\(\int \frac {(a+b x)^{10}}{x^{10}} \, dx\) [144]

Optimal result

Integrand size = 11, antiderivative size = 114 \[ \int \frac {(a+b x)^{10}}{x^{10}} \, dx=-\frac {a^{10}}{9 x^9}-\frac {5 a^9 b}{4 x^8}-\frac {45 a^8 b^2}{7 x^7}-\frac {20 a^7 b^3}{x^6}-\frac {42 a^6 b^4}{x^5}-\frac {63 a^5 b^5}{x^4}-\frac {70 a^4 b^6}{x^3}-\frac {60 a^3 b^7}{x^2}-\frac {45 a^2 b^8}{x}+b^{10} x+10 a b^9 \log (x) \]

[Out]

-1/9*a^10/x^9-5/4*a^9*b/x^8-45/7*a^8*b^2/x^7-20*a^7*b^3/x^6-42*a^6*b^4/x^5-63*a^5*b^5/x^4-70*a^4*b^6/x^3-60*a^
3*b^7/x^2-45*a^2*b^8/x+b^10*x+10*a*b^9*ln(x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^{10}}{x^{10}} \, dx=-\frac {a^{10}}{9 x^9}-\frac {5 a^9 b}{4 x^8}-\frac {45 a^8 b^2}{7 x^7}-\frac {20 a^7 b^3}{x^6}-\frac {42 a^6 b^4}{x^5}-\frac {63 a^5 b^5}{x^4}-\frac {70 a^4 b^6}{x^3}-\frac {60 a^3 b^7}{x^2}-\frac {45 a^2 b^8}{x}+10 a b^9 \log (x)+b^{10} x \]

[In]

Int[(a + b*x)^10/x^10,x]

[Out]

-1/9*a^10/x^9 - (5*a^9*b)/(4*x^8) - (45*a^8*b^2)/(7*x^7) - (20*a^7*b^3)/x^6 - (42*a^6*b^4)/x^5 - (63*a^5*b^5)/
x^4 - (70*a^4*b^6)/x^3 - (60*a^3*b^7)/x^2 - (45*a^2*b^8)/x + b^10*x + 10*a*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*} \text {integral}& = \int \left (b^{10}+\frac {a^{10}}{x^{10}}+\frac {10 a^9 b}{x^9}+\frac {45 a^8 b^2}{x^8}+\frac {120 a^7 b^3}{x^7}+\frac {210 a^6 b^4}{x^6}+\frac {252 a^5 b^5}{x^5}+\frac {210 a^4 b^6}{x^4}+\frac {120 a^3 b^7}{x^3}+\frac {45 a^2 b^8}{x^2}+\frac {10 a b^9}{x}\right ) \, dx \\ & = -\frac {a^{10}}{9 x^9}-\frac {5 a^9 b}{4 x^8}-\frac {45 a^8 b^2}{7 x^7}-\frac {20 a^7 b^3}{x^6}-\frac {42 a^6 b^4}{x^5}-\frac {63 a^5 b^5}{x^4}-\frac {70 a^4 b^6}{x^3}-\frac {60 a^3 b^7}{x^2}-\frac {45 a^2 b^8}{x}+b^{10} x+10 a b^9 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^{10}} \, dx=-\frac {a^{10}}{9 x^9}-\frac {5 a^9 b}{4 x^8}-\frac {45 a^8 b^2}{7 x^7}-\frac {20 a^7 b^3}{x^6}-\frac {42 a^6 b^4}{x^5}-\frac {63 a^5 b^5}{x^4}-\frac {70 a^4 b^6}{x^3}-\frac {60 a^3 b^7}{x^2}-\frac {45 a^2 b^8}{x}+b^{10} x+10 a b^9 \log (x) \]

[In]

Integrate[(a + b*x)^10/x^10,x]

[Out]

-1/9*a^10/x^9 - (5*a^9*b)/(4*x^8) - (45*a^8*b^2)/(7*x^7) - (20*a^7*b^3)/x^6 - (42*a^6*b^4)/x^5 - (63*a^5*b^5)/
x^4 - (70*a^4*b^6)/x^3 - (60*a^3*b^7)/x^2 - (45*a^2*b^8)/x + b^10*x + 10*a*b^9*Log[x]

