\(\int \frac {(a+b x)^{10}}{x^3} \, dx\) [137]

Optimal result

Integrand size = 11, antiderivative size = 119 \[ \int \frac {(a+b x)^{10}}{x^3} \, dx=-\frac {a^{10}}{2 x^2}-\frac {10 a^9 b}{x}+120 a^7 b^3 x+105 a^6 b^4 x^2+84 a^5 b^5 x^3+\frac {105}{2} a^4 b^6 x^4+24 a^3 b^7 x^5+\frac {15}{2} a^2 b^8 x^6+\frac {10}{7} a b^9 x^7+\frac {b^{10} x^8}{8}+45 a^8 b^2 \log (x) \]

[Out]

-1/2*a^10/x^2-10*a^9*b/x+120*a^7*b^3*x+105*a^6*b^4*x^2+84*a^5*b^5*x^3+105/2*a^4*b^6*x^4+24*a^3*b^7*x^5+15/2*a^
2*b^8*x^6+10/7*a*b^9*x^7+1/8*b^10*x^8+45*a^8*b^2*ln(x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 119, 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} \[ \int \frac {(a+b x)^{10}}{x^3} \, dx=-\frac {a^{10}}{2 x^2}-\frac {10 a^9 b}{x}+45 a^8 b^2 \log (x)+120 a^7 b^3 x+105 a^6 b^4 x^2+84 a^5 b^5 x^3+\frac {105}{2} a^4 b^6 x^4+24 a^3 b^7 x^5+\frac {15}{2} a^2 b^8 x^6+\frac {10}{7} a b^9 x^7+\frac {b^{10} x^8}{8} \]

[In]

Int[(a + b*x)^10/x^3,x]

[Out]

-1/2*a^10/x^2 - (10*a^9*b)/x + 120*a^7*b^3*x + 105*a^6*b^4*x^2 + 84*a^5*b^5*x^3 + (105*a^4*b^6*x^4)/2 + 24*a^3
*b^7*x^5 + (15*a^2*b^8*x^6)/2 + (10*a*b^9*x^7)/7 + (b^10*x^8)/8 + 45*a^8*b^2*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 (120 a^7 b^3+\frac {a^{10}}{x^3}+\frac {10 a^9 b}{x^2}+\frac {45 a^8 b^2}{x}+210 a^6 b^4 x+252 a^5 b^5 x^2+210 a^4 b^6 x^3+120 a^3 b^7 x^4+45 a^2 b^8 x^5+10 a b^9 x^6+b^{10} x^7\right ) \, dx \\ & = -\frac {a^{10}}{2 x^2}-\frac {10 a^9 b}{x}+120 a^7 b^3 x+105 a^6 b^4 x^2+84 a^5 b^5 x^3+\frac {105}{2} a^4 b^6 x^4+24 a^3 b^7 x^5+\frac {15}{2} a^2 b^8 x^6+\frac {10}{7} a b^9 x^7+\frac {b^{10} x^8}{8}+45 a^8 b^2 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^3} \, dx=-\frac {a^{10}}{2 x^2}-\frac {10 a^9 b}{x}+120 a^7 b^3 x+105 a^6 b^4 x^2+84 a^5 b^5 x^3+\frac {105}{2} a^4 b^6 x^4+24 a^3 b^7 x^5+\frac {15}{2} a^2 b^8 x^6+\frac {10}{7} a b^9 x^7+\frac {b^{10} x^8}{8}+45 a^8 b^2 \log (x) \]

[In]

Integrate[(a + b*x)^10/x^3,x]

[Out]

-1/2*a^10/x^2 - (10*a^9*b)/x + 120*a^7*b^3*x + 105*a^6*b^4*x^2 + 84*a^5*b^5*x^3 + (105*a^4*b^6*x^4)/2 + 24*a^3
*b^7*x^5 + (15*a^2*b^8*x^6)/2 + (10*a*b^9*x^7)/7 + (b^10*x^8)/8 + 45*a^8*b^2*Log[x]

