Optimal. Leaf size=38 \[ \frac {a}{2 b c^5 (a-b x)^4}-\frac {1}{3 b c^5 (a-b x)^3} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \begin {gather*} \frac {a}{2 b c^5 (a-b x)^4}-\frac {1}{3 b c^5 (a-b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {a+b x}{(a c-b c x)^5} \, dx &=\int \left (\frac {2 a}{c^5 (a-b x)^5}-\frac {1}{c^5 (a-b x)^4}\right ) \, dx\\ &=\frac {a}{2 b c^5 (a-b x)^4}-\frac {1}{3 b c^5 (a-b x)^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.63 \begin {gather*} \frac {a+2 b x}{6 b c^5 (a-b x)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{(a c-b c x)^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.41, size = 67, normalized size = 1.76 \begin {gather*} \frac {2 \, b x + a}{6 \, {\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 40, normalized size = 1.05 \begin {gather*} \frac {a}{2 \, {\left (b c x - a c\right )}^{4} b c} + \frac {1}{3 \, {\left (b c x - a c\right )}^{3} b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.92 \begin {gather*} \frac {\frac {a}{2 \left (b x -a \right )^{4} b}+\frac {1}{3 \left (b x -a \right )^{3} b}}{c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 67, normalized size = 1.76 \begin {gather*} \frac {2 \, b x + a}{6 \, {\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 67, normalized size = 1.76 \begin {gather*} \frac {\frac {x}{3}+\frac {a}{6\,b}}{a^4\,c^5-4\,a^3\,b\,c^5\,x+6\,a^2\,b^2\,c^5\,x^2-4\,a\,b^3\,c^5\,x^3+b^4\,c^5\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.39, size = 73, normalized size = 1.92 \begin {gather*} - \frac {- a - 2 b x}{6 a^{4} b c^{5} - 24 a^{3} b^{2} c^{5} x + 36 a^{2} b^{3} c^{5} x^{2} - 24 a b^{4} c^{5} x^{3} + 6 b^{5} c^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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