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {a^{10}}{9 x^{9}}-\frac {5 a^{9} b}{4 x^{8}}-\frac {45 a^{8} b^{2}}{7 x^{7}}-\frac {20 a^{7} b^{3}}{x^{6}}-\frac {42 a^{6} b^{4}}{x^{5}}-\frac {63 a^{5} b^{5}}{x^{4}}-\frac {70 a^{4} b^{6}}{x^{3}}-\frac {60 a^{3} b^{7}}{x^{2}}-\frac {45 a^{2} b^{8}}{x}+b^{10} x +10 a \,b^{9} \ln \left (x \right )\)	\(109\)
risch	\(b^{10} x +\frac {-45 a^{2} b^{8} x^{8}-60 a^{3} b^{7} x^{7}-70 a^{4} b^{6} x^{6}-63 a^{5} b^{5} x^{5}-42 a^{6} b^{4} x^{4}-20 a^{7} b^{3} x^{3}-\frac {45}{7} a^{8} b^{2} x^{2}-\frac {5}{4} a^{9} b x -\frac {1}{9} a^{10}}{x^{9}}+10 a \,b^{9} \ln \left (x \right )\)	\(109\)
norman	\(\frac {b^{10} x^{10}-\frac {1}{9} a^{10}-45 a^{2} b^{8} x^{8}-60 a^{3} b^{7} x^{7}-70 a^{4} b^{6} x^{6}-63 a^{5} b^{5} x^{5}-42 a^{6} b^{4} x^{4}-20 a^{7} b^{3} x^{3}-\frac {45}{7} a^{8} b^{2} x^{2}-\frac {5}{4} a^{9} b x}{x^{9}}+10 a \,b^{9} \ln \left (x \right )\)	\(111\)
parallelrisch	\(\frac {2520 a \,b^{9} \ln \left (x \right ) x^{9}+252 b^{10} x^{10}-11340 a^{2} b^{8} x^{8}-15120 a^{3} b^{7} x^{7}-17640 a^{4} b^{6} x^{6}-15876 a^{5} b^{5} x^{5}-10584 a^{6} b^{4} x^{4}-5040 a^{7} b^{3} x^{3}-1620 a^{8} b^{2} x^{2}-315 a^{9} b x -28 a^{10}}{252 x^{9}}\)	\(115\)

[In]

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

[Out]

-1/9*a^10/x^9-5/4*a^9*b/x^8-45/7*a^8*b^2/x^7-20*a^7*b^3/x^6-42*a^6*b^4/x^5-63*a^5*b^5/x^4-70*a^4*b^6/x^3-60*a^
3*b^7/x^2-45*a^2*b^8/x+b^10*x+10*a*b^9*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^{10}} \, dx=\frac {252 \, b^{10} x^{10} + 2520 \, a b^{9} x^{9} \log \left (x\right ) - 11340 \, a^{2} b^{8} x^{8} - 15120 \, a^{3} b^{7} x^{7} - 17640 \, a^{4} b^{6} x^{6} - 15876 \, a^{5} b^{5} x^{5} - 10584 \, a^{6} b^{4} x^{4} - 5040 \, a^{7} b^{3} x^{3} - 1620 \, a^{8} b^{2} x^{2} - 315 \, a^{9} b x - 28 \, a^{10}}{252 \, x^{9}} \]

[In]

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

[Out]

1/252*(252*b^10*x^10 + 2520*a*b^9*x^9*log(x) - 11340*a^2*b^8*x^8 - 15120*a^3*b^7*x^7 - 17640*a^4*b^6*x^6 - 158
76*a^5*b^5*x^5 - 10584*a^6*b^4*x^4 - 5040*a^7*b^3*x^3 - 1620*a^8*b^2*x^2 - 315*a^9*b*x - 28*a^10)/x^9

