Optimal. Leaf size=66 \[ -\frac {\left (b^2-4 a c\right )^2}{96 c^3 d^4 (b+2 c x)^3}+\frac {b^2-4 a c}{16 c^3 d^4 (b+2 c x)}+\frac {x}{16 c^2 d^4} \]
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Rubi [A] time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {683} \begin {gather*} -\frac {\left (b^2-4 a c\right )^2}{96 c^3 d^4 (b+2 c x)^3}+\frac {b^2-4 a c}{16 c^3 d^4 (b+2 c x)}+\frac {x}{16 c^2 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^4} \, dx &=\int \left (\frac {1}{16 c^2 d^4}+\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^4 (b+2 c x)^4}+\frac {-b^2+4 a c}{8 c^2 d^4 (b+2 c x)^2}\right ) \, dx\\ &=\frac {x}{16 c^2 d^4}-\frac {\left (b^2-4 a c\right )^2}{96 c^3 d^4 (b+2 c x)^3}+\frac {b^2-4 a c}{16 c^3 d^4 (b+2 c x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 53, normalized size = 0.80 \begin {gather*} \frac {-\frac {\left (b^2-4 a c\right )^2}{(b+2 c x)^3}+\frac {6 \left (b^2-4 a c\right )}{b+2 c x}+6 c x}{96 c^3 d^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 125, normalized size = 1.89 \begin {gather*} \frac {48 \, c^{4} x^{4} + 72 \, b c^{3} x^{3} + 5 \, b^{4} - 16 \, a b^{2} c - 16 \, a^{2} c^{2} + 12 \, {\left (5 \, b^{2} c^{2} - 8 \, a c^{3}\right )} x^{2} + 6 \, {\left (5 \, b^{3} c - 16 \, a b c^{2}\right )} x}{96 \, {\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 82, normalized size = 1.24 \begin {gather*} \frac {x}{16 \, c^{2} d^{4}} + \frac {24 \, b^{2} c^{2} x^{2} - 96 \, a c^{3} x^{2} + 24 \, b^{3} c x - 96 \, a b c^{2} x + 5 \, b^{4} - 16 \, a b^{2} c - 16 \, a^{2} c^{2}}{96 \, {\left (2 \, c x + b\right )}^{3} c^{3} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 67, normalized size = 1.02 \begin {gather*} \frac {\frac {x}{16 c^{2}}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{96 \left (2 c x +b \right )^{3} c^{3}}-\frac {4 a c -b^{2}}{16 \left (2 c x +b \right ) c^{3}}}{d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 116, normalized size = 1.76 \begin {gather*} \frac {5 \, b^{4} - 16 \, a b^{2} c - 16 \, a^{2} c^{2} + 24 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 24 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x}{96 \, {\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )}} + \frac {x}{16 \, c^{2} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 67, normalized size = 1.02 \begin {gather*} -\frac {\frac {a^2\,c^2}{6}-b\,\left (\frac {c^3\,x^3}{3}-a\,c^2\,x\right )-\frac {c^4\,x^4}{2}+a\,c^3\,x^2+\frac {a\,b^2\,c}{6}}{c^3\,d^4\,{\left (b+2\,c\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.86, size = 117, normalized size = 1.77 \begin {gather*} \frac {- 16 a^{2} c^{2} - 16 a b^{2} c + 5 b^{4} + x^{2} \left (- 96 a c^{3} + 24 b^{2} c^{2}\right ) + x \left (- 96 a b c^{2} + 24 b^{3} c\right )}{96 b^{3} c^{3} d^{4} + 576 b^{2} c^{4} d^{4} x + 1152 b c^{5} d^{4} x^{2} + 768 c^{6} d^{4} x^{3}} + \frac {x}{16 c^{2} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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