\(\int \frac {(a+b x)^4}{x^{10}} \, dx\) [247]

Optimal result

Integrand size = 11, antiderivative size = 56 \[ \int \frac {(a+b x)^4}{x^{10}} \, dx=-\frac {a^4}{9 x^9}-\frac {a^3 b}{2 x^8}-\frac {6 a^2 b^2}{7 x^7}-\frac {2 a b^3}{3 x^6}-\frac {b^4}{5 x^5} \]

[Out]

-1/9*a^4/x^9-1/2*a^3*b/x^8-6/7*a^2*b^2/x^7-2/3*a*b^3/x^6-1/5*b^4/x^5

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, 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} \[ \int \frac {(a+b x)^4}{x^{10}} \, dx=-\frac {a^4}{9 x^9}-\frac {a^3 b}{2 x^8}-\frac {6 a^2 b^2}{7 x^7}-\frac {2 a b^3}{3 x^6}-\frac {b^4}{5 x^5} \]

[In]

Int[(a + b*x)^4/x^10,x]

[Out]

-1/9*a^4/x^9 - (a^3*b)/(2*x^8) - (6*a^2*b^2)/(7*x^7) - (2*a*b^3)/(3*x^6) - b^4/(5*x^5)

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 (\frac {a^4}{x^{10}}+\frac {4 a^3 b}{x^9}+\frac {6 a^2 b^2}{x^8}+\frac {4 a b^3}{x^7}+\frac {b^4}{x^6}\right ) \, dx \\ & = -\frac {a^4}{9 x^9}-\frac {a^3 b}{2 x^8}-\frac {6 a^2 b^2}{7 x^7}-\frac {2 a b^3}{3 x^6}-\frac {b^4}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^4}{x^{10}} \, dx=-\frac {a^4}{9 x^9}-\frac {a^3 b}{2 x^8}-\frac {6 a^2 b^2}{7 x^7}-\frac {2 a b^3}{3 x^6}-\frac {b^4}{5 x^5} \]

[In]

Integrate[(a + b*x)^4/x^10,x]

[Out]

-1/9*a^4/x^9 - (a^3*b)/(2*x^8) - (6*a^2*b^2)/(7*x^7) - (2*a*b^3)/(3*x^6) - b^4/(5*x^5)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82

method	result	size
norman	\(\frac {-\frac {1}{5} b^{4} x^{4}-\frac {2}{3} a \,b^{3} x^{3}-\frac {6}{7} a^{2} b^{2} x^{2}-\frac {1}{2} a^{3} b x -\frac {1}{9} a^{4}}{x^{9}}\)	\(46\)
risch	\(\frac {-\frac {1}{5} b^{4} x^{4}-\frac {2}{3} a \,b^{3} x^{3}-\frac {6}{7} a^{2} b^{2} x^{2}-\frac {1}{2} a^{3} b x -\frac {1}{9} a^{4}}{x^{9}}\)	\(46\)
gosper	\(-\frac {126 b^{4} x^{4}+420 a \,b^{3} x^{3}+540 a^{2} b^{2} x^{2}+315 a^{3} b x +70 a^{4}}{630 x^{9}}\)	\(47\)
default	\(-\frac {a^{4}}{9 x^{9}}-\frac {a^{3} b}{2 x^{8}}-\frac {6 a^{2} b^{2}}{7 x^{7}}-\frac {2 a \,b^{3}}{3 x^{6}}-\frac {b^{4}}{5 x^{5}}\)	\(47\)
parallelrisch	\(\frac {-126 b^{4} x^{4}-420 a \,b^{3} x^{3}-540 a^{2} b^{2} x^{2}-315 a^{3} b x -70 a^{4}}{630 x^{9}}\)	\(47\)

[In]

int((b*x+a)^4/x^10,x,method=_RETURNVERBOSE)

[Out]

1/x^9*(-1/5*b^4*x^4-2/3*a*b^3*x^3-6/7*a^2*b^2*x^2-1/2*a^3*b*x-1/9*a^4)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^4}{x^{10}} \, dx=-\frac {126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \]

[In]

integrate((b*x+a)^4/x^10,x, algorithm="fricas")

[Out]

-1/630*(126*b^4*x^4 + 420*a*b^3*x^3 + 540*a^2*b^2*x^2 + 315*a^3*b*x + 70*a^4)/x^9

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^4}{x^{10}} \, dx=\frac {- 70 a^{4} - 315 a^{3} b x - 540 a^{2} b^{2} x^{2} - 420 a b^{3} x^{3} - 126 b^{4} x^{4}}{630 x^{9}} \]

[In]

integrate((b*x+a)**4/x**10,x)

[Out]

(-70*a**4 - 315*a**3*b*x - 540*a**2*b**2*x**2 - 420*a*b**3*x**3 - 126*b**4*x**4)/(630*x**9)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^4}{x^{10}} \, dx=-\frac {126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \]

[In]

integrate((b*x+a)^4/x^10,x, algorithm="maxima")

[Out]

-1/630*(126*b^4*x^4 + 420*a*b^3*x^3 + 540*a^2*b^2*x^2 + 315*a^3*b*x + 70*a^4)/x^9

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^4}{x^{10}} \, dx=-\frac {126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \]

[In]

integrate((b*x+a)^4/x^10,x, algorithm="giac")

[Out]

-1/630*(126*b^4*x^4 + 420*a*b^3*x^3 + 540*a^2*b^2*x^2 + 315*a^3*b*x + 70*a^4)/x^9

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^4}{x^{10}} \, dx=-\frac {\frac {a^4}{9}+\frac {a^3\,b\,x}{2}+\frac {6\,a^2\,b^2\,x^2}{7}+\frac {2\,a\,b^3\,x^3}{3}+\frac {b^4\,x^4}{5}}{x^9} \]

[In]

int((a + b*x)^4/x^10,x)

[Out]

-(a^4/9 + (b^4*x^4)/5 + (2*a*b^3*x^3)/3 + (6*a^2*b^2*x^2)/7 + (a^3*b*x)/2)/x^9