*x^2 + 2835*a^8*b*x + 280*a^9)/x^9

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Mupad [B]
time = 0.08, size = 100, normalized size = 0.92 \begin {gather*} b^9\,\ln \left (x\right )-\frac {\frac {a^9}{9}+\frac {9\,a^8\,b\,x}{8}+\frac {36\,a^7\,b^2\,x^2}{7}+14\,a^6\,b^3\,x^3+\frac {126\,a^5\,b^4\,x^4}{5}+\frac {63\,a^4\,b^5\,x^5}{2}+28\,a^3\,b^6\,x^6+18\,a^2\,b^7\,x^7+9\,a\,b^8\,x^8}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^9/x^10,x)

[Out]

b^9*log(x) - (a^9/9 + 9*a*b^8*x^8 + (36*a^7*b^2*x^2)/7 + 14*a^6*b^3*x^3 + (126*a^5*b^4*x^4)/5 + (63*a^4*b^5*x^
5)/2 + 28*a^3*b^6*x^6 + 18*a^2*b^7*x^7 + (9*a^8*b*x)/8)/x^9

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