Optimal. Leaf size=58 \[ \frac{d (c+d x)^4}{20 (a+b x)^4 (b c-a d)^2}-\frac{(c+d x)^4}{5 (a+b x)^5 (b c-a d)} \]
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Rubi [A] time = 0.0189528, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {626, 45, 37} \[ \frac{d (c+d x)^4}{20 (a+b x)^4 (b c-a d)^2}-\frac{(c+d x)^4}{5 (a+b x)^5 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 45
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)^9} \, dx &=\int \frac{(c+d x)^3}{(a+b x)^6} \, dx\\ &=-\frac{(c+d x)^4}{5 (b c-a d) (a+b x)^5}-\frac{d \int \frac{(c+d x)^3}{(a+b x)^5} \, dx}{5 (b c-a d)}\\ &=-\frac{(c+d x)^4}{5 (b c-a d) (a+b x)^5}+\frac{d (c+d x)^4}{20 (b c-a d)^2 (a+b x)^4}\\ \end{align*}
Mathematica [A] time = 0.033927, size = 97, normalized size = 1.67 \[ -\frac{a^2 b d^2 (2 c+5 d x)+a^3 d^3+a b^2 d \left (3 c^2+10 c d x+10 d^2 x^2\right )+b^3 \left (15 c^2 d x+4 c^3+20 c d^2 x^2+10 d^3 x^3\right )}{20 b^4 (a+b x)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 121, normalized size = 2.1 \begin{align*} -{\frac{{d}^{3}}{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}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}-{\frac{3\,d \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}+{\frac{{d}^{2} \left ( ad-bc \right ) }{{b}^{4} \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00864, size = 216, normalized size = 3.72 \begin{align*} -\frac{10 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3} + 10 \,{\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.55489, size = 328, normalized size = 5.66 \begin{align*} -\frac{10 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3} + 10 \,{\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.88685, size = 170, normalized size = 2.93 \begin{align*} - \frac{a^{3} d^{3} + 2 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + 4 b^{3} c^{3} + 10 b^{3} d^{3} x^{3} + x^{2} \left (10 a b^{2} d^{3} + 20 b^{3} c d^{2}\right ) + x \left (5 a^{2} b d^{3} + 10 a b^{2} c d^{2} + 15 b^{3} c^{2} d\right )}{20 a^{5} b^{4} + 100 a^{4} b^{5} x + 200 a^{3} b^{6} x^{2} + 200 a^{2} b^{7} x^{3} + 100 a b^{8} x^{4} + 20 b^{9} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2378, size = 154, normalized size = 2.66 \begin{align*} -\frac{10 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c d^{2} x^{2} + 10 \, a b^{2} d^{3} x^{2} + 15 \, b^{3} c^{2} d x + 10 \, a b^{2} c d^{2} x + 5 \, a^{2} b d^{3} x + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}}{20 \,{\left (b x + a\right )}^{5} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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