3.2.45 \(\int \frac {(a+b x)^{10}}{x^{11}} \, dx\) [145]

Optimal. Leaf size=124 \[ -\frac {a^{10}}{10 x^{10}}-\frac {10 a^9 b}{9 x^9}-\frac {45 a^8 b^2}{8 x^8}-\frac {120 a^7 b^3}{7 x^7}-\frac {35 a^6 b^4}{x^6}-\frac {252 a^5 b^5}{5 x^5}-\frac {105 a^4 b^6}{2 x^4}-\frac {40 a^3 b^7}{x^3}-\frac {45 a^2 b^8}{2 x^2}-\frac {10 a b^9}{x}+b^{10} \log (x) \]

[Out]

-1/10*a^10/x^10-10/9*a^9*b/x^9-45/8*a^8*b^2/x^8-120/7*a^7*b^3/x^7-35*a^6*b^4/x^6-252/5*a^5*b^5/x^5-105/2*a^4*b
^6/x^4-40*a^3*b^7/x^3-45/2*a^2*b^8/x^2-10*a*b^9/x+b^10*ln(x)

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Rubi [A]
time = 0.03, antiderivative size = 124, 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^{10}}{10 x^{10}}-\frac {10 a^9 b}{9 x^9}-\frac {45 a^8 b^2}{8 x^8}-\frac {120 a^7 b^3}{7 x^7}-\frac {35 a^6 b^4}{x^6}-\frac {252 a^5 b^5}{5 x^5}-\frac {105 a^4 b^6}{2 x^4}-\frac {40 a^3 b^7}{x^3}-\frac {45 a^2 b^8}{2 x^2}-\frac {10 a b^9}{x}+b^{10} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^11,x]

[Out]

-1/10*a^10/x^10 - (10*a^9*b)/(9*x^9) - (45*a^8*b^2)/(8*x^8) - (120*a^7*b^3)/(7*x^7) - (35*a^6*b^4)/x^6 - (252*
a^5*b^5)/(5*x^5) - (105*a^4*b^6)/(2*x^4) - (40*a^3*b^7)/x^3 - (45*a^2*b^8)/(2*x^2) - (10*a*b^9)/x + b^10*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)^{10}}{x^{11}} \, dx &=\int \left (\frac {a^{10}}{x^{11}}+\frac {10 a^9 b}{x^{10}}+\frac {45 a^8 b^2}{x^9}+\frac {120 a^7 b^3}{x^8}+\frac {210 a^6 b^4}{x^7}+\frac {252 a^5 b^5}{x^6}+\frac {210 a^4 b^6}{x^5}+\frac {120 a^3 b^7}{x^4}+\frac {45 a^2 b^8}{x^3}+\frac {10 a b^9}{x^2}+\frac {b^{10}}{x}\right ) \, dx\\ &=-\frac {a^{10}}{10 x^{10}}-\frac {10 a^9 b}{9 x^9}-\frac {45 a^8 b^2}{8 x^8}-\frac {120 a^7 b^3}{7 x^7}-\frac {35 a^6 b^4}{x^6}-\frac {252 a^5 b^5}{5 x^5}-\frac {105 a^4 b^6}{2 x^4}-\frac {40 a^3 b^7}{x^3}-\frac {45 a^2 b^8}{2 x^2}-\frac {10 a b^9}{x}+b^{10} \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 124, normalized size = 1.00 \begin {gather*} -\frac {a^{10}}{10 x^{10}}-\frac {10 a^9 b}{9 x^9}-\frac {45 a^8 b^2}{8 x^8}-\frac {120 a^7 b^3}{7 x^7}-\frac {35 a^6 b^4}{x^6}-\frac {252 a^5 b^5}{5 x^5}-\frac {105 a^4 b^6}{2 x^4}-\frac {40 a^3 b^7}{x^3}-\frac {45 a^2 b^8}{2 x^2}-\frac {10 a b^9}{x}+b^{10} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^11,x]

[Out]

-1/10*a^10/x^10 - (10*a^9*b)/(9*x^9) - (45*a^8*b^2)/(8*x^8) - (120*a^7*b^3)/(7*x^7) - (35*a^6*b^4)/x^6 - (252*
a^5*b^5)/(5*x^5) - (105*a^4*b^6)/(2*x^4) - (40*a^3*b^7)/x^3 - (45*a^2*b^8)/(2*x^2) - (10*a*b^9)/x + b^10*Log[x
]

