Optimal. Leaf size=55 \[ \frac{b^2-4 a c}{4 c^2 d \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{3/2}}{12 c^2 d^3} \]
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Rubi [A] time = 0.0233488, antiderivative size = 55, normalized size of antiderivative = 1., 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} \[ \frac{b^2-4 a c}{4 c^2 d \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{3/2}}{12 c^2 d^3} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx &=\int \left (\frac{-b^2+4 a c}{4 c (b d+2 c d x)^{3/2}}+\frac{\sqrt{b d+2 c d x}}{4 c d^2}\right ) \, dx\\ &=\frac{b^2-4 a c}{4 c^2 d \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{3/2}}{12 c^2 d^3}\\ \end{align*}
Mathematica [A] time = 0.0249194, size = 41, normalized size = 0.75 \[ \frac{c \left (c x^2-3 a\right )+b^2+b c x}{3 c^2 d \sqrt{d (b+2 c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 46, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( -{c}^{2}{x}^{2}-bcx+3\,ac-{b}^{2} \right ) }{3\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0142, size = 69, normalized size = 1.25 \begin{align*} \frac{\frac{3 \,{\left (b^{2} - 4 \, a c\right )}}{\sqrt{2 \, c d x + b d} c} + \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{c d^{2}}}{12 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95868, size = 112, normalized size = 2.04 \begin{align*} \frac{{\left (c^{2} x^{2} + b c x + b^{2} - 3 \, a c\right )} \sqrt{2 \, c d x + b d}}{3 \,{\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.5489, size = 49, normalized size = 0.89 \begin{align*} - \frac{4 a c - b^{2}}{4 c^{2} d \sqrt{b d + 2 c d x}} + \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}}}{12 c^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11439, size = 63, normalized size = 1.15 \begin{align*} \frac{b^{2} - 4 \, a c}{4 \, \sqrt{2 \, c d x + b d} c^{2} d} + \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{12 \, c^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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