\(\int \frac {(a+b x)^3}{x^6} \, dx\) [74]

Optimal result

Integrand size = 11, antiderivative size = 36 \[ \int \frac {(a+b x)^3}{x^6} \, dx=-\frac {(a+b x)^4}{5 a x^5}+\frac {b (a+b x)^4}{20 a^2 x^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^3}{x^6} \, dx=\frac {b (a+b x)^4}{20 a^2 x^4}-\frac {(a+b x)^4}{5 a x^5} \]

[In]

Int[(a + b*x)^3/x^6,x]

[Out]

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

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)^4}{5 a x^5}-\frac {b \int \frac {(a+b x)^3}{x^5} \, dx}{5 a} \\ & = -\frac {(a+b x)^4}{5 a x^5}+\frac {b (a+b x)^4}{20 a^2 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^3}{x^6} \, dx=-\frac {a^3}{5 x^5}-\frac {3 a^2 b}{4 x^4}-\frac {a b^2}{x^3}-\frac {b^3}{2 x^2} \]

[In]

Integrate[(a + b*x)^3/x^6,x]

[Out]

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97

method	result	size
norman	\(\frac {-\frac {1}{2} b^{3} x^{3}-a \,b^{2} x^{2}-\frac {3}{4} a^{2} b x -\frac {1}{5} a^{3}}{x^{5}}\)	\(35\)
risch	\(\frac {-\frac {1}{2} b^{3} x^{3}-a \,b^{2} x^{2}-\frac {3}{4} a^{2} b x -\frac {1}{5} a^{3}}{x^{5}}\)	\(35\)
gosper	\(-\frac {10 b^{3} x^{3}+20 a \,b^{2} x^{2}+15 a^{2} b x +4 a^{3}}{20 x^{5}}\)	\(36\)
default	\(-\frac {a \,b^{2}}{x^{3}}-\frac {b^{3}}{2 x^{2}}-\frac {3 a^{2} b}{4 x^{4}}-\frac {a^{3}}{5 x^{5}}\)	\(36\)
parallelrisch	\(\frac {-10 b^{3} x^{3}-20 a \,b^{2} x^{2}-15 a^{2} b x -4 a^{3}}{20 x^{5}}\)	\(36\)

[In]

int((b*x+a)^3/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^3}{x^6} \, dx=-\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \]

[In]

integrate((b*x+a)^3/x^6,x, algorithm="fricas")

[Out]

-1/20*(10*b^3*x^3 + 20*a*b^2*x^2 + 15*a^2*b*x + 4*a^3)/x^5

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^3}{x^6} \, dx=\frac {- 4 a^{3} - 15 a^{2} b x - 20 a b^{2} x^{2} - 10 b^{3} x^{3}}{20 x^{5}} \]

[In]

integrate((b*x+a)**3/x**6,x)

[Out]

(-4*a**3 - 15*a**2*b*x - 20*a*b**2*x**2 - 10*b**3*x**3)/(20*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^3}{x^6} \, dx=-\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \]

[In]

integrate((b*x+a)^3/x^6,x, algorithm="maxima")

[Out]

-1/20*(10*b^3*x^3 + 20*a*b^2*x^2 + 15*a^2*b*x + 4*a^3)/x^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^3}{x^6} \, dx=-\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \]

[In]

integrate((b*x+a)^3/x^6,x, algorithm="giac")

[Out]

-1/20*(10*b^3*x^3 + 20*a*b^2*x^2 + 15*a^2*b*x + 4*a^3)/x^5

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^3}{x^6} \, dx=-\frac {\frac {a^3}{5}+\frac {3\,a^2\,b\,x}{4}+a\,b^2\,x^2+\frac {b^3\,x^3}{2}}{x^5} \]

[In]

int((a + b*x)^3/x^6,x)

[Out]

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