Optimal. Leaf size=36 \[ -\frac {(a+b x)^4}{5 a x^5}+\frac {b (a+b x)^4}{20 a^2 x^4} \]
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Rubi [A]
time = 0.00, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37}
\begin {gather*} \frac {b (a+b x)^4}{20 a^2 x^4}-\frac {(a+b x)^4}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{x^6} \, dx &=-\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*}
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Mathematica [A]
time = 0.00, size = 41, normalized size = 1.14 \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 36, normalized size = 1.00
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 | \(-\frac {a \,b^{2}}{x^{3}}-\frac {3 a^{2} b}{4 x^{4}}-\frac {b^{3}}{2 x^{2}}-\frac {a^{3}}{5 x^{5}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 35, normalized size = 0.97 \begin {gather*} -\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.98, size = 35, normalized size = 0.97 \begin {gather*} -\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 37, normalized size = 1.03 \begin {gather*} \frac {- 4 a^{3} - 15 a^{2} b x - 20 a b^{2} x^{2} - 10 b^{3} x^{3}}{20 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.65, size = 35, normalized size = 0.97 \begin {gather*} -\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 34, normalized size = 0.94 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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