Optimal. Leaf size=55 \[ \frac {(b d+2 c d x)^{9/2}}{36 c^2 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{20 c^2 d} \]
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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} \[ \frac {(b d+2 c d x)^{9/2}}{36 c^2 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{20 c^2 d} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^{3/2}}{4 c}+\frac {(b d+2 c d x)^{7/2}}{4 c d^2}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{20 c^2 d}+\frac {(b d+2 c d x)^{9/2}}{36 c^2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 0.82 \[ \frac {\left (c \left (9 a+5 c x^2\right )-b^2+5 b c x\right ) (d (b+2 c x))^{5/2}}{45 c^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 87, normalized size = 1.58 \[ \frac {{\left (20 \, c^{4} d x^{4} + 40 \, b c^{3} d x^{3} + 3 \, {\left (7 \, b^{2} c^{2} + 12 \, a c^{3}\right )} d x^{2} + {\left (b^{3} c + 36 \, a b c^{2}\right )} d x - {\left (b^{4} - 9 \, a b^{2} c\right )} d\right )} \sqrt {2 \, c d x + b d}}{45 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 376, normalized size = 6.84 \[ \frac {1260 \, \sqrt {2 \, c d x + b d} a b^{2} d - 840 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} a b - \frac {210 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} b^{3}}{c} + \frac {84 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} a}{d} + \frac {105 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} b^{2}}{c d} - \frac {36 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b}{c d^{2}} + \frac {315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{c d^{3}}}{1260 \, c} \]
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 \[ \frac {\left (2 c x +b \right ) \left (5 c^{2} x^{2}+5 b c x +9 a c -b^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{45 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 46, normalized size = 0.84 \[ -\frac {9 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{180 \, c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 39, normalized size = 0.71 \[ \frac {{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (36\,a\,c+5\,{\left (b+2\,c\,x\right )}^2-9\,b^2\right )}{180\,c^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.76, size = 274, normalized size = 4.98 \[ a b d \left (\begin {cases} x \sqrt {b d} & \text {for}\: c = 0 \\0 & \text {for}\: d = 0 \\\frac {\left (b d + 2 c d x\right )^{\frac {3}{2}}}{3 c d} & \text {otherwise} \end {cases}\right ) + \frac {a \left (- \frac {b d \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} + \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}}}{5}\right )}{c d} + \frac {b^{2} \left (- \frac {b d \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} + \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}}}{5}\right )}{2 c^{2} d} + \frac {3 b \left (\frac {b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} - \frac {2 b d \left (b d + 2 c d x\right )^{\frac {5}{2}}}{5} + \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}}}{7}\right )}{4 c^{2} d^{2}} + \frac {- \frac {b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac {3}{2}}}{3} + \frac {3 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac {5}{2}}}{5} - \frac {3 b d \left (b d + 2 c d x\right )^{\frac {7}{2}}}{7} + \frac {\left (b d + 2 c d x\right )^{\frac {9}{2}}}{9}}{4 c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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