Optimal. Leaf size=55 \[ \frac {(b d+2 c d x)^{7/2}}{28 c^2 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d} \]
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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} \[ \frac {(b d+2 c d x)^{7/2}}{28 c^2 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (-b^2+4 a c\right ) \sqrt {b d+2 c d x}}{4 c}+\frac {(b d+2 c d x)^{5/2}}{4 c d^2}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d}+\frac {(b d+2 c d x)^{7/2}}{28 c^2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.82 \[ \frac {\left (c \left (7 a+3 c x^2\right )-b^2+3 b c x\right ) (d (b+2 c x))^{3/2}}{21 c^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 58, normalized size = 1.05 \[ \frac {{\left (6 \, c^{3} x^{3} + 9 \, b c^{2} x^{2} - b^{3} + 7 \, a b c + {\left (b^{2} c + 14 \, a c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d}}{21 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 228, normalized size = 4.15 \[ \frac {420 \, \sqrt {2 \, c d x + b d} a b - \frac {140 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} a}{d} - \frac {70 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} b^{2}}{c d} + \frac {21 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} b}{c d^{2}} - \frac {3 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )}}{c d^{3}}}{420 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 46, normalized size = 0.84 \[ \frac {\left (2 c x +b \right ) \left (3 c^{2} x^{2}+3 b c x +7 a c -b^{2}\right ) \sqrt {2 c d x +b d}}{21 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 46, normalized size = 0.84 \[ -\frac {7 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - 3 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}}{84 \, c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.48, size = 39, normalized size = 0.71 \[ \frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (28\,a\,c+3\,{\left (b+2\,c\,x\right )}^2-7\,b^2\right )}{84\,c^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.90, size = 48, normalized size = 0.87 \[ \frac {\frac {\left (4 a c - b^{2}\right ) \left (b d + 2 c d x\right )^{\frac {3}{2}}}{12 c} + \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}}}{28 c d^{2}}}{c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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