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {a^{10}}{2 x^{2}}-\frac {10 a^{9} b}{x}+120 a^{7} b^{3} x +105 a^{6} b^{4} x^{2}+84 a^{5} b^{5} x^{3}+\frac {105 a^{4} b^{6} x^{4}}{2}+24 a^{3} b^{7} x^{5}+\frac {15 a^{2} b^{8} x^{6}}{2}+\frac {10 a \,b^{9} x^{7}}{7}+\frac {b^{10} x^{8}}{8}+45 a^{8} b^{2} \ln \left (x \right )\)	\(110\)
risch	\(\frac {b^{10} x^{8}}{8}+\frac {10 a \,b^{9} x^{7}}{7}+\frac {15 a^{2} b^{8} x^{6}}{2}+24 a^{3} b^{7} x^{5}+\frac {105 a^{4} b^{6} x^{4}}{2}+84 a^{5} b^{5} x^{3}+105 a^{6} b^{4} x^{2}+120 a^{7} b^{3} x +\frac {-10 a^{9} b x -\frac {1}{2} a^{10}}{x^{2}}+45 a^{8} b^{2} \ln \left (x \right )\)	\(110\)
norman	\(\frac {-\frac {1}{2} a^{10}+\frac {1}{8} b^{10} x^{10}+\frac {10}{7} a \,b^{9} x^{9}+\frac {15}{2} a^{2} b^{8} x^{8}+24 a^{3} b^{7} x^{7}+\frac {105}{2} a^{4} b^{6} x^{6}+84 a^{5} b^{5} x^{5}+105 a^{6} b^{4} x^{4}+120 a^{7} b^{3} x^{3}-10 a^{9} b x}{x^{2}}+45 a^{8} b^{2} \ln \left (x \right )\)	\(112\)
parallelrisch	\(\frac {7 b^{10} x^{10}+80 a \,b^{9} x^{9}+420 a^{2} b^{8} x^{8}+1344 a^{3} b^{7} x^{7}+2940 a^{4} b^{6} x^{6}+4704 a^{5} b^{5} x^{5}+5880 a^{6} b^{4} x^{4}+2520 a^{8} b^{2} \ln \left (x \right ) x^{2}+6720 a^{7} b^{3} x^{3}-560 a^{9} b x -28 a^{10}}{56 x^{2}}\)	\(115\)

[In]

int((b*x+a)^10/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*a^10/x^2-10*a^9*b/x+120*a^7*b^3*x+105*a^6*b^4*x^2+84*a^5*b^5*x^3+105/2*a^4*b^6*x^4+24*a^3*b^7*x^5+15/2*a^
2*b^8*x^6+10/7*a*b^9*x^7+1/8*b^10*x^8+45*a^8*b^2*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^3} \, dx=\frac {7 \, b^{10} x^{10} + 80 \, a b^{9} x^{9} + 420 \, a^{2} b^{8} x^{8} + 1344 \, a^{3} b^{7} x^{7} + 2940 \, a^{4} b^{6} x^{6} + 4704 \, a^{5} b^{5} x^{5} + 5880 \, a^{6} b^{4} x^{4} + 6720 \, a^{7} b^{3} x^{3} + 2520 \, a^{8} b^{2} x^{2} \log \left (x\right ) - 560 \, a^{9} b x - 28 \, a^{10}}{56 \, x^{2}} \]

[In]

integrate((b*x+a)^10/x^3,x, algorithm="fricas")

[Out]

1/56*(7*b^10*x^10 + 80*a*b^9*x^9 + 420*a^2*b^8*x^8 + 1344*a^3*b^7*x^7 + 2940*a^4*b^6*x^6 + 4704*a^5*b^5*x^5 +
5880*a^6*b^4*x^4 + 6720*a^7*b^3*x^3 + 2520*a^8*b^2*x^2*log(x) - 560*a^9*b*x - 28*a^10)/x^2

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^{10}}{x^3} \, dx=45 a^{8} b^{2} \log {\left (x \right )} + 120 a^{7} b^{3} x + 105 a^{6} b^{4} x^{2} + 84 a^{5} b^{5} x^{3} + \frac {105 a^{4} b^{6} x^{4}}{2} + 24 a^{3} b^{7} x^{5} + \frac {15 a^{2} b^{8} x^{6}}{2} + \frac {10 a b^{9} x^{7}}{7} + \frac {b^{10} x^{8}}{8} + \frac {- a^{10} - 20 a^{9} b x}{2 x^{2}} \]

