Optimal. Leaf size=18 \[ -\frac {c^3 (a-b x)^4}{2 x^2} \]
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Rubi [A] time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {74} \begin {gather*} -\frac {c^3 (a-b x)^4}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^3}{x^3} \, dx &=-\frac {c^3 (a-b x)^4}{2 x^2}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 41, normalized size = 2.28 \begin {gather*} c^3 \left (-\frac {a^4}{2 x^2}+\frac {2 a^3 b}{x}+2 a b^3 x-\frac {b^4 x^2}{2}\right ) \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)^3}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.16, size = 45, normalized size = 2.50 \begin {gather*} -\frac {b^{4} c^{3} x^{4} - 4 \, a b^{3} c^{3} x^{3} - 4 \, a^{3} b c^{3} x + a^{4} c^{3}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.88, size = 46, normalized size = 2.56 \begin {gather*} -\frac {1}{2} \, b^{4} c^{3} x^{2} + 2 \, a b^{3} c^{3} x + \frac {4 \, a^{3} b c^{3} x - a^{4} c^{3}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 38, normalized size = 2.11 \begin {gather*} \left (-\frac {b^{4} x^{2}}{2}+2 a \,b^{3} x +\frac {2 a^{3} b}{x}-\frac {a^{4}}{2 x^{2}}\right ) c^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.08, size = 46, normalized size = 2.56 \begin {gather*} -\frac {1}{2} \, b^{4} c^{3} x^{2} + 2 \, a b^{3} c^{3} x + \frac {4 \, a^{3} b c^{3} x - a^{4} c^{3}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 35, normalized size = 1.94 \begin {gather*} -\frac {c^3\,\left (a^4-4\,a^3\,b\,x-4\,a\,b^3\,x^3+b^4\,x^4\right )}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.16, size = 46, normalized size = 2.56 \begin {gather*} 2 a b^{3} c^{3} x - \frac {b^{4} c^{3} x^{2}}{2} - \frac {a^{4} c^{3} - 4 a^{3} b c^{3} x}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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