\(\int \frac {(a+b x)^{12}}{x^{10}} \, dx\) [239]

Optimal result

Integrand size = 11, antiderivative size = 141 \[ \int \frac {(a+b x)^{12}}{x^{10}} \, dx=-\frac {a^{12}}{9 x^9}-\frac {3 a^{11} b}{2 x^8}-\frac {66 a^{10} b^2}{7 x^7}-\frac {110 a^9 b^3}{3 x^6}-\frac {99 a^8 b^4}{x^5}-\frac {198 a^7 b^5}{x^4}-\frac {308 a^6 b^6}{x^3}-\frac {396 a^5 b^7}{x^2}-\frac {495 a^4 b^8}{x}+66 a^2 b^{10} x+6 a b^{11} x^2+\frac {b^{12} x^3}{3}+220 a^3 b^9 \log (x) \]

[Out]

-1/9*a^12/x^9-3/2*a^11*b/x^8-66/7*a^10*b^2/x^7-110/3*a^9*b^3/x^6-99*a^8*b^4/x^5-198*a^7*b^5/x^4-308*a^6*b^6/x^
3-396*a^5*b^7/x^2-495*a^4*b^8/x+66*a^2*b^10*x+6*a*b^11*x^2+1/3*b^12*x^3+220*a^3*b^9*ln(x)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 141, 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} \[ \int \frac {(a+b x)^{12}}{x^{10}} \, dx=-\frac {a^{12}}{9 x^9}-\frac {3 a^{11} b}{2 x^8}-\frac {66 a^{10} b^2}{7 x^7}-\frac {110 a^9 b^3}{3 x^6}-\frac {99 a^8 b^4}{x^5}-\frac {198 a^7 b^5}{x^4}-\frac {308 a^6 b^6}{x^3}-\frac {396 a^5 b^7}{x^2}-\frac {495 a^4 b^8}{x}+220 a^3 b^9 \log (x)+66 a^2 b^{10} x+6 a b^{11} x^2+\frac {b^{12} x^3}{3} \]

[In]

Int[(a + b*x)^12/x^10,x]

[Out]

-1/9*a^12/x^9 - (3*a^11*b)/(2*x^8) - (66*a^10*b^2)/(7*x^7) - (110*a^9*b^3)/(3*x^6) - (99*a^8*b^4)/x^5 - (198*a
^7*b^5)/x^4 - (308*a^6*b^6)/x^3 - (396*a^5*b^7)/x^2 - (495*a^4*b^8)/x + 66*a^2*b^10*x + 6*a*b^11*x^2 + (b^12*x
^3)/3 + 220*a^3*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 (66 a^2 b^{10}+\frac {a^{12}}{x^{10}}+\frac {12 a^{11} b}{x^9}+\frac {66 a^{10} b^2}{x^8}+\frac {220 a^9 b^3}{x^7}+\frac {495 a^8 b^4}{x^6}+\frac {792 a^7 b^5}{x^5}+\frac {924 a^6 b^6}{x^4}+\frac {792 a^5 b^7}{x^3}+\frac {495 a^4 b^8}{x^2}+\frac {220 a^3 b^9}{x}+12 a b^{11} x+b^{12} x^2\right ) \, dx \\ & = -\frac {a^{12}}{9 x^9}-\frac {3 a^{11} b}{2 x^8}-\frac {66 a^{10} b^2}{7 x^7}-\frac {110 a^9 b^3}{3 x^6}-\frac {99 a^8 b^4}{x^5}-\frac {198 a^7 b^5}{x^4}-\frac {308 a^6 b^6}{x^3}-\frac {396 a^5 b^7}{x^2}-\frac {495 a^4 b^8}{x}+66 a^2 b^{10} x+6 a b^{11} x^2+\frac {b^{12} x^3}{3}+220 a^3 b^9 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{12}}{x^{10}} \, dx=-\frac {a^{12}}{9 x^9}-\frac {3 a^{11} b}{2 x^8}-\frac {66 a^{10} b^2}{7 x^7}-\frac {110 a^9 b^3}{3 x^6}-\frac {99 a^8 b^4}{x^5}-\frac {198 a^7 b^5}{x^4}-\frac {308 a^6 b^6}{x^3}-\frac {396 a^5 b^7}{x^2}-\frac {495 a^4 b^8}{x}+66 a^2 b^{10} x+6 a b^{11} x^2+\frac {b^{12} x^3}{3}+220 a^3 b^9 \log (x) \]

[In]

Integrate[(a + b*x)^12/x^10,x]

[Out]

