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 = 0.0116204, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {626, 37} \[ -\frac{(c+d x)^4}{4 (a+b x)^4 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx &=\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 = 0.0294395, 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 = 0.045, size = 122, normalized size = 4.4 \begin{align*}{\frac{3\,{d}^{2} \left ( ad-bc \right ) }{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{-{a}^{3}{d}^{3}+3\,cb{a}^{2}{d}^{2}-3\,a{c}^{2}d{b}^{2}+{c}^{3}{b}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{d \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{3}}{{b}^{4} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.23457, 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 = 1.46081, 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 = 2.90295, 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 = 1.16293, size = 150, normalized size = 5.36 \begin{align*} -\frac{4 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, a b^{2} d^{3} x^{2} + 4 \, b^{3} c^{2} d x + 4 \, a b^{2} c d^{2} x + 4 \, a^{2} b d^{3} x + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}}{4 \,{\left (b x + a\right )}^{4} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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