\(\int \frac {(a+b x)^3}{x^5} \, dx\) [73]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {(a+b x)^3}{x^5} \, dx=-\frac {(a+b x)^4}{4 a x^4} \]

[Out]

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

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} \[ \int \frac {(a+b x)^3}{x^5} \, dx=-\frac {(a+b x)^4}{4 a x^4} \]

[In]

Int[(a + b*x)^3/x^5,x]

[Out]

-1/4*(a + b*x)^4/(a*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]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^4}{4 a x^4} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(17)=34\).

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.29 \[ \int \frac {(a+b x)^3}{x^5} \, dx=-\frac {a^3}{4 x^4}-\frac {a^2 b}{x^3}-\frac {3 a b^2}{2 x^2}-\frac {b^3}{x} \]

[In]

Integrate[(a + b*x)^3/x^5,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(33\) vs. \(2(15)=30\).

Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00

method	result	size
gosper	\(-\frac {4 b^{3} x^{3}+6 a \,b^{2} x^{2}+4 a^{2} b x +a^{3}}{4 x^{4}}\)	\(34\)
norman	\(\frac {-b^{3} x^{3}-\frac {3}{2} a \,b^{2} x^{2}-a^{2} b x -\frac {1}{4} a^{3}}{x^{4}}\)	\(35\)
risch	\(\frac {-b^{3} x^{3}-\frac {3}{2} a \,b^{2} x^{2}-a^{2} b x -\frac {1}{4} a^{3}}{x^{4}}\)	\(35\)
default	\(-\frac {a^{2} b}{x^{3}}-\frac {b^{3}}{x}-\frac {3 a \,b^{2}}{2 x^{2}}-\frac {a^{3}}{4 x^{4}}\)	\(36\)
parallelrisch	\(\frac {-4 b^{3} x^{3}-6 a \,b^{2} x^{2}-4 a^{2} b x -a^{3}}{4 x^{4}}\)	\(36\)

[In]

int((b*x+a)^3/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x)^3}{x^5} \, dx=-\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \]

[In]

integrate((b*x+a)^3/x^5,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).

Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b x)^3}{x^5} \, dx=\frac {- a^{3} - 4 a^{2} b x - 6 a b^{2} x^{2} - 4 b^{3} x^{3}}{4 x^{4}} \]

[In]

integrate((b*x+a)**3/x**5,x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x)^3}{x^5} \, dx=-\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \]

[In]

integrate((b*x+a)^3/x^5,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x)^3}{x^5} \, dx=-\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \]

[In]

integrate((b*x+a)^3/x^5,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x)^3}{x^5} \, dx=-\frac {\frac {a^3}{4}+a^2\,b\,x+\frac {3\,a\,b^2\,x^2}{2}+b^3\,x^3}{x^4} \]

[In]

int((a + b*x)^3/x^5,x)

[Out]

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