-1/9*a^12/x^9 - (3*a^11*b)/(2*x^8) - (66*a^10*b^2)/(7*x^7) - (110*a^9*b^3)/(3*x^6) - (99*a^8*b^4)/x^5 - (198*a
^7*b^5)/x^4 - (308*a^6*b^6)/x^3 - (396*a^5*b^7)/x^2 - (495*a^4*b^8)/x + 66*a^2*b^10*x + 6*a*b^11*x^2 + (b^12*x
^3)/3 + 220*a^3*b^9*Log[x]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {a^{12}}{9 x^{9}}-\frac {3 a^{11} b}{2 x^{8}}-\frac {66 a^{10} b^{2}}{7 x^{7}}-\frac {110 a^{9} b^{3}}{3 x^{6}}-\frac {99 a^{8} b^{4}}{x^{5}}-\frac {198 a^{7} b^{5}}{x^{4}}-\frac {308 a^{6} b^{6}}{x^{3}}-\frac {396 a^{5} b^{7}}{x^{2}}-\frac {495 a^{4} b^{8}}{x}+66 a^{2} b^{10} x +6 a \,b^{11} x^{2}+\frac {b^{12} x^{3}}{3}+220 a^{3} b^{9} \ln \left (x \right )\)	\(132\)
risch	\(\frac {b^{12} x^{3}}{3}+6 a \,b^{11} x^{2}+66 a^{2} b^{10} x +\frac {-495 a^{4} x^{8} b^{8}-396 a^{5} x^{7} b^{7}-308 a^{6} x^{6} b^{6}-198 a^{7} x^{5} b^{5}-99 a^{8} x^{4} b^{4}-\frac {110}{3} a^{9} x^{3} b^{3}-\frac {66}{7} a^{10} x^{2} b^{2}-\frac {3}{2} a^{11} x b -\frac {1}{9} a^{12}}{x^{9}}+220 a^{3} b^{9} \ln \left (x \right )\)	\(132\)
norman	\(\frac {-\frac {1}{9} a^{12}+\frac {1}{3} b^{12} x^{12}+6 a \,x^{11} b^{11}+66 a^{2} x^{10} b^{10}-495 a^{4} x^{8} b^{8}-396 a^{5} x^{7} b^{7}-308 a^{6} x^{6} b^{6}-198 a^{7} x^{5} b^{5}-99 a^{8} x^{4} b^{4}-\frac {110}{3} a^{9} x^{3} b^{3}-\frac {66}{7} a^{10} x^{2} b^{2}-\frac {3}{2} a^{11} x b}{x^{9}}+220 a^{3} b^{9} \ln \left (x \right )\)	\(134\)
parallelrisch	\(\frac {42 b^{12} x^{12}+756 a \,x^{11} b^{11}+27720 \ln \left (x \right ) x^{9} a^{3} b^{9}+8316 a^{2} x^{10} b^{10}-62370 a^{4} x^{8} b^{8}-49896 a^{5} x^{7} b^{7}-38808 a^{6} x^{6} b^{6}-24948 a^{7} x^{5} b^{5}-12474 a^{8} x^{4} b^{4}-4620 a^{9} x^{3} b^{3}-1188 a^{10} x^{2} b^{2}-189 a^{11} x b -14 a^{12}}{126 x^{9}}\)	\(137\)

[In]

int((b*x+a)^12/x^10,x,method=_RETURNVERBOSE)

[Out]

-1/9*a^12/x^9-3/2*a^11*b/x^8-66/7*a^10*b^2/x^7-110/3*a^9*b^3/x^6-99*a^8*b^4/x^5-198*a^7*b^5/x^4-308*a^6*b^6/x^
3-396*a^5*b^7/x^2-495*a^4*b^8/x+66*a^2*b^10*x+6*a*b^11*x^2+1/3*b^12*x^3+220*a^3*b^9*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{12}}{x^{10}} \, dx=\frac {42 \, b^{12} x^{12} + 756 \, a b^{11} x^{11} + 8316 \, a^{2} b^{10} x^{10} + 27720 \, a^{3} b^{9} x^{9} \log \left (x\right ) - 62370 \, a^{4} b^{8} x^{8} - 49896 \, a^{5} b^{7} x^{7} - 38808 \, a^{6} b^{6} x^{6} - 24948 \, a^{7} b^{5} x^{5} - 12474 \, a^{8} b^{4} x^{4} - 4620 \, a^{9} b^{3} x^{3} - 1188 \, a^{10} b^{2} x^{2} - 189 \, a^{11} b x - 14 \, a^{12}}{126 \, x^{9}} \]

[In]

integrate((b*x+a)^12/x^10,x, algorithm="fricas")

[Out]

1/126*(42*b^12*x^12 + 756*a*b^11*x^11 + 8316*a^2*b^10*x^10 + 27720*a^3*b^9*x^9*log(x) - 62370*a^4*b^8*x^8 - 49
896*a^5*b^7*x^7 - 38808*a^6*b^6*x^6 - 24948*a^7*b^5*x^5 - 12474*a^8*b^4*x^4 - 4620*a^9*b^3*x^3 - 1188*a^10*b^2
*x^2 - 189*a^11*b*x - 14*a^12)/x^9

