Optimal. Leaf size=59 \[ \frac{4}{3} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}+\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2} \]
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Rubi [A] time = 0.0259978, 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} \[ \frac{4}{3} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}+\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2} \]
Antiderivative was successfully verified.
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Rule 692
Rule 629
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^3}{\sqrt{a+b x+c x^2}} \, dx &=\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}+\frac{1}{3} \left (2 \left (b^2-4 a c\right ) d^2\right ) \int \frac{b d+2 c d x}{\sqrt{a+b x+c x^2}} \, dx\\ &=\frac{4}{3} \left (b^2-4 a c\right ) d^3 \sqrt{a+b x+c x^2}+\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}\\ \end{align*}
Mathematica [A] time = 0.0355574, size = 43, normalized size = 0.73 \[ \frac{2}{3} d^3 \sqrt{a+x (b+c x)} \left (4 c \left (c x^2-2 a\right )+3 b^2+4 b c x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 41, normalized size = 0.7 \begin{align*} -{\frac{2\,{d}^{3} \left ( -4\,{c}^{2}{x}^{2}-4\,bcx+8\,ac-3\,{b}^{2} \right ) }{3}\sqrt{c{x}^{2}+bx+a}} \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 = 2.84848, size = 107, normalized size = 1.81 \begin{align*} \frac{2}{3} \,{\left (4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (3 \, b^{2} - 8 \, a 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 [A] time = 0.387048, size = 97, normalized size = 1.64 \begin{align*} - \frac{16 a c d^{3} \sqrt{a + b x + c x^{2}}}{3} + 2 b^{2} d^{3} \sqrt{a + b x + c x^{2}} + \frac{8 b c d^{3} x \sqrt{a + b x + c x^{2}}}{3} + \frac{8 c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18905, size = 78, normalized size = 1.32 \begin{align*} \frac{2}{3} \, \sqrt{c x^{2} + b x + a}{\left (4 \,{\left (c^{2} d^{3} x + b c d^{3}\right )} x + \frac{3 \, b^{2} c^{2} d^{3} - 8 \, a c^{3} d^{3}}{c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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