Optimal. Leaf size=59 \[ \frac{4}{15} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}+\frac{2}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2} \]
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Rubi [A] time = 0.0262283, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {692, 629} \[ \frac{4}{15} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}+\frac{2}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 692
Rule 629
Rubi steps
\begin{align*} \int (b d+2 c d x)^3 \sqrt{a+b x+c x^2} \, dx &=\frac{2}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}+\frac{1}{5} \left (2 \left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \sqrt{a+b x+c x^2} \, dx\\ &=\frac{4}{15} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^{3/2}+\frac{2}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0378078, size = 44, normalized size = 0.75 \[ \frac{2}{15} d^3 (a+x (b+c x))^{3/2} \left (4 c \left (3 c x^2-2 a\right )+5 b^2+12 b c x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 41, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -24\,{c}^{2}{x}^{2}-24\,bcx+16\,ac-10\,{b}^{2} \right ){d}^{3}}{15} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25002, size = 198, normalized size = 3.36 \begin{align*} \frac{2}{15} \,{\left (12 \, c^{3} d^{3} x^{4} + 24 \, b c^{2} d^{3} x^{3} +{\left (17 \, b^{2} c + 4 \, a c^{2}\right )} d^{3} x^{2} +{\left (5 \, b^{3} + 4 \, a b c\right )} d^{3} x +{\left (5 \, a b^{2} - 8 \, a^{2} c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.497091, size = 216, normalized size = 3.66 \begin{align*} - \frac{16 a^{2} c d^{3} \sqrt{a + b x + c x^{2}}}{15} + \frac{2 a b^{2} d^{3} \sqrt{a + b x + c x^{2}}}{3} + \frac{8 a b c d^{3} x \sqrt{a + b x + c x^{2}}}{15} + \frac{8 a c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{15} + \frac{2 b^{3} d^{3} x \sqrt{a + b x + c x^{2}}}{3} + \frac{34 b^{2} c d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{15} + \frac{16 b c^{2} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{5} + \frac{8 c^{3} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19079, size = 163, normalized size = 2.76 \begin{align*} \frac{2}{15} \, \sqrt{c x^{2} + b x + a}{\left ({\left ({\left (12 \,{\left (c^{3} d^{3} x + 2 \, b c^{2} d^{3}\right )} x + \frac{17 \, b^{2} c^{5} d^{3} + 4 \, a c^{6} d^{3}}{c^{4}}\right )} x + \frac{5 \, b^{3} c^{4} d^{3} + 4 \, a b c^{5} d^{3}}{c^{4}}\right )} x + \frac{5 \, a b^{2} c^{4} d^{3} - 8 \, a^{2} c^{5} d^{3}}{c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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