[In]

integrate((b*x+a)**10/x**3,x)

[Out]

45*a**8*b**2*log(x) + 120*a**7*b**3*x + 105*a**6*b**4*x**2 + 84*a**5*b**5*x**3 + 105*a**4*b**6*x**4/2 + 24*a**
3*b**7*x**5 + 15*a**2*b**8*x**6/2 + 10*a*b**9*x**7/7 + b**10*x**8/8 + (-a**10 - 20*a**9*b*x)/(2*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^{10}}{x^3} \, dx=\frac {1}{8} \, b^{10} x^{8} + \frac {10}{7} \, a b^{9} x^{7} + \frac {15}{2} \, a^{2} b^{8} x^{6} + 24 \, a^{3} b^{7} x^{5} + \frac {105}{2} \, a^{4} b^{6} x^{4} + 84 \, a^{5} b^{5} x^{3} + 105 \, a^{6} b^{4} x^{2} + 120 \, a^{7} b^{3} x + 45 \, a^{8} b^{2} \log \left (x\right ) - \frac {20 \, a^{9} b x + a^{10}}{2 \, x^{2}} \]

[In]

integrate((b*x+a)^10/x^3,x, algorithm="maxima")

[Out]

1/8*b^10*x^8 + 10/7*a*b^9*x^7 + 15/2*a^2*b^8*x^6 + 24*a^3*b^7*x^5 + 105/2*a^4*b^6*x^4 + 84*a^5*b^5*x^3 + 105*a
^6*b^4*x^2 + 120*a^7*b^3*x + 45*a^8*b^2*log(x) - 1/2*(20*a^9*b*x + a^10)/x^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{10}}{x^3} \, dx=\frac {1}{8} \, b^{10} x^{8} + \frac {10}{7} \, a b^{9} x^{7} + \frac {15}{2} \, a^{2} b^{8} x^{6} + 24 \, a^{3} b^{7} x^{5} + \frac {105}{2} \, a^{4} b^{6} x^{4} + 84 \, a^{5} b^{5} x^{3} + 105 \, a^{6} b^{4} x^{2} + 120 \, a^{7} b^{3} x + 45 \, a^{8} b^{2} \log \left ({\left | x \right |}\right ) - \frac {20 \, a^{9} b x + a^{10}}{2 \, x^{2}} \]

[In]

integrate((b*x+a)^10/x^3,x, algorithm="giac")

[Out]

1/8*b^10*x^8 + 10/7*a*b^9*x^7 + 15/2*a^2*b^8*x^6 + 24*a^3*b^7*x^5 + 105/2*a^4*b^6*x^4 + 84*a^5*b^5*x^3 + 105*a
^6*b^4*x^2 + 120*a^7*b^3*x + 45*a^8*b^2*log(abs(x)) - 1/2*(20*a^9*b*x + a^10)/x^2

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{10}}{x^3} \, dx=\frac {b^{10}\,x^8}{8}-\frac {\frac {a^{10}}{2}+10\,b\,x\,a^9}{x^2}+120\,a^7\,b^3\,x+\frac {10\,a\,b^9\,x^7}{7}+105\,a^6\,b^4\,x^2+84\,a^5\,b^5\,x^3+\frac {105\,a^4\,b^6\,x^4}{2}+24\,a^3\,b^7\,x^5+\frac {15\,a^2\,b^8\,x^6}{2}+45\,a^8\,b^2\,\ln \left (x\right ) \]

[In]

int((a + b*x)^10/x^3,x)

[Out]

(b^10*x^8)/8 - (a^10/2 + 10*a^9*b*x)/x^2 + 120*a^7*b^3*x + (10*a*b^9*x^7)/7 + 105*a^6*b^4*x^2 + 84*a^5*b^5*x^3
+ (105*a^4*b^6*x^4)/2 + 24*a^3*b^7*x^5 + (15*a^2*b^8*x^6)/2 + 45*a^8*b^2*log(x)