\(\int \frac {(a+b x)^5}{x^7} \, dx\) [90]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {(a+b x)^5}{x^7} \, dx=-\frac {(a+b x)^6}{6 a x^6} \]

[Out]

-1/6*(b*x+a)^6/a/x^6

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} \[ \int \frac {(a+b x)^5}{x^7} \, dx=-\frac {(a+b x)^6}{6 a x^6} \]

[In]

Int[(a + b*x)^5/x^7,x]

[Out]

-1/6*(a + b*x)^6/(a*x^6)

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)^6}{6 a x^6} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(17)=34\).

Time = 0.00 (sec) , antiderivative size = 65, normalized size of antiderivative = 3.82 \[ \int \frac {(a+b x)^5}{x^7} \, dx=-\frac {a^5}{6 x^6}-\frac {a^4 b}{x^5}-\frac {5 a^3 b^2}{2 x^4}-\frac {10 a^2 b^3}{3 x^3}-\frac {5 a b^4}{2 x^2}-\frac {b^5}{x} \]

[In]

Integrate[(a + b*x)^5/x^7,x]

[Out]

-1/6*a^5/x^6 - (a^4*b)/x^5 - (5*a^3*b^2)/(2*x^4) - (10*a^2*b^3)/(3*x^3) - (5*a*b^4)/(2*x^2) - b^5/x

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(15)=30\).

Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.29

method	result	size
gosper	\(-\frac {6 b^{5} x^{5}+15 a \,b^{4} x^{4}+20 a^{2} b^{3} x^{3}+15 a^{3} b^{2} x^{2}+6 a^{4} b x +a^{5}}{6 x^{6}}\)	\(56\)
norman	\(\frac {-b^{5} x^{5}-\frac {5}{2} a \,b^{4} x^{4}-\frac {10}{3} a^{2} b^{3} x^{3}-\frac {5}{2} a^{3} b^{2} x^{2}-a^{4} b x -\frac {1}{6} a^{5}}{x^{6}}\)	\(57\)
risch	\(\frac {-b^{5} x^{5}-\frac {5}{2} a \,b^{4} x^{4}-\frac {10}{3} a^{2} b^{3} x^{3}-\frac {5}{2} a^{3} b^{2} x^{2}-a^{4} b x -\frac {1}{6} a^{5}}{x^{6}}\)	\(57\)
default	\(-\frac {a^{5}}{6 x^{6}}-\frac {10 a^{2} b^{3}}{3 x^{3}}-\frac {b^{5}}{x}-\frac {5 a \,b^{4}}{2 x^{2}}-\frac {5 a^{3} b^{2}}{2 x^{4}}-\frac {a^{4} b}{x^{5}}\)	\(58\)
parallelrisch	\(\frac {-6 b^{5} x^{5}-15 a \,b^{4} x^{4}-20 a^{2} b^{3} x^{3}-15 a^{3} b^{2} x^{2}-6 a^{4} b x -a^{5}}{6 x^{6}}\)	\(58\)

[In]

int((b*x+a)^5/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*(6*b^5*x^5+15*a*b^4*x^4+20*a^2*b^3*x^3+15*a^3*b^2*x^2+6*a^4*b*x+a^5)/x^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.24 \[ \int \frac {(a+b x)^5}{x^7} \, dx=-\frac {6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, x^{6}} \]

[In]

integrate((b*x+a)^5/x^7,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/x^6

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (14) = 28\).

Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^5}{x^7} \, dx=\frac {- a^{5} - 6 a^{4} b x - 15 a^{3} b^{2} x^{2} - 20 a^{2} b^{3} x^{3} - 15 a b^{4} x^{4} - 6 b^{5} x^{5}}{6 x^{6}} \]

[In]

integrate((b*x+a)**5/x**7,x)

[Out]

(-a**5 - 6*a**4*b*x - 15*a**3*b**2*x**2 - 20*a**2*b**3*x**3 - 15*a*b**4*x**4 - 6*b**5*x**5)/(6*x**6)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (15) = 30\).

Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.24 \[ \int \frac {(a+b x)^5}{x^7} \, dx=-\frac {6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, x^{6}} \]

[In]

integrate((b*x+a)^5/x^7,x, algorithm="maxima")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/x^6

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.24 \[ \int \frac {(a+b x)^5}{x^7} \, dx=-\frac {6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, x^{6}} \]

[In]

integrate((b*x+a)^5/x^7,x, algorithm="giac")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/x^6

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.24 \[ \int \frac {(a+b x)^5}{x^7} \, dx=-\frac {\frac {a^5}{6}+a^4\,b\,x+\frac {5\,a^3\,b^2\,x^2}{2}+\frac {10\,a^2\,b^3\,x^3}{3}+\frac {5\,a\,b^4\,x^4}{2}+b^5\,x^5}{x^6} \]

[In]

int((a + b*x)^5/x^7,x)

[Out]

-(a^5/6 + b^5*x^5 + (5*a*b^4*x^4)/2 + (5*a^3*b^2*x^2)/2 + (10*a^2*b^3*x^3)/3 + a^4*b*x)/x^6