\(\int \frac {(a+b x)^7}{x^9} \, dx\) [115]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {(a+b x)^7}{x^9} \, dx=-\frac {(a+b x)^8}{8 a x^8} \]

[Out]

-1/8*(b*x+a)^8/a/x^8

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {37} \[ \int \frac {(a+b x)^7}{x^9} \, dx=-\frac {(a+b x)^8}{8 a x^8} \]

[In]

Int[(a + b*x)^7/x^9,x]

[Out]

-1/8*(a + b*x)^8/(a*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]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^8}{8 a x^8} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(17)=34\).

Time = 0.00 (sec) , antiderivative size = 87, normalized size of antiderivative = 5.12 \[ \int \frac {(a+b x)^7}{x^9} \, dx=-\frac {a^7}{8 x^8}-\frac {a^6 b}{x^7}-\frac {7 a^5 b^2}{2 x^6}-\frac {7 a^4 b^3}{x^5}-\frac {35 a^3 b^4}{4 x^4}-\frac {7 a^2 b^5}{x^3}-\frac {7 a b^6}{2 x^2}-\frac {b^7}{x} \]

[In]

Integrate[(a + b*x)^7/x^9,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs. \(2(15)=30\).

Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 4.59

method	result	size
gosper	\(-\frac {8 b^{7} x^{7}+28 a \,b^{6} x^{6}+56 a^{2} b^{5} x^{5}+70 a^{3} b^{4} x^{4}+56 a^{4} b^{3} x^{3}+28 a^{5} b^{2} x^{2}+8 a^{6} b x +a^{7}}{8 x^{8}}\)	\(78\)
norman	\(\frac {-b^{7} x^{7}-\frac {7}{2} a \,b^{6} x^{6}-7 a^{2} b^{5} x^{5}-\frac {35}{4} a^{3} b^{4} x^{4}-7 a^{4} b^{3} x^{3}-\frac {7}{2} a^{5} b^{2} x^{2}-a^{6} b x -\frac {1}{8} a^{7}}{x^{8}}\)	\(79\)
risch	\(\frac {-b^{7} x^{7}-\frac {7}{2} a \,b^{6} x^{6}-7 a^{2} b^{5} x^{5}-\frac {35}{4} a^{3} b^{4} x^{4}-7 a^{4} b^{3} x^{3}-\frac {7}{2} a^{5} b^{2} x^{2}-a^{6} b x -\frac {1}{8} a^{7}}{x^{8}}\)	\(79\)
default	\(-\frac {7 a^{5} b^{2}}{2 x^{6}}-\frac {a^{6} b}{x^{7}}-\frac {7 a^{2} b^{5}}{x^{3}}-\frac {b^{7}}{x}-\frac {7 a \,b^{6}}{2 x^{2}}-\frac {35 a^{3} b^{4}}{4 x^{4}}-\frac {7 a^{4} b^{3}}{x^{5}}-\frac {a^{7}}{8 x^{8}}\)	\(80\)
parallelrisch	\(\frac {-8 b^{7} x^{7}-28 a \,b^{6} x^{6}-56 a^{2} b^{5} x^{5}-70 a^{3} b^{4} x^{4}-56 a^{4} b^{3} x^{3}-28 a^{5} b^{2} x^{2}-8 a^{6} b x -a^{7}}{8 x^{8}}\)	\(80\)

[In]

int((b*x+a)^7/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/8*(8*b^7*x^7+28*a*b^6*x^6+56*a^2*b^5*x^5+70*a^3*b^4*x^4+56*a^4*b^3*x^3+28*a^5*b^2*x^2+8*a^6*b*x+a^7)/x^8

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 4.53 \[ \int \frac {(a+b x)^7}{x^9} \, dx=-\frac {8 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 56 \, a^{2} b^{5} x^{5} + 70 \, a^{3} b^{4} x^{4} + 56 \, a^{4} b^{3} x^{3} + 28 \, a^{5} b^{2} x^{2} + 8 \, a^{6} b x + a^{7}}{8 \, x^{8}} \]

[In]

integrate((b*x+a)^7/x^9,x, algorithm="fricas")

[Out]

-1/8*(8*b^7*x^7 + 28*a*b^6*x^6 + 56*a^2*b^5*x^5 + 70*a^3*b^4*x^4 + 56*a^4*b^3*x^3 + 28*a^5*b^2*x^2 + 8*a^6*b*x
+ a^7)/x^8

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (14) = 28\).

Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.88 \[ \int \frac {(a+b x)^7}{x^9} \, dx=\frac {- a^{7} - 8 a^{6} b x - 28 a^{5} b^{2} x^{2} - 56 a^{4} b^{3} x^{3} - 70 a^{3} b^{4} x^{4} - 56 a^{2} b^{5} x^{5} - 28 a b^{6} x^{6} - 8 b^{7} x^{7}}{8 x^{8}} \]

[In]

integrate((b*x+a)**7/x**9,x)

[Out]

(-a**7 - 8*a**6*b*x - 28*a**5*b**2*x**2 - 56*a**4*b**3*x**3 - 70*a**3*b**4*x**4 - 56*a**2*b**5*x**5 - 28*a*b**
6*x**6 - 8*b**7*x**7)/(8*x**8)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (15) = 30\).

Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 4.53 \[ \int \frac {(a+b x)^7}{x^9} \, dx=-\frac {8 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 56 \, a^{2} b^{5} x^{5} + 70 \, a^{3} b^{4} x^{4} + 56 \, a^{4} b^{3} x^{3} + 28 \, a^{5} b^{2} x^{2} + 8 \, a^{6} b x + a^{7}}{8 \, x^{8}} \]

[In]

integrate((b*x+a)^7/x^9,x, algorithm="maxima")

[Out]

-1/8*(8*b^7*x^7 + 28*a*b^6*x^6 + 56*a^2*b^5*x^5 + 70*a^3*b^4*x^4 + 56*a^4*b^3*x^3 + 28*a^5*b^2*x^2 + 8*a^6*b*x
+ a^7)/x^8

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (15) = 30\).

Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 4.53 \[ \int \frac {(a+b x)^7}{x^9} \, dx=-\frac {8 \, b^{7} x^{7} + 28 \, a b^{6} x^{6} + 56 \, a^{2} b^{5} x^{5} + 70 \, a^{3} b^{4} x^{4} + 56 \, a^{4} b^{3} x^{3} + 28 \, a^{5} b^{2} x^{2} + 8 \, a^{6} b x + a^{7}}{8 \, x^{8}} \]

[In]

integrate((b*x+a)^7/x^9,x, algorithm="giac")

[Out]

-1/8*(8*b^7*x^7 + 28*a*b^6*x^6 + 56*a^2*b^5*x^5 + 70*a^3*b^4*x^4 + 56*a^4*b^3*x^3 + 28*a^5*b^2*x^2 + 8*a^6*b*x
+ a^7)/x^8

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 4.53 \[ \int \frac {(a+b x)^7}{x^9} \, dx=-\frac {\frac {a^7}{8}+a^6\,b\,x+\frac {7\,a^5\,b^2\,x^2}{2}+7\,a^4\,b^3\,x^3+\frac {35\,a^3\,b^4\,x^4}{4}+7\,a^2\,b^5\,x^5+\frac {7\,a\,b^6\,x^6}{2}+b^7\,x^7}{x^8} \]

[In]

int((a + b*x)^7/x^9,x)

[Out]

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