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^{10}}{x^{10}} \, dx=10 a b^{9} \log {\left (x \right )} + b^{10} x + \frac {- 28 a^{10} - 315 a^{9} b x - 1620 a^{8} b^{2} x^{2} - 5040 a^{7} b^{3} x^{3} - 10584 a^{6} b^{4} x^{4} - 15876 a^{5} b^{5} x^{5} - 17640 a^{4} b^{6} x^{6} - 15120 a^{3} b^{7} x^{7} - 11340 a^{2} b^{8} x^{8}}{252 x^{9}} \]

[In]

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

[Out]

10*a*b**9*log(x) + b**10*x + (-28*a**10 - 315*a**9*b*x - 1620*a**8*b**2*x**2 - 5040*a**7*b**3*x**3 - 10584*a**
6*b**4*x**4 - 15876*a**5*b**5*x**5 - 17640*a**4*b**6*x**6 - 15120*a**3*b**7*x**7 - 11340*a**2*b**8*x**8)/(252*
x**9)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^{10}} \, dx=b^{10} x + 10 \, a b^{9} \log \left (x\right ) - \frac {11340 \, a^{2} b^{8} x^{8} + 15120 \, a^{3} b^{7} x^{7} + 17640 \, a^{4} b^{6} x^{6} + 15876 \, a^{5} b^{5} x^{5} + 10584 \, a^{6} b^{4} x^{4} + 5040 \, a^{7} b^{3} x^{3} + 1620 \, a^{8} b^{2} x^{2} + 315 \, a^{9} b x + 28 \, a^{10}}{252 \, x^{9}} \]

[In]

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

[Out]

b^10*x + 10*a*b^9*log(x) - 1/252*(11340*a^2*b^8*x^8 + 15120*a^3*b^7*x^7 + 17640*a^4*b^6*x^6 + 15876*a^5*b^5*x^
5 + 10584*a^6*b^4*x^4 + 5040*a^7*b^3*x^3 + 1620*a^8*b^2*x^2 + 315*a^9*b*x + 28*a^10)/x^9

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^{10}} \, dx=b^{10} x + 10 \, a b^{9} \log \left ({\left | x \right |}\right ) - \frac {11340 \, a^{2} b^{8} x^{8} + 15120 \, a^{3} b^{7} x^{7} + 17640 \, a^{4} b^{6} x^{6} + 15876 \, a^{5} b^{5} x^{5} + 10584 \, a^{6} b^{4} x^{4} + 5040 \, a^{7} b^{3} x^{3} + 1620 \, a^{8} b^{2} x^{2} + 315 \, a^{9} b x + 28 \, a^{10}}{252 \, x^{9}} \]

[In]

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

[Out]

b^10*x + 10*a*b^9*log(abs(x)) - 1/252*(11340*a^2*b^8*x^8 + 15120*a^3*b^7*x^7 + 17640*a^4*b^6*x^6 + 15876*a^5*b
^5*x^5 + 10584*a^6*b^4*x^4 + 5040*a^7*b^3*x^3 + 1620*a^8*b^2*x^2 + 315*a^9*b*x + 28*a^10)/x^9

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^{10}} \, dx=-\frac {\frac {a^{10}}{9}-b^{10}\,x^{10}+\frac {45\,a^8\,b^2\,x^2}{7}+20\,a^7\,b^3\,x^3+42\,a^6\,b^4\,x^4+63\,a^5\,b^5\,x^5+70\,a^4\,b^6\,x^6+60\,a^3\,b^7\,x^7+45\,a^2\,b^8\,x^8+\frac {5\,a^9\,b\,x}{4}-10\,a\,b^9\,x^9\,\ln \left (x\right )}{x^9} \]

[In]

int((a + b*x)^10/x^10,x)

[Out]

-(a^10/9 - b^10*x^10 + (45*a^8*b^2*x^2)/7 + 20*a^7*b^3*x^3 + 42*a^6*b^4*x^4 + 63*a^5*b^5*x^5 + 70*a^4*b^6*x^6
+ 60*a^3*b^7*x^7 + 45*a^2*b^8*x^8 + (5*a^9*b*x)/4 - 10*a*b^9*x^9*log(x))/x^9