Optimal. Leaf size=92 \[ \frac{d^2 (a+b x)^6 (b c-a d)}{2 b^4}+\frac{3 d (a+b x)^5 (b c-a d)^2}{5 b^4}+\frac{(a+b x)^4 (b c-a d)^3}{4 b^4}+\frac{d^3 (a+b x)^7}{7 b^4} \]
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Rubi [A] time = 0.0826116, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{d^2 (a+b x)^6 (b c-a d)}{2 b^4}+\frac{3 d (a+b x)^5 (b c-a d)^2}{5 b^4}+\frac{(a+b x)^4 (b c-a d)^3}{4 b^4}+\frac{d^3 (a+b x)^7}{7 b^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^3 (c+d x)^3 \, dx &=\int \left (\frac{(b c-a d)^3 (a+b x)^3}{b^3}+\frac{3 d (b c-a d)^2 (a+b x)^4}{b^3}+\frac{3 d^2 (b c-a d) (a+b x)^5}{b^3}+\frac{d^3 (a+b x)^6}{b^3}\right ) \, dx\\ &=\frac{(b c-a d)^3 (a+b x)^4}{4 b^4}+\frac{3 d (b c-a d)^2 (a+b x)^5}{5 b^4}+\frac{d^2 (b c-a d) (a+b x)^6}{2 b^4}+\frac{d^3 (a+b x)^7}{7 b^4}\\ \end{align*}
Mathematica [A] time = 0.0212754, 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., size = 177, normalized size = 1.9 \begin{align*}{\frac{{b}^{3}{d}^{3}{x}^{7}}{7}}+{\frac{ \left ( 3\,a{b}^{2}{d}^{3}+3\,{b}^{3}c{d}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,{a}^{2}b{d}^{3}+9\,a{b}^{2}c{d}^{2}+3\,{b}^{3}{c}^{2}d \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{3}{d}^{3}+9\,{a}^{2}bc{d}^{2}+9\,a{b}^{2}{c}^{2}d+{b}^{3}{c}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{3}c{d}^{2}+9\,{a}^{2}b{c}^{2}d+3\,a{b}^{2}{c}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{3}{c}^{2}d+3\,{a}^{2}b{c}^{3} \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 = 0.961389, size = 225, normalized size = 2.45 \begin{align*} \frac{1}{7} \, b^{3} d^{3} x^{7} + a^{3} c^{3} x + \frac{1}{2} \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{6} + \frac{3}{5} \,{\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{4} +{\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{3} + \frac{3}{2} \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.803, 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 = 0.086416, 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 = 1.0964, 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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