Optimal. Leaf size=17 \[ -\frac {b^2}{5 d^3 (a+b x)^5} \]
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Rubi [A] time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \[ -\frac {b^2}{5 d^3 (a+b x)^5} \]
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^3 \left (\frac {a d}{b}+d x\right )^3} \, dx &=\frac {b^3 \int \frac {1}{(a+b x)^6} \, dx}{d^3}\\ &=-\frac {b^2}{5 d^3 (a+b x)^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ -\frac {b^2}{5 d^3 (a+b x)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 75, normalized size = 4.41 \[ -\frac {b^{2}}{5 \, {\left (b^{5} d^{3} x^{5} + 5 \, a b^{4} d^{3} x^{4} + 10 \, a^{2} b^{3} d^{3} x^{3} + 10 \, a^{3} b^{2} d^{3} x^{2} + 5 \, a^{4} b d^{3} x + a^{5} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 15, normalized size = 0.88 \[ -\frac {b^{2}}{5 \, {\left (b x + a\right )}^{5} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.94 \[ -\frac {b^{2}}{5 \left (b x +a \right )^{5} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.33, size = 75, normalized size = 4.41 \[ -\frac {b^{2}}{5 \, {\left (b^{5} d^{3} x^{5} + 5 \, a b^{4} d^{3} x^{4} + 10 \, a^{2} b^{3} d^{3} x^{3} + 10 \, a^{3} b^{2} d^{3} x^{2} + 5 \, a^{4} b d^{3} x + a^{5} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 77, normalized size = 4.53 \[ -\frac {b^2}{5\,\left (a^5\,d^3+5\,a^4\,b\,d^3\,x+10\,a^3\,b^2\,d^3\,x^2+10\,a^2\,b^3\,d^3\,x^3+5\,a\,b^4\,d^3\,x^4+b^5\,d^3\,x^5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.42, size = 83, normalized size = 4.88 \[ - \frac {b^{3}}{5 a^{5} b d^{3} + 25 a^{4} b^{2} d^{3} x + 50 a^{3} b^{3} d^{3} x^{2} + 50 a^{2} b^{4} d^{3} x^{3} + 25 a b^{5} d^{3} x^{4} + 5 b^{6} d^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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