Optimal. Leaf size=43 \[ -\frac {b^3}{7 x^7}-\frac {a b^2}{2 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {a^3}{4 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 45}
\begin {gather*} -\frac {a^3}{4 x^4}-\frac {3 a^2 b}{5 x^5}-\frac {a b^2}{2 x^6}-\frac {b^3}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 269
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^3}{x^5} \, dx &=\int \frac {(b+a x)^3}{x^8} \, dx\\ &=\int \left (\frac {b^3}{x^8}+\frac {3 a b^2}{x^7}+\frac {3 a^2 b}{x^6}+\frac {a^3}{x^5}\right ) \, dx\\ &=-\frac {b^3}{7 x^7}-\frac {a b^2}{2 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {a^3}{4 x^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} -\frac {b^3}{7 x^7}-\frac {a b^2}{2 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {a^3}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 36, normalized size = 0.84
method | result | size |
norman | \(\frac {-\frac {1}{4} a^{3} x^{3}-\frac {3}{5} a^{2} b \,x^{2}-\frac {1}{2} a \,b^{2} x -\frac {1}{7} b^{3}}{x^{7}}\) | \(35\) |
risch | \(\frac {-\frac {1}{4} a^{3} x^{3}-\frac {3}{5} a^{2} b \,x^{2}-\frac {1}{2} a \,b^{2} x -\frac {1}{7} b^{3}}{x^{7}}\) | \(35\) |
gosper | \(-\frac {35 a^{3} x^{3}+84 a^{2} b \,x^{2}+70 a \,b^{2} x +20 b^{3}}{140 x^{7}}\) | \(36\) |
default | \(-\frac {b^{3}}{7 x^{7}}-\frac {a \,b^{2}}{2 x^{6}}-\frac {3 a^{2} b}{5 x^{5}}-\frac {a^{3}}{4 x^{4}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 35, normalized size = 0.81 \begin {gather*} -\frac {35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 35, normalized size = 0.81 \begin {gather*} -\frac {35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.11, size = 37, normalized size = 0.86 \begin {gather*} \frac {- 35 a^{3} x^{3} - 84 a^{2} b x^{2} - 70 a b^{2} x - 20 b^{3}}{140 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.16, size = 35, normalized size = 0.81 \begin {gather*} -\frac {35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 35, normalized size = 0.81 \begin {gather*} -\frac {\frac {a^3\,x^3}{4}+\frac {3\,a^2\,b\,x^2}{5}+\frac {a\,b^2\,x}{2}+\frac {b^3}{7}}{x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________