Optimal. Leaf size=55 \[ \frac {b^2-4 a c}{12 c^2 d (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{4 c^2 d^3} \]
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Rubi [A] time = 0.02, antiderivative size = 55, 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} \[ \frac {b^2-4 a c}{12 c^2 d (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{4 c^2 d^3} \]
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)^{5/2}} \, dx &=\int \left (\frac {-b^2+4 a c}{4 c (b d+2 c d x)^{5/2}}+\frac {1}{4 c d^2 \sqrt {b d+2 c d x}}\right ) \, dx\\ &=\frac {b^2-4 a c}{12 c^2 d (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{4 c^2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 0.78 \[ \frac {c \left (3 c x^2-a\right )+b^2+3 b c x}{3 c^2 d (d (b+2 c x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 68, normalized size = 1.24 \[ \frac {{\left (3 \, c^{2} x^{2} + 3 \, b c x + b^{2} - a c\right )} \sqrt {2 \, c d x + b d}}{3 \, {\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + b^{2} c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 47, normalized size = 0.85 \[ \frac {b^{2} - 4 \, a c}{12 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} c^{2} d} + \frac {\sqrt {2 \, c d x + b d}}{4 \, c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 45, normalized size = 0.82 \[ -\frac {\left (2 c x +b \right ) \left (-3 c^{2} x^{2}-3 b c x +a c -b^{2}\right )}{3 \left (2 c d x +b d \right )^{\frac {5}{2}} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 51, normalized size = 0.93 \[ \frac {\frac {b^{2} - 4 \, a c}{{\left (2 \, c d x + b d\right )}^{\frac {3}{2}} c} + \frac {3 \, \sqrt {2 \, c d x + b d}}{c d^{2}}}{12 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 37, normalized size = 0.67 \[ \frac {3\,{\left (b+2\,c\,x\right )}^2-4\,a\,c+b^2}{12\,c^2\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.28, size = 235, normalized size = 4.27 \[ \begin {cases} - \frac {a c \sqrt {b d + 2 c d x}}{3 b^{2} c^{2} d^{3} + 12 b c^{3} d^{3} x + 12 c^{4} d^{3} x^{2}} + \frac {b^{2} \sqrt {b d + 2 c d x}}{3 b^{2} c^{2} d^{3} + 12 b c^{3} d^{3} x + 12 c^{4} d^{3} x^{2}} + \frac {3 b c x \sqrt {b d + 2 c d x}}{3 b^{2} c^{2} d^{3} + 12 b c^{3} d^{3} x + 12 c^{4} d^{3} x^{2}} + \frac {3 c^{2} x^{2} \sqrt {b d + 2 c d x}}{3 b^{2} c^{2} d^{3} + 12 b c^{3} d^{3} x + 12 c^{4} d^{3} x^{2}} & \text {for}\: c \neq 0 \\\frac {a x + \frac {b x^{2}}{2}}{\left (b d\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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