Optimal. Leaf size=55 \[ \frac{b^2-4 a c}{20 c^2 d (b d+2 c d x)^{5/2}}-\frac{1}{4 c^2 d^3 \sqrt{b d+2 c d x}} \]
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Rubi [A] time = 0.0223381, 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} \[ \frac{b^2-4 a c}{20 c^2 d (b d+2 c d x)^{5/2}}-\frac{1}{4 c^2 d^3 \sqrt{b d+2 c d x}} \]
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)^{7/2}} \, dx &=\int \left (\frac{-b^2+4 a c}{4 c (b d+2 c d x)^{7/2}}+\frac{1}{4 c d^2 (b d+2 c d x)^{3/2}}\right ) \, dx\\ &=\frac{b^2-4 a c}{20 c^2 d (b d+2 c d x)^{5/2}}-\frac{1}{4 c^2 d^3 \sqrt{b d+2 c d x}}\\ \end{align*}
Mathematica [A] time = 0.0293239, size = 44, normalized size = 0.8 \[ \frac{-c \left (a+5 c x^2\right )-b^2-5 b c x}{5 c^2 d (d (b+2 c x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 43, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( 5\,{c}^{2}{x}^{2}+5\,bcx+ac+{b}^{2} \right ) }{5\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.026, size = 61, normalized size = 1.11 \begin{align*} \frac{{\left (b^{2} - 4 \, a c\right )} d^{2} - 5 \,{\left (2 \, c d x + b d\right )}^{2}}{20 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01084, size = 171, normalized size = 3.11 \begin{align*} -\frac{{\left (5 \, c^{2} x^{2} + 5 \, b c x + b^{2} + a c\right )} \sqrt{2 \, c d x + b d}}{5 \,{\left (8 \, c^{5} d^{4} x^{3} + 12 \, b c^{4} d^{4} x^{2} + 6 \, b^{2} c^{3} d^{4} x + b^{3} c^{2} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.4656, size = 298, normalized size = 5.42 \begin{align*} \begin{cases} - \frac{a c \sqrt{b d + 2 c d x}}{5 b^{3} c^{2} d^{4} + 30 b^{2} c^{3} d^{4} x + 60 b c^{4} d^{4} x^{2} + 40 c^{5} d^{4} x^{3}} - \frac{b^{2} \sqrt{b d + 2 c d x}}{5 b^{3} c^{2} d^{4} + 30 b^{2} c^{3} d^{4} x + 60 b c^{4} d^{4} x^{2} + 40 c^{5} d^{4} x^{3}} - \frac{5 b c x \sqrt{b d + 2 c d x}}{5 b^{3} c^{2} d^{4} + 30 b^{2} c^{3} d^{4} x + 60 b c^{4} d^{4} x^{2} + 40 c^{5} d^{4} x^{3}} - \frac{5 c^{2} x^{2} \sqrt{b d + 2 c d x}}{5 b^{3} c^{2} d^{4} + 30 b^{2} c^{3} d^{4} x + 60 b c^{4} d^{4} x^{2} + 40 c^{5} d^{4} x^{3}} & \text{for}\: c \neq 0 \\\frac{a x + \frac{b x^{2}}{2}}{\left (b d\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1477, size = 63, normalized size = 1.15 \begin{align*} \frac{b^{2} d^{2} - 4 \, a c d^{2} - 5 \,{\left (2 \, c d x + b d\right )}^{2}}{20 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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