\(\int \frac {(a+b x)^{10}}{x} \, dx\) [135]

Optimal result

Integrand size = 11, antiderivative size = 122 \[ \int \frac {(a+b x)^{10}}{x} \, dx=10 a^9 b x+\frac {45}{2} a^8 b^2 x^2+40 a^7 b^3 x^3+\frac {105}{2} a^6 b^4 x^4+\frac {252}{5} a^5 b^5 x^5+35 a^4 b^6 x^6+\frac {120}{7} a^3 b^7 x^7+\frac {45}{8} a^2 b^8 x^8+\frac {10}{9} a b^9 x^9+\frac {b^{10} x^{10}}{10}+a^{10} \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 122, 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} \[ \int \frac {(a+b x)^{10}}{x} \, dx=a^{10} \log (x)+10 a^9 b x+\frac {45}{2} a^8 b^2 x^2+40 a^7 b^3 x^3+\frac {105}{2} a^6 b^4 x^4+\frac {252}{5} a^5 b^5 x^5+35 a^4 b^6 x^6+\frac {120}{7} a^3 b^7 x^7+\frac {45}{8} a^2 b^8 x^8+\frac {10}{9} a b^9 x^9+\frac {b^{10} x^{10}}{10} \]

[In]

Int[(a + b*x)^10/x,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x} \, dx=10 a^9 b x+\frac {45}{2} a^8 b^2 x^2+40 a^7 b^3 x^3+\frac {105}{2} a^6 b^4 x^4+\frac {252}{5} a^5 b^5 x^5+35 a^4 b^6 x^6+\frac {120}{7} a^3 b^7 x^7+\frac {45}{8} a^2 b^8 x^8+\frac {10}{9} a b^9 x^9+\frac {b^{10} x^{10}}{10}+a^{10} \log (x) \]

[In]

Integrate[(a + b*x)^10/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89

method	result	size
default	\(10 a^{9} b x +\frac {45 a^{8} b^{2} x^{2}}{2}+40 a^{7} b^{3} x^{3}+\frac {105 a^{6} b^{4} x^{4}}{2}+\frac {252 a^{5} b^{5} x^{5}}{5}+35 a^{4} b^{6} x^{6}+\frac {120 a^{3} b^{7} x^{7}}{7}+\frac {45 a^{2} b^{8} x^{8}}{8}+\frac {10 a \,b^{9} x^{9}}{9}+\frac {b^{10} x^{10}}{10}+a^{10} \ln \left (x \right )\)	\(109\)
norman	\(10 a^{9} b x +\frac {45 a^{8} b^{2} x^{2}}{2}+40 a^{7} b^{3} x^{3}+\frac {105 a^{6} b^{4} x^{4}}{2}+\frac {252 a^{5} b^{5} x^{5}}{5}+35 a^{4} b^{6} x^{6}+\frac {120 a^{3} b^{7} x^{7}}{7}+\frac {45 a^{2} b^{8} x^{8}}{8}+\frac {10 a \,b^{9} x^{9}}{9}+\frac {b^{10} x^{10}}{10}+a^{10} \ln \left (x \right )\)	\(109\)
risch	\(10 a^{9} b x +\frac {45 a^{8} b^{2} x^{2}}{2}+40 a^{7} b^{3} x^{3}+\frac {105 a^{6} b^{4} x^{4}}{2}+\frac {252 a^{5} b^{5} x^{5}}{5}+35 a^{4} b^{6} x^{6}+\frac {120 a^{3} b^{7} x^{7}}{7}+\frac {45 a^{2} b^{8} x^{8}}{8}+\frac {10 a \,b^{9} x^{9}}{9}+\frac {b^{10} x^{10}}{10}+a^{10} \ln \left (x \right )\)	\(109\)
parallelrisch	\(10 a^{9} b x +\frac {45 a^{8} b^{2} x^{2}}{2}+40 a^{7} b^{3} x^{3}+\frac {105 a^{6} b^{4} x^{4}}{2}+\frac {252 a^{5} b^{5} x^{5}}{5}+35 a^{4} b^{6} x^{6}+\frac {120 a^{3} b^{7} x^{7}}{7}+\frac {45 a^{2} b^{8} x^{8}}{8}+\frac {10 a \,b^{9} x^{9}}{9}+\frac {b^{10} x^{10}}{10}+a^{10} \ln \left (x \right )\)	\(109\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x} \, dx=\frac {1}{10} \, b^{10} x^{10} + \frac {10}{9} \, a b^{9} x^{9} + \frac {45}{8} \, a^{2} b^{8} x^{8} + \frac {120}{7} \, a^{3} b^{7} x^{7} + 35 \, a^{4} b^{6} x^{6} + \frac {252}{5} \, a^{5} b^{5} x^{5} + \frac {105}{2} \, a^{6} b^{4} x^{4} + 40 \, a^{7} b^{3} x^{3} + \frac {45}{2} \, a^{8} b^{2} x^{2} + 10 \, a^{9} b x + a^{10} \log \left (x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^{10}}{x} \, dx=a^{10} \log {\left (x \right )} + 10 a^{9} b x + \frac {45 a^{8} b^{2} x^{2}}{2} + 40 a^{7} b^{3} x^{3} + \frac {105 a^{6} b^{4} x^{4}}{2} + \frac {252 a^{5} b^{5} x^{5}}{5} + 35 a^{4} b^{6} x^{6} + \frac {120 a^{3} b^{7} x^{7}}{7} + \frac {45 a^{2} b^{8} x^{8}}{8} + \frac {10 a b^{9} x^{9}}{9} + \frac {b^{10} x^{10}}{10} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x} \, dx=\frac {1}{10} \, b^{10} x^{10} + \frac {10}{9} \, a b^{9} x^{9} + \frac {45}{8} \, a^{2} b^{8} x^{8} + \frac {120}{7} \, a^{3} b^{7} x^{7} + 35 \, a^{4} b^{6} x^{6} + \frac {252}{5} \, a^{5} b^{5} x^{5} + \frac {105}{2} \, a^{6} b^{4} x^{4} + 40 \, a^{7} b^{3} x^{3} + \frac {45}{2} \, a^{8} b^{2} x^{2} + 10 \, a^{9} b x + a^{10} \log \left (x\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x} \, dx=\frac {1}{10} \, b^{10} x^{10} + \frac {10}{9} \, a b^{9} x^{9} + \frac {45}{8} \, a^{2} b^{8} x^{8} + \frac {120}{7} \, a^{3} b^{7} x^{7} + 35 \, a^{4} b^{6} x^{6} + \frac {252}{5} \, a^{5} b^{5} x^{5} + \frac {105}{2} \, a^{6} b^{4} x^{4} + 40 \, a^{7} b^{3} x^{3} + \frac {45}{2} \, a^{8} b^{2} x^{2} + 10 \, a^{9} b x + a^{10} \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x} \, dx=a^{10}\,\ln \left (x\right )+\frac {b^{10}\,x^{10}}{10}+\frac {10\,a\,b^9\,x^9}{9}+\frac {45\,a^8\,b^2\,x^2}{2}+40\,a^7\,b^3\,x^3+\frac {105\,a^6\,b^4\,x^4}{2}+\frac {252\,a^5\,b^5\,x^5}{5}+35\,a^4\,b^6\,x^6+\frac {120\,a^3\,b^7\,x^7}{7}+\frac {45\,a^2\,b^8\,x^8}{8}+10\,a^9\,b\,x \]

[In]

int((a + b*x)^10/x,x)

[Out]

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