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^{12}}{x^{10}} \, dx=220 a^{3} b^{9} \log {\left (x \right )} + 66 a^{2} b^{10} x + 6 a b^{11} x^{2} + \frac {b^{12} x^{3}}{3} + \frac {- 14 a^{12} - 189 a^{11} b x - 1188 a^{10} b^{2} x^{2} - 4620 a^{9} b^{3} x^{3} - 12474 a^{8} b^{4} x^{4} - 24948 a^{7} b^{5} x^{5} - 38808 a^{6} b^{6} x^{6} - 49896 a^{5} b^{7} x^{7} - 62370 a^{4} b^{8} x^{8}}{126 x^{9}} \]

[In]

integrate((b*x+a)**12/x**10,x)

[Out]

220*a**3*b**9*log(x) + 66*a**2*b**10*x + 6*a*b**11*x**2 + b**12*x**3/3 + (-14*a**12 - 189*a**11*b*x - 1188*a**
10*b**2*x**2 - 4620*a**9*b**3*x**3 - 12474*a**8*b**4*x**4 - 24948*a**7*b**5*x**5 - 38808*a**6*b**6*x**6 - 4989
6*a**5*b**7*x**7 - 62370*a**4*b**8*x**8)/(126*x**9)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^{12}}{x^{10}} \, dx=\frac {1}{3} \, b^{12} x^{3} + 6 \, a b^{11} x^{2} + 66 \, a^{2} b^{10} x + 220 \, a^{3} b^{9} \log \left (x\right ) - \frac {62370 \, a^{4} b^{8} x^{8} + 49896 \, a^{5} b^{7} x^{7} + 38808 \, a^{6} b^{6} x^{6} + 24948 \, a^{7} b^{5} x^{5} + 12474 \, a^{8} b^{4} x^{4} + 4620 \, a^{9} b^{3} x^{3} + 1188 \, a^{10} b^{2} x^{2} + 189 \, a^{11} b x + 14 \, a^{12}}{126 \, x^{9}} \]

[In]

integrate((b*x+a)^12/x^10,x, algorithm="maxima")

[Out]

1/3*b^12*x^3 + 6*a*b^11*x^2 + 66*a^2*b^10*x + 220*a^3*b^9*log(x) - 1/126*(62370*a^4*b^8*x^8 + 49896*a^5*b^7*x^
7 + 38808*a^6*b^6*x^6 + 24948*a^7*b^5*x^5 + 12474*a^8*b^4*x^4 + 4620*a^9*b^3*x^3 + 1188*a^10*b^2*x^2 + 189*a^1
1*b*x + 14*a^12)/x^9

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^{12}}{x^{10}} \, dx=\frac {1}{3} \, b^{12} x^{3} + 6 \, a b^{11} x^{2} + 66 \, a^{2} b^{10} x + 220 \, a^{3} b^{9} \log \left ({\left | x \right |}\right ) - \frac {62370 \, a^{4} b^{8} x^{8} + 49896 \, a^{5} b^{7} x^{7} + 38808 \, a^{6} b^{6} x^{6} + 24948 \, a^{7} b^{5} x^{5} + 12474 \, a^{8} b^{4} x^{4} + 4620 \, a^{9} b^{3} x^{3} + 1188 \, a^{10} b^{2} x^{2} + 189 \, a^{11} b x + 14 \, a^{12}}{126 \, x^{9}} \]

[In]

integrate((b*x+a)^12/x^10,x, algorithm="giac")

[Out]

1/3*b^12*x^3 + 6*a*b^11*x^2 + 66*a^2*b^10*x + 220*a^3*b^9*log(abs(x)) - 1/126*(62370*a^4*b^8*x^8 + 49896*a^5*b
^7*x^7 + 38808*a^6*b^6*x^6 + 24948*a^7*b^5*x^5 + 12474*a^8*b^4*x^4 + 4620*a^9*b^3*x^3 + 1188*a^10*b^2*x^2 + 18
9*a^11*b*x + 14*a^12)/x^9

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^{12}}{x^{10}} \, dx=\frac {b^{12}\,x^3}{3}-\frac {\frac {a^{12}}{9}+\frac {3\,a^{11}\,b\,x}{2}+\frac {66\,a^{10}\,b^2\,x^2}{7}+\frac {110\,a^9\,b^3\,x^3}{3}+99\,a^8\,b^4\,x^4+198\,a^7\,b^5\,x^5+308\,a^6\,b^6\,x^6+396\,a^5\,b^7\,x^7+495\,a^4\,b^8\,x^8}{x^9}+66\,a^2\,b^{10}\,x+6\,a\,b^{11}\,x^2+220\,a^3\,b^9\,\ln \left (x\right ) \]

[In]

int((a + b*x)^12/x^10,x)

[Out]

(b^12*x^3)/3 - (a^12/9 + (66*a^10*b^2*x^2)/7 + (110*a^9*b^3*x^3)/3 + 99*a^8*b^4*x^4 + 198*a^7*b^5*x^5 + 308*a^
6*b^6*x^6 + 396*a^5*b^7*x^7 + 495*a^4*b^8*x^8 + (3*a^11*b*x)/2)/x^9 + 66*a^2*b^10*x + 6*a*b^11*x^2 + 220*a^3*b
^9*log(x)