Optimal. Leaf size=92 \[ -\frac{b^2 (c+d x)^6 (b c-a d)}{2 d^4}+\frac{3 b (c+d x)^5 (b c-a d)^2}{5 d^4}-\frac{(c+d x)^4 (b c-a d)^3}{4 d^4}+\frac{b^3 (c+d x)^7}{7 d^4} \]
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Rubi [A] time = 0.117105, antiderivative size = 92, 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} \[ -\frac{b^2 (c+d x)^6 (b c-a d)}{2 d^4}+\frac{3 b (c+d x)^5 (b c-a d)^2}{5 d^4}-\frac{(c+d x)^4 (b c-a d)^3}{4 d^4}+\frac{b^3 (c+d x)^7}{7 d^4} \]
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 )^3 \, dx &=\frac{\int (b c+b d x)^3 (a d+b d x)^3 \, dx}{b^3 d^3}\\ &=\frac{\int \left (-(b c-a d)^3 (b c+b d x)^3+3 (b c-a d)^2 (b c+b d x)^4-3 (b c-a d) (b c+b d x)^5+(b c+b d x)^6\right ) \, dx}{b^3 d^3}\\ &=-\frac{(b c-a d)^3 (c+d x)^4}{4 d^4}+\frac{3 b (b c-a d)^2 (c+d x)^5}{5 d^4}-\frac{b^2 (b c-a d) (c+d x)^6}{2 d^4}+\frac{b^3 (c+d x)^7}{7 d^4}\\ \end{align*}
Mathematica [A] time = 0.0179502, size = 161, normalized size = 1.75 \[ \frac{3}{5} b d x^5 \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{4} x^4 \left (9 a^2 b c d^2+a^3 d^3+9 a b^2 c^2 d+b^3 c^3\right )+a c x^3 \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{3}{2} a^2 c^2 x^2 (a d+b c)+a^3 c^3 x+\frac{1}{2} b^2 d^2 x^6 (a d+b c)+\frac{1}{7} b^3 d^3 x^7 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 194, normalized size = 2.1 \begin{align*}{\frac{{b}^{3}{d}^{3}{x}^{7}}{7}}+{\frac{ \left ( ad+bc \right ){b}^{2}{d}^{2}{x}^{6}}{2}}+{\frac{ \left ( a{b}^{2}c{d}^{2}+2\, \left ( ad+bc \right ) ^{2}bd+bd \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,ac \left ( ad+bc \right ) bd+ \left ( ad+bc \right ) \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( ac \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\, \left ( ad+bc \right ) ^{2}ac+bd{a}^{2}{c}^{2} \right ){x}^{3}}{3}}+{\frac{3\,{a}^{2}{c}^{2} \left ( ad+bc \right ){x}^{2}}{2}}+{a}^{3}{c}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0822, size = 189, normalized size = 2.05 \begin{align*} \frac{1}{7} \, b^{3} d^{3} x^{7} + \frac{1}{2} \,{\left (b c + a d\right )} b^{2} d^{2} x^{6} + \frac{3}{5} \,{\left (b c + a d\right )}^{2} b d x^{5} + a^{3} c^{3} x + \frac{1}{4} \,{\left (b c + a d\right )}^{3} x^{4} + \frac{1}{2} \,{\left (2 \, b d x^{3} + 3 \,{\left (b c + a d\right )} x^{2}\right )} a^{2} c^{2} + \frac{1}{10} \,{\left (6 \, b^{2} d^{2} x^{5} + 15 \,{\left (b c + a d\right )} b d x^{4} + 10 \,{\left (b c + a d\right )}^{2} x^{3}\right )} a c \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.39292, size = 409, normalized size = 4.45 \begin{align*} \frac{1}{7} x^{7} d^{3} b^{3} + \frac{1}{2} x^{6} d^{2} c b^{3} + \frac{1}{2} x^{6} d^{3} b^{2} a + \frac{3}{5} x^{5} d c^{2} b^{3} + \frac{9}{5} x^{5} d^{2} c b^{2} a + \frac{3}{5} x^{5} d^{3} b a^{2} + \frac{1}{4} x^{4} c^{3} b^{3} + \frac{9}{4} x^{4} d c^{2} b^{2} a + \frac{9}{4} x^{4} d^{2} c b a^{2} + \frac{1}{4} x^{4} d^{3} a^{3} + x^{3} c^{3} b^{2} a + 3 x^{3} d c^{2} b a^{2} + x^{3} d^{2} c a^{3} + \frac{3}{2} x^{2} c^{3} b a^{2} + \frac{3}{2} x^{2} d c^{2} a^{3} + x c^{3} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.219996, size = 190, normalized size = 2.07 \begin{align*} a^{3} c^{3} x + \frac{b^{3} d^{3} x^{7}}{7} + x^{6} \left (\frac{a b^{2} d^{3}}{2} + \frac{b^{3} c d^{2}}{2}\right ) + x^{5} \left (\frac{3 a^{2} b d^{3}}{5} + \frac{9 a b^{2} c d^{2}}{5} + \frac{3 b^{3} c^{2} d}{5}\right ) + x^{4} \left (\frac{a^{3} d^{3}}{4} + \frac{9 a^{2} b c d^{2}}{4} + \frac{9 a b^{2} c^{2} d}{4} + \frac{b^{3} c^{3}}{4}\right ) + x^{3} \left (a^{3} c d^{2} + 3 a^{2} b c^{2} d + a b^{2} c^{3}\right ) + x^{2} \left (\frac{3 a^{3} c^{2} d}{2} + \frac{3 a^{2} b c^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19604, size = 254, normalized size = 2.76 \begin{align*} \frac{1}{7} \, b^{3} d^{3} x^{7} + \frac{1}{2} \, b^{3} c d^{2} x^{6} + \frac{1}{2} \, a b^{2} d^{3} x^{6} + \frac{3}{5} \, b^{3} c^{2} d x^{5} + \frac{9}{5} \, a b^{2} c d^{2} x^{5} + \frac{3}{5} \, a^{2} b d^{3} x^{5} + \frac{1}{4} \, b^{3} c^{3} x^{4} + \frac{9}{4} \, a b^{2} c^{2} d x^{4} + \frac{9}{4} \, a^{2} b c d^{2} x^{4} + \frac{1}{4} \, a^{3} d^{3} x^{4} + a b^{2} c^{3} x^{3} + 3 \, a^{2} b c^{2} d x^{3} + a^{3} c d^{2} x^{3} + \frac{3}{2} \, a^{2} b c^{3} x^{2} + \frac{3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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