\(\int \frac {(a+b x)^{10}}{x^2} \, dx\) [136]

Optimal result

Integrand size = 11, antiderivative size = 115 \[ \int \frac {(a+b x)^{10}}{x^2} \, dx=-\frac {a^{10}}{x}+45 a^8 b^2 x+60 a^7 b^3 x^2+70 a^6 b^4 x^3+63 a^5 b^5 x^4+42 a^4 b^6 x^5+20 a^3 b^7 x^6+\frac {45}{7} a^2 b^8 x^7+\frac {5}{4} a b^9 x^8+\frac {b^{10} x^9}{9}+10 a^9 b \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 115, 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} \[ \int \frac {(a+b x)^{10}}{x^2} \, dx=-\frac {a^{10}}{x}+10 a^9 b \log (x)+45 a^8 b^2 x+60 a^7 b^3 x^2+70 a^6 b^4 x^3+63 a^5 b^5 x^4+42 a^4 b^6 x^5+20 a^3 b^7 x^6+\frac {45}{7} a^2 b^8 x^7+\frac {5}{4} a b^9 x^8+\frac {b^{10} x^9}{9} \]

[In]

Int[(a + b*x)^10/x^2,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^2} \, dx=-\frac {a^{10}}{x}+45 a^8 b^2 x+60 a^7 b^3 x^2+70 a^6 b^4 x^3+63 a^5 b^5 x^4+42 a^4 b^6 x^5+20 a^3 b^7 x^6+\frac {45}{7} a^2 b^8 x^7+\frac {5}{4} a b^9 x^8+\frac {b^{10} x^9}{9}+10 a^9 b \log (x) \]

[In]

Integrate[(a + b*x)^10/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {a^{10}}{x}+45 a^{8} b^{2} x +60 a^{7} b^{3} x^{2}+70 a^{6} b^{4} x^{3}+63 a^{5} b^{5} x^{4}+42 a^{4} b^{6} x^{5}+20 a^{3} b^{7} x^{6}+\frac {45 a^{2} b^{8} x^{7}}{7}+\frac {5 a \,b^{9} x^{8}}{4}+\frac {b^{10} x^{9}}{9}+10 a^{9} b \ln \left (x \right )\)	\(110\)
risch	\(-\frac {a^{10}}{x}+45 a^{8} b^{2} x +60 a^{7} b^{3} x^{2}+70 a^{6} b^{4} x^{3}+63 a^{5} b^{5} x^{4}+42 a^{4} b^{6} x^{5}+20 a^{3} b^{7} x^{6}+\frac {45 a^{2} b^{8} x^{7}}{7}+\frac {5 a \,b^{9} x^{8}}{4}+\frac {b^{10} x^{9}}{9}+10 a^{9} b \ln \left (x \right )\)	\(110\)
norman	\(\frac {-a^{10}+\frac {1}{9} b^{10} x^{10}+\frac {5}{4} a \,b^{9} x^{9}+\frac {45}{7} a^{2} b^{8} x^{8}+20 a^{3} b^{7} x^{7}+42 a^{4} b^{6} x^{6}+63 a^{5} b^{5} x^{5}+70 a^{6} b^{4} x^{4}+60 a^{7} b^{3} x^{3}+45 a^{8} b^{2} x^{2}}{x}+10 a^{9} b \ln \left (x \right )\)	\(114\)
parallelrisch	\(\frac {28 b^{10} x^{10}+315 a \,b^{9} x^{9}+1620 a^{2} b^{8} x^{8}+5040 a^{3} b^{7} x^{7}+10584 a^{4} b^{6} x^{6}+15876 a^{5} b^{5} x^{5}+17640 a^{6} b^{4} x^{4}+15120 a^{7} b^{3} x^{3}+2520 a^{9} b \ln \left (x \right ) x +11340 a^{8} b^{2} x^{2}-252 a^{10}}{252 x}\)	\(115\)

[In]

int((b*x+a)^10/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{10}}{x^2} \, dx=\frac {28 \, b^{10} x^{10} + 315 \, a b^{9} x^{9} + 1620 \, a^{2} b^{8} x^{8} + 5040 \, a^{3} b^{7} x^{7} + 10584 \, a^{4} b^{6} x^{6} + 15876 \, a^{5} b^{5} x^{5} + 17640 \, a^{6} b^{4} x^{4} + 15120 \, a^{7} b^{3} x^{3} + 11340 \, a^{8} b^{2} x^{2} + 2520 \, a^{9} b x \log \left (x\right ) - 252 \, a^{10}}{252 \, x} \]

[In]

integrate((b*x+a)^10/x^2,x, algorithm="fricas")

[Out]

1/252*(28*b^10*x^10 + 315*a*b^9*x^9 + 1620*a^2*b^8*x^8 + 5040*a^3*b^7*x^7 + 10584*a^4*b^6*x^6 + 15876*a^5*b^5*
x^5 + 17640*a^6*b^4*x^4 + 15120*a^7*b^3*x^3 + 11340*a^8*b^2*x^2 + 2520*a^9*b*x*log(x) - 252*a^10)/x

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^{10}}{x^2} \, dx=- \frac {a^{10}}{x} + 10 a^{9} b \log {\left (x \right )} + 45 a^{8} b^{2} x + 60 a^{7} b^{3} x^{2} + 70 a^{6} b^{4} x^{3} + 63 a^{5} b^{5} x^{4} + 42 a^{4} b^{6} x^{5} + 20 a^{3} b^{7} x^{6} + \frac {45 a^{2} b^{8} x^{7}}{7} + \frac {5 a b^{9} x^{8}}{4} + \frac {b^{10} x^{9}}{9} \]

[In]

integrate((b*x+a)**10/x**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{10}}{x^2} \, dx=\frac {1}{9} \, b^{10} x^{9} + \frac {5}{4} \, a b^{9} x^{8} + \frac {45}{7} \, a^{2} b^{8} x^{7} + 20 \, a^{3} b^{7} x^{6} + 42 \, a^{4} b^{6} x^{5} + 63 \, a^{5} b^{5} x^{4} + 70 \, a^{6} b^{4} x^{3} + 60 \, a^{7} b^{3} x^{2} + 45 \, a^{8} b^{2} x + 10 \, a^{9} b \log \left (x\right ) - \frac {a^{10}}{x} \]

[In]

integrate((b*x+a)^10/x^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^2} \, dx=\frac {1}{9} \, b^{10} x^{9} + \frac {5}{4} \, a b^{9} x^{8} + \frac {45}{7} \, a^{2} b^{8} x^{7} + 20 \, a^{3} b^{7} x^{6} + 42 \, a^{4} b^{6} x^{5} + 63 \, a^{5} b^{5} x^{4} + 70 \, a^{6} b^{4} x^{3} + 60 \, a^{7} b^{3} x^{2} + 45 \, a^{8} b^{2} x + 10 \, a^{9} b \log \left ({\left | x \right |}\right ) - \frac {a^{10}}{x} \]

[In]

integrate((b*x+a)^10/x^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{10}}{x^2} \, dx=\frac {b^{10}\,x^9}{9}-\frac {a^{10}}{x}+45\,a^8\,b^2\,x+\frac {5\,a\,b^9\,x^8}{4}+10\,a^9\,b\,\ln \left (x\right )+60\,a^7\,b^3\,x^2+70\,a^6\,b^4\,x^3+63\,a^5\,b^5\,x^4+42\,a^4\,b^6\,x^5+20\,a^3\,b^7\,x^6+\frac {45\,a^2\,b^8\,x^7}{7} \]

[In]

int((a + b*x)^10/x^2,x)

[Out]

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