Optimal. Leaf size=193 \[ \frac{(a+b