\(\int \frac {(a+b x)^7}{x^{11}} \, dx\) [117]

Optimal result

Integrand size = 11, antiderivative size = 56 \[ \int \frac {(a+b x)^7}{x^{11}} \, dx=-\frac {(a+b x)^8}{10 a x^{10}}+\frac {b (a+b x)^8}{45 a^2 x^9}-\frac {b^2 (a+b x)^8}{360 a^3 x^8} \]

[Out]

-1/10*(b*x+a)^8/a/x^10+1/45*b*(b*x+a)^8/a^2/x^9-1/360*b^2*(b*x+a)^8/a^3/x^8

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^7}{x^{11}} \, dx=-\frac {b^2 (a+b x)^8}{360 a^3 x^8}+\frac {b (a+b x)^8}{45 a^2 x^9}-\frac {(a+b x)^8}{10 a x^{10}} \]

[In]

Int[(a + b*x)^7/x^11,x]

[Out]

-1/10*(a + b*x)^8/(a*x^10) + (b*(a + b*x)^8)/(45*a^2*x^9) - (b^2*(a + b*x)^8)/(360*a^3*x^8)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^8}{10 a x^{10}}-\frac {b \int \frac {(a+b x)^7}{x^{10}} \, dx}{5 a} \\ & = -\frac {(a+b x)^8}{10 a x^{10}}+\frac {b (a+b x)^8}{45 a^2 x^9}+\frac {b^2 \int \frac {(a+b x)^7}{x^9} \, dx}{45 a^2} \\ & = -\frac {(a+b x)^8}{10 a x^{10}}+\frac {b (a+b x)^8}{45 a^2 x^9}-\frac {b^2 (a+b x)^8}{360 a^3 x^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x)^7}{x^{11}} \, dx=-\frac {a^7}{10 x^{10}}-\frac {7 a^6 b}{9 x^9}-\frac {21 a^5 b^2}{8 x^8}-\frac {5 a^4 b^3}{x^7}-\frac {35 a^3 b^4}{6 x^6}-\frac {21 a^2 b^5}{5 x^5}-\frac {7 a b^6}{4 x^4}-\frac {b^7}{3 x^3} \]

[In]

Integrate[(a + b*x)^7/x^11,x]

[Out]

-1/10*a^7/x^10 - (7*a^6*b)/(9*x^9) - (21*a^5*b^2)/(8*x^8) - (5*a^4*b^3)/x^7 - (35*a^3*b^4)/(6*x^6) - (21*a^2*b
^5)/(5*x^5) - (7*a*b^6)/(4*x^4) - b^7/(3*x^3)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41

method	result	size
norman	\(\frac {-\frac {1}{3} b^{7} x^{7}-\frac {7}{4} a \,b^{6} x^{6}-\frac {21}{5} a^{2} b^{5} x^{5}-\frac {35}{6} a^{3} b^{4} x^{4}-5 a^{4} b^{3} x^{3}-\frac {21}{8} a^{5} b^{2} x^{2}-\frac {7}{9} a^{6} b x -\frac {1}{10} a^{7}}{x^{10}}\)	\(79\)
risch	\(\frac {-\frac {1}{3} b^{7} x^{7}-\frac {7}{4} a \,b^{6} x^{6}-\frac {21}{5} a^{2} b^{5} x^{5}-\frac {35}{6} a^{3} b^{4} x^{4}-5 a^{4} b^{3} x^{3}-\frac {21}{8} a^{5} b^{2} x^{2}-\frac {7}{9} a^{6} b x -\frac {1}{10} a^{7}}{x^{10}}\)	\(79\)
gosper	\(-\frac {120 b^{7} x^{7}+630 a \,b^{6} x^{6}+1512 a^{2} b^{5} x^{5}+2100 a^{3} b^{4} x^{4}+1800 a^{4} b^{3} x^{3}+945 a^{5} b^{2} x^{2}+280 a^{6} b x +36 a^{7}}{360 x^{10}}\)	\(80\)
default	\(-\frac {a^{7}}{10 x^{10}}-\frac {35 a^{3} b^{4}}{6 x^{6}}-\frac {5 a^{4} b^{3}}{x^{7}}-\frac {7 a^{6} b}{9 x^{9}}-\frac {b^{7}}{3 x^{3}}-\frac {7 a \,b^{6}}{4 x^{4}}-\frac {21 a^{2} b^{5}}{5 x^{5}}-\frac {21 a^{5} b^{2}}{8 x^{8}}\)	\(80\)
parallelrisch	\(\frac {-120 b^{7} x^{7}-630 a \,b^{6} x^{6}-1512 a^{2} b^{5} x^{5}-2100 a^{3} b^{4} x^{4}-1800 a^{4} b^{3} x^{3}-945 a^{5} b^{2} x^{2}-280 a^{6} b x -36 a^{7}}{360 x^{10}}\)	\(80\)

