Optimal. Leaf size=65 \[ -\frac{d (b c-a d)}{2 b^3 (a+b x)^4}-\frac{(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac{d^2}{3 b^3 (a+b x)^3} \]
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Rubi [A] time = 0.0435076, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {626, 43} \[ -\frac{d (b c-a d)}{2 b^3 (a+b x)^4}-\frac{(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac{d^2}{3 b^3 (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^8} \, dx &=\int \frac{(c+d x)^2}{(a+b x)^6} \, dx\\ &=\int \left (\frac{(b c-a d)^2}{b^2 (a+b x)^6}+\frac{2 d (b c-a d)}{b^2 (a+b x)^5}+\frac{d^2}{b^2 (a+b x)^4}\right ) \, dx\\ &=-\frac{(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac{d (b c-a d)}{2 b^3 (a+b x)^4}-\frac{d^2}{3 b^3 (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0241188, size = 57, normalized size = 0.88 \[ -\frac{a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )}{30 b^3 (a+b x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 71, normalized size = 1.1 \begin{align*} -{\frac{{a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2}}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}+{\frac{ \left ( ad-bc \right ) d}{2\,{b}^{3} \left ( bx+a \right ) ^{4}}}-{\frac{{d}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0295, size = 147, normalized size = 2.26 \begin{align*} -\frac{10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \,{\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42936, size = 227, normalized size = 3.49 \begin{align*} -\frac{10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \,{\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \,{\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.6324, size = 116, normalized size = 1.78 \begin{align*} - \frac{a^{2} d^{2} + 3 a b c d + 6 b^{2} c^{2} + 10 b^{2} d^{2} x^{2} + x \left (5 a b d^{2} + 15 b^{2} c d\right )}{30 a^{5} b^{3} + 150 a^{4} b^{4} x + 300 a^{3} b^{5} x^{2} + 300 a^{2} b^{6} x^{3} + 150 a b^{7} x^{4} + 30 b^{8} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20867, size = 82, normalized size = 1.26 \begin{align*} -\frac{10 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c d x + 5 \, a b d^{2} x + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2}}{30 \,{\left (b x + a\right )}^{5} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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