Optimal. Leaf size=79 \[ \frac{4 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac{2 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]
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Rubi [A] time = 0.0319232, antiderivative size = 79, 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 = {693, 682} \[ \frac{4 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac{2 \sqrt{a+b x+c x^2}}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3} \]
Antiderivative was successfully verified.
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Rule 693
Rule 682
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^4 \sqrt{a+b x+c x^2}} \, dx &=\frac{2 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac{2 \int \frac{1}{(b d+2 c d x)^2 \sqrt{a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right ) d^2}\\ &=\frac{2 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3}+\frac{4 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.0275424, size = 60, normalized size = 0.76 \[ \frac{2 \sqrt{a+x (b+c x)} \left (-4 c \left (a-2 c x^2\right )+3 b^2+8 b c x\right )}{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 70, normalized size = 0.9 \begin{align*} -{\frac{-16\,{c}^{2}{x}^{2}-16\,bcx+8\,ac-6\,{b}^{2}}{3\, \left ( 2\,cx+b \right ) ^{3}{d}^{4} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \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 = 7.41221, size = 346, normalized size = 4.38 \begin{align*} \frac{2 \,{\left (8 \, c^{2} x^{2} + 8 \, b c x + 3 \, b^{2} - 4 \, a c\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (8 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 12 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 6 \,{\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{4} x +{\left (b^{7} - 8 \, a b^{5} c + 16 \, a^{2} b^{3} c^{2}\right )} d^{4}\right )}} \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}{b^{4} \sqrt{a + b x + c x^{2}} + 8 b^{3} c x \sqrt{a + b x + c x^{2}} + 24 b^{2} c^{2} x^{2} \sqrt{a + b x + c x^{2}} + 32 b c^{3} x^{3} \sqrt{a + b x + c x^{2}} + 16 c^{4} x^{4} \sqrt{a + b x + c x^{2}}}\, dx}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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