Optimal. Leaf size=28 \[ -\frac{(c+d x)^4}{4 (a+b x)^4 (b c-a d)} \]
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Rubi [A] time = 0.0029669, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {37} \[ -\frac{(c+d x)^4}{4 (a+b x)^4 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{(a+b x)^5} \, dx &=-\frac{(c+d x)^4}{4 (b c-a d) (a+b x)^4}\\ \end{align*}
Mathematica [B] time = 0.0317511, size = 91, normalized size = 3.25 \[ -\frac{a^2 b d^2 (c+4 d x)+a^3 d^3+a b^2 d \left (c^2+4 c d x+6 d^2 x^2\right )+b^3 \left (4 c^2 d x+c^3+6 c d^2 x^2+4 d^3 x^3\right )}{4 b^4 (a+b x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 122, normalized size = 4.4 \begin{align*} -{\frac{{d}^{3}}{{b}^{4} \left ( bx+a \right ) }}+{\frac{3\,{d}^{2} \left ( ad-bc \right ) }{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{d \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{-{a}^{3}{d}^{3}+3\,{a}^{2}bc{d}^{2}-3\,a{b}^{2}{c}^{2}d+{b}^{3}{c}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.979592, size = 193, normalized size = 6.89 \begin{align*} -\frac{4 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3} + 6 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{4 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.28969, size = 284, normalized size = 10.14 \begin{align*} -\frac{4 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3} + 6 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{4 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.76393, size = 153, normalized size = 5.46 \begin{align*} - \frac{a^{3} d^{3} + a^{2} b c d^{2} + a b^{2} c^{2} d + b^{3} c^{3} + 4 b^{3} d^{3} x^{3} + x^{2} \left (6 a b^{2} d^{3} + 6 b^{3} c d^{2}\right ) + x \left (4 a^{2} b d^{3} + 4 a b^{2} c d^{2} + 4 b^{3} c^{2} d\right )}{4 a^{4} b^{4} + 16 a^{3} b^{5} x + 24 a^{2} b^{6} x^{2} + 16 a b^{7} x^{3} + 4 b^{8} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05459, size = 215, normalized size = 7.68 \begin{align*} -\frac{\frac{b^{2} c^{3}}{{\left (b x + a\right )}^{4}} + \frac{4 \, b c^{2} d}{{\left (b x + a\right )}^{3}} - \frac{3 \, a b c^{2} d}{{\left (b x + a\right )}^{4}} + \frac{6 \, c d^{2}}{{\left (b x + a\right )}^{2}} - \frac{8 \, a c d^{2}}{{\left (b x + a\right )}^{3}} + \frac{3 \, a^{2} c d^{2}}{{\left (b x + a\right )}^{4}} + \frac{4 \, d^{3}}{{\left (b x + a\right )} b} - \frac{6 \, a d^{3}}{{\left (b x + a\right )}^{2} b} + \frac{4 \, a^{2} d^{3}}{{\left (b x + a\right )}^{3} b} - \frac{a^{3} d^{3}}{{\left (b x + a\right )}^{4} b}}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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