Optimal. Leaf size=76 \[ -\frac{16 c \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)}-\frac{2}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.0340286, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {687, 682} \[ -\frac{16 c \sqrt{a+b x+c x^2}}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)}-\frac{2}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 682
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt{a+b x+c x^2}}-\frac{(8 c) \int \frac{1}{(b d+2 c d x)^2 \sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{2}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \sqrt{a+b x+c x^2}}-\frac{16 c \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.0280347, size = 56, normalized size = 0.74 \[ -\frac{2 \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 68, normalized size = 0.9 \begin{align*} -2\,{\frac{8\,{c}^{2}{x}^{2}+8\,bcx+4\,ac+{b}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ){d}^{2} \left ( 2\,cx+b \right ) \sqrt{c{x}^{2}+bx+a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 5.74985, size = 327, normalized size = 4.3 \begin{align*} -\frac{2 \,{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c x^{2} + b x + a}}{2 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} x^{3} + 3 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{2} x^{2} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d^{2} x +{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a b^{2} \sqrt{a + b x + c x^{2}} + 4 a b c x \sqrt{a + b x + c x^{2}} + 4 a c^{2} x^{2} \sqrt{a + b x + c x^{2}} + b^{3} x \sqrt{a + b x + c x^{2}} + 5 b^{2} c x^{2} \sqrt{a + b x + c x^{2}} + 8 b c^{2} x^{3} \sqrt{a + b x + c x^{2}} + 4 c^{3} x^{4} \sqrt{a + b x + c x^{2}}}\, dx}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (2 \, c d x + b d\right )}^{2}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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