[In]

int((b*x+a)^7/x^11,x,method=_RETURNVERBOSE)

[Out]

1/x^10*(-1/3*b^7*x^7-7/4*a*b^6*x^6-21/5*a^2*b^5*x^5-35/6*a^3*b^4*x^4-5*a^4*b^3*x^3-21/8*a^5*b^2*x^2-7/9*a^6*b*
x-1/10*a^7)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x)^7}{x^{11}} \, dx=-\frac {120 \, b^{7} x^{7} + 630 \, a b^{6} x^{6} + 1512 \, a^{2} b^{5} x^{5} + 2100 \, a^{3} b^{4} x^{4} + 1800 \, a^{4} b^{3} x^{3} + 945 \, a^{5} b^{2} x^{2} + 280 \, a^{6} b x + 36 \, a^{7}}{360 \, x^{10}} \]

[In]

integrate((b*x+a)^7/x^11,x, algorithm="fricas")

[Out]

-1/360*(120*b^7*x^7 + 630*a*b^6*x^6 + 1512*a^2*b^5*x^5 + 2100*a^3*b^4*x^4 + 1800*a^4*b^3*x^3 + 945*a^5*b^2*x^2
+ 280*a^6*b*x + 36*a^7)/x^10

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x)^7}{x^{11}} \, dx=\frac {- 36 a^{7} - 280 a^{6} b x - 945 a^{5} b^{2} x^{2} - 1800 a^{4} b^{3} x^{3} - 2100 a^{3} b^{4} x^{4} - 1512 a^{2} b^{5} x^{5} - 630 a b^{6} x^{6} - 120 b^{7} x^{7}}{360 x^{10}} \]

[In]

integrate((b*x+a)**7/x**11,x)

[Out]

(-36*a**7 - 280*a**6*b*x - 945*a**5*b**2*x**2 - 1800*a**4*b**3*x**3 - 2100*a**3*b**4*x**4 - 1512*a**2*b**5*x**
5 - 630*a*b**6*x**6 - 120*b**7*x**7)/(360*x**10)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x)^7}{x^{11}} \, dx=-\frac {120 \, b^{7} x^{7} + 630 \, a b^{6} x^{6} + 1512 \, a^{2} b^{5} x^{5} + 2100 \, a^{3} b^{4} x^{4} + 1800 \, a^{4} b^{3} x^{3} + 945 \, a^{5} b^{2} x^{2} + 280 \, a^{6} b x + 36 \, a^{7}}{360 \, x^{10}} \]

[In]

integrate((b*x+a)^7/x^11,x, algorithm="maxima")

[Out]

-1/360*(120*b^7*x^7 + 630*a*b^6*x^6 + 1512*a^2*b^5*x^5 + 2100*a^3*b^4*x^4 + 1800*a^4*b^3*x^3 + 945*a^5*b^2*x^2
+ 280*a^6*b*x + 36*a^7)/x^10

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x)^7}{x^{11}} \, dx=-\frac {120 \, b^{7} x^{7} + 630 \, a b^{6} x^{6} + 1512 \, a^{2} b^{5} x^{5} + 2100 \, a^{3} b^{4} x^{4} + 1800 \, a^{4} b^{3} x^{3} + 945 \, a^{5} b^{2} x^{2} + 280 \, a^{6} b x + 36 \, a^{7}}{360 \, x^{10}} \]

[In]

integrate((b*x+a)^7/x^11,x, algorithm="giac")

[Out]

-1/360*(120*b^7*x^7 + 630*a*b^6*x^6 + 1512*a^2*b^5*x^5 + 2100*a^3*b^4*x^4 + 1800*a^4*b^3*x^3 + 945*a^5*b^2*x^2
+ 280*a^6*b*x + 36*a^7)/x^10

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x)^7}{x^{11}} \, dx=-\frac {\frac {a^7}{10}+\frac {7\,a^6\,b\,x}{9}+\frac {21\,a^5\,b^2\,x^2}{8}+5\,a^4\,b^3\,x^3+\frac {35\,a^3\,b^4\,x^4}{6}+\frac {21\,a^2\,b^5\,x^5}{5}+\frac {7\,a\,b^6\,x^6}{4}+\frac {b^7\,x^7}{3}}{x^{10}} \]

[In]

int((a + b*x)^7/x^11,x)

[Out]

-(a^7/10 + (b^7*x^7)/3 + (7*a*b^6*x^6)/4 + (21*a^5*b^2*x^2)/8 + 5*a^4*b^3*x^3 + (35*a^3*b^4*x^4)/6 + (21*a^2*b
^5*x^5)/5 + (7*a^6*b*x)/9)/x^10