Optimal. Leaf size=55 \[ -\frac {a^4 c^3}{7 x^7}+\frac {a^3 b c^3}{3 x^6}-\frac {a b^3 c^3}{2 x^4}+\frac {b^4 c^3}{3 x^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76}
\begin {gather*} -\frac {a^4 c^3}{7 x^7}+\frac {a^3 b c^3}{3 x^6}-\frac {a b^3 c^3}{2 x^4}+\frac {b^4 c^3}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^3}{x^8} \, dx &=\int \left (\frac {a^4 c^3}{x^8}-\frac {2 a^3 b c^3}{x^7}+\frac {2 a b^3 c^3}{x^5}-\frac {b^4 c^3}{x^4}\right ) \, dx\\ &=-\frac {a^4 c^3}{7 x^7}+\frac {a^3 b c^3}{3 x^6}-\frac {a b^3 c^3}{2 x^4}+\frac {b^4 c^3}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 0.85 \begin {gather*} c^3 \left (-\frac {a^4}{7 x^7}+\frac {a^3 b}{3 x^6}-\frac {a b^3}{2 x^4}+\frac {b^4}{3 x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 40, normalized size = 0.73
method | result | size |
gosper | \(-\frac {c^{3} \left (-14 b^{4} x^{4}+21 a \,b^{3} x^{3}-14 a^{3} b x +6 a^{4}\right )}{42 x^{7}}\) | \(39\) |
default | \(c^{3} \left (-\frac {a \,b^{3}}{2 x^{4}}+\frac {a^{3} b}{3 x^{6}}+\frac {b^{4}}{3 x^{3}}-\frac {a^{4}}{7 x^{7}}\right )\) | \(40\) |
norman | \(\frac {-\frac {1}{7} a^{4} c^{3}+\frac {1}{3} b^{4} c^{3} x^{4}-\frac {1}{2} a \,b^{3} c^{3} x^{3}+\frac {1}{3} a^{3} b \,c^{3} x}{x^{7}}\) | \(47\) |
risch | \(\frac {-\frac {1}{7} a^{4} c^{3}+\frac {1}{3} b^{4} c^{3} x^{4}-\frac {1}{2} a \,b^{3} c^{3} x^{3}+\frac {1}{3} a^{3} b \,c^{3} x}{x^{7}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 0.85 \begin {gather*} \frac {14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.03, size = 47, normalized size = 0.85 \begin {gather*} \frac {14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 51, normalized size = 0.93 \begin {gather*} - \frac {6 a^{4} c^{3} - 14 a^{3} b c^{3} x + 21 a b^{3} c^{3} x^{3} - 14 b^{4} c^{3} x^{4}}{42 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.44, size = 47, normalized size = 0.85 \begin {gather*} \frac {14 \, b^{4} c^{3} x^{4} - 21 \, a b^{3} c^{3} x^{3} + 14 \, a^{3} b c^{3} x - 6 \, a^{4} c^{3}}{42 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 47, normalized size = 0.85 \begin {gather*} -\frac {\frac {a^4\,c^3}{7}-\frac {a^3\,b\,c^3\,x}{3}+\frac {a\,b^3\,c^3\,x^3}{2}-\frac {b^4\,c^3\,x^4}{3}}{x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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