Optimal. Leaf size=65 \[ -\frac{b (c+d x)^4 (b c-a d)}{2 d^3}+\frac{(c+d x)^3 (b c-a d)^2}{3 d^3}+\frac{b^2 (c+d x)^5}{5 d^3} \]
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Rubi [A] time = 0.0692441, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {610, 43} \[ -\frac{b (c+d x)^4 (b c-a d)}{2 d^3}+\frac{(c+d x)^3 (b c-a d)^2}{3 d^3}+\frac{b^2 (c+d x)^5}{5 d^3} \]
Antiderivative was successfully verified.
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Rule 610
Rule 43
Rubi steps
\begin{align*} \int \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx &=\frac{\int (b c+b d x)^2 (a d+b d x)^2 \, dx}{b^2 d^2}\\ &=\frac{\int \left ((b c-a d)^2 (b c+b d x)^2-2 (b c-a d) (b c+b d x)^3+(b c+b d x)^4\right ) \, dx}{b^2 d^2}\\ &=\frac{(b c-a d)^2 (c+d x)^3}{3 d^3}-\frac{b (b c-a d) (c+d x)^4}{2 d^3}+\frac{b^2 (c+d x)^5}{5 d^3}\\ \end{align*}
Mathematica [A] time = 0.0099373, size = 79, normalized size = 1.22 \[ \frac{1}{3} x^3 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{1}{2} b d x^4 (a d+b c)+a c x^2 (a d+b c)+\frac{1}{5} b^2 d^2 x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 69, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{5}}{5}}+{\frac{ \left ( ad+bc \right ) bd{x}^{4}}{2}}+{\frac{ \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ){x}^{3}}{3}}+ac \left ( ad+bc \right ){x}^{2}+{a}^{2}{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04471, size = 97, normalized size = 1.49 \begin{align*} \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{1}{2} \,{\left (b c + a d\right )} b d x^{4} + a^{2} c^{2} x + \frac{1}{3} \,{\left (b c + a d\right )}^{2} x^{3} + \frac{1}{3} \,{\left (2 \, b d x^{3} + 3 \,{\left (b c + a d\right )} x^{2}\right )} a c \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4657, size = 198, normalized size = 3.05 \begin{align*} \frac{1}{5} x^{5} d^{2} b^{2} + \frac{1}{2} x^{4} d c b^{2} + \frac{1}{2} x^{4} d^{2} b a + \frac{1}{3} x^{3} c^{2} b^{2} + \frac{4}{3} x^{3} d c b a + \frac{1}{3} x^{3} d^{2} a^{2} + x^{2} c^{2} b a + x^{2} d c a^{2} + x c^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.086148, size = 87, normalized size = 1.34 \begin{align*} a^{2} c^{2} x + \frac{b^{2} d^{2} x^{5}}{5} + x^{4} \left (\frac{a b d^{2}}{2} + \frac{b^{2} c d}{2}\right ) + x^{3} \left (\frac{a^{2} d^{2}}{3} + \frac{4 a b c d}{3} + \frac{b^{2} c^{2}}{3}\right ) + x^{2} \left (a^{2} c d + a b c^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2152, size = 120, normalized size = 1.85 \begin{align*} \frac{1}{5} \, b^{2} d^{2} x^{5} + \frac{1}{2} \, b^{2} c d x^{4} + \frac{1}{2} \, a b d^{2} x^{4} + \frac{1}{3} \, b^{2} c^{2} x^{3} + \frac{4}{3} \, a b c d x^{3} + \frac{1}{3} \, a^{2} d^{2} x^{3} + a b c^{2} x^{2} + a^{2} c d x^{2} + a^{2} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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