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Mathics [A]
time = 2.87, size = 113, normalized size = 0.91 \begin {gather*} \frac {-\frac {a \left (252 a^9+2800 a^8 b x+14175 a^7 b^2 x^2+43200 a^6 b^3 x^3+88200 a^5 b^4 x^4+127008 a^4 b^5 x^5+132300 a^3 b^6 x^6+100800 a^2 b^7 x^7+56700 a b^8 x^8+25200 b^9 x^9\right )}{2520}+b^{10} x^{10} \text {Log}\left [x\right ]}{x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^10/x^11,x]')

[Out]

(-a (252 a ^ 9 + 2800 a ^ 8 b x + 14175 a ^ 7 b ^ 2 x ^ 2 + 43200 a ^ 6 b ^ 3 x ^ 3 + 88200 a ^ 5 b ^ 4 x ^ 4
+ 127008 a ^ 4 b ^ 5 x ^ 5 + 132300 a ^ 3 b ^ 6 x ^ 6 + 100800 a ^ 2 b ^ 7 x ^ 7 + 56700 a b ^ 8 x ^ 8 + 25200
b ^ 9 x ^ 9) / 2520 + b ^ 10 x ^ 10 Log[x]) / x ^ 10

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Maple [A]
time = 0.08, size = 111, normalized size = 0.90

method	result	size
default	\(-\frac {a^{10}}{10 x^{10}}-\frac {10 a^{9} b}{9 x^{9}}-\frac {45 a^{8} b^{2}}{8 x^{8}}-\frac {120 a^{7} b^{3}}{7 x^{7}}-\frac {35 a^{6} b^{4}}{x^{6}}-\frac {252 a^{5} b^{5}}{5 x^{5}}-\frac {105 a^{4} b^{6}}{2 x^{4}}-\frac {40 a^{3} b^{7}}{x^{3}}-\frac {45 a^{2} b^{8}}{2 x^{2}}-\frac {10 a \,b^{9}}{x}+b^{10} \ln \left (x \right )\)	\(111\)
norman	\(\frac {-\frac {1}{10} a^{10}-10 a \,b^{9} x^{9}-\frac {45}{2} a^{2} b^{8} x^{8}-40 a^{3} b^{7} x^{7}-\frac {105}{2} a^{4} b^{6} x^{6}-\frac {252}{5} a^{5} b^{5} x^{5}-35 a^{6} b^{4} x^{4}-\frac {120}{7} a^{7} b^{3} x^{3}-\frac {45}{8} a^{8} b^{2} x^{2}-\frac {10}{9} a^{9} b x}{x^{10}}+b^{10} \ln \left (x \right )\)	\(111\)
risch	\(\frac {-\frac {1}{10} a^{10}-10 a \,b^{9} x^{9}-\frac {45}{2} a^{2} b^{8} x^{8}-40 a^{3} b^{7} x^{7}-\frac {105}{2} a^{4} b^{6} x^{6}-\frac {252}{5} a^{5} b^{5} x^{5}-35 a^{6} b^{4} x^{4}-\frac {120}{7} a^{7} b^{3} x^{3}-\frac {45}{8} a^{8} b^{2} x^{2}-\frac {10}{9} a^{9} b x}{x^{10}}+b^{10} \ln \left (x \right )\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^11,x,method=_RETURNVERBOSE)

[Out]

-1/10*a^10/x^10-10/9*a^9*b/x^9-45/8*a^8*b^2/x^8-120/7*a^7*b^3/x^7-35*a^6*b^4/x^6-252/5*a^5*b^5/x^5-105/2*a^4*b
^6/x^4-40*a^3*b^7/x^3-45/2*a^2*b^8/x^2-10*a*b^9/x+b^10*ln(x)

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Maxima [A]
time = 0.24, size = 111, normalized size = 0.90 \begin {gather*} b^{10} \log \left (x\right ) - \frac {25200 \, a b^{9} x^{9} + 56700 \, a^{2} b^{8} x^{8} + 100800 \, a^{3} b^{7} x^{7} + 132300 \, a^{4} b^{6} x^{6} + 127008 \, a^{5} b^{5} x^{5} + 88200 \, a^{6} b^{4} x^{4} + 43200 \, a^{7} b^{3} x^{3} + 14175 \, a^{8} b^{2} x^{2} + 2800 \, a^{9} b x + 252 \, a^{10}}{2520 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^11,x, algorithm="maxima")

[Out]

b^10*log(x) - 1/2520*(25200*a*b^9*x^9 + 56700*a^2*b^8*x^8 + 100800*a^3*b^7*x^7 + 132300*a^4*b^6*x^6 + 127008*a
^5*b^5*x^5 + 88200*a^6*b^4*x^4 + 43200*a^7*b^3*x^3 + 14175*a^8*b^2*x^2 + 2800*a^9*b*x + 252*a^10)/x^10

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Fricas [A]
time = 0.30, size = 114, normalized size = 0.92 \begin {gather*} \frac {2520 \, b^{10} x^{10} \log \left (x\right ) - 25200 \, a b^{9} x^{9} - 56700 \, a^{2} b^{8} x^{8} - 100800 \, a^{3} b^{7} x^{7} - 132300 \, a^{4} b^{6} x^{6} - 127008 \, a^{5} b^{5} x^{5} - 88200 \, a^{6} b^{4} x^{4} - 43200 \, a^{7} b^{3} x^{3} - 14175 \, a^{8} b^{2} x^{2} - 2800 \, a^{9} b x - 252 \, a^{10}}{2520 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^11,x, algorithm="fricas")

[Out]

1/2520*(2520*b^10*x^10*log(x) - 25200*a*b^9*x^9 - 56700*a^2*b^8*x^8 - 100800*a^3*b^7*x^7 - 132300*a^4*b^6*x^6
- 127008*a^5*b^5*x^5 - 88200*a^6*b^4*x^4 - 43200*a^7*b^3*x^3 - 14175*a^8*b^2*x^2 - 2800*a^9*b*x - 252*a^10)/x^
10

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Sympy [A]
time = 0.42, size = 119, normalized size = 0.96 \begin {gather*} b^{10} \log {\left (x \right )} + \frac {- 252 a^{10} - 2800 a^{9} b x - 14175 a^{8} b^{2} x^{2} - 43200 a^{7} b^{3} x^{3} - 88200 a^{6} b^{4} x^{4} - 127008 a^{5} b^{5} x^{5} - 132300 a^{4} b^{6} x^{6} - 100800 a^{3} b^{7} x^{7} - 56700 a^{2} b^{8} x^{8} - 25200 a b^{9} x^{9}}{2520 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**11,x)

[Out]

b**10*log(x) + (-252*a**10 - 2800*a**9*b*x - 14175*a**8*b**2*x**2 - 43200*a**7*b**3*x**3 - 88200*a**6*b**4*x**
4 - 127008*a**5*b**5*x**5 - 132300*a**4*b**6*x**6 - 100800*a**3*b**7*x**7 - 56700*a**2*b**8*x**8 - 25200*a*b**
9*x**9)/(2520*x**10)

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Giac [A]
time = 0.00, size = 125, normalized size = 1.01 \begin {gather*} \frac {\frac {1}{2520} \left (-25200 b^{9} a x^{9}-56700 b^{8} a^{2} x^{8}-100800 b^{7} a^{3} x^{7}-132300 b^{6} a^{4} x^{6}-127008 b^{5} a^{5} x^{5}-88200 b^{4} a^{6} x^{4}-43200 b^{3} a^{7} x^{3}-14175 b^{2} a^{8} x^{2}-2800 b a^{9} x-252 a^{10}\right )}{x^{10}}+b^{10} \ln \left |x\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^11,x)

[Out]

b^10*log(abs(x)) - 1/2520*(25200*a*b^9*x^9 + 56700*a^2*b^8*x^8 + 100800*a^3*b^7*x^7 + 132300*a^4*b^6*x^6 + 127
008*a^5*b^5*x^5 + 88200*a^6*b^4*x^4 + 43200*a^7*b^3*x^3 + 14175*a^8*b^2*x^2 + 2800*a^9*b*x + 252*a^10)/x^10

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Mupad [B]
time = 0.07, size = 111, normalized size = 0.90 \begin {gather*} b^{10}\,\ln \left (x\right )-\frac {\frac {a^{10}}{10}+\frac {10\,a^9\,b\,x}{9}+\frac {45\,a^8\,b^2\,x^2}{8}+\frac {120\,a^7\,b^3\,x^3}{7}+35\,a^6\,b^4\,x^4+\frac {252\,a^5\,b^5\,x^5}{5}+\frac {105\,a^4\,b^6\,x^6}{2}+40\,a^3\,b^7\,x^7+\frac {45\,a^2\,b^8\,x^8}{2}+10\,a\,b^9\,x^9}{x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^10/x^11,x)

[Out]

b^10*log(x) - (a^10/10 + 10*a*b^9*x^9 + (45*a^8*b^2*x^2)/8 + (120*a^7*b^3*x^3)/7 + 35*a^6*b^4*x^4 + (252*a^5*b
^5*x^5)/5 + (105*a^4*b^6*x^6)/2 + 40*a^3*b^7*x^7 + (45*a^2*b^8*x^8)/2 + (10*a^9*b*x)/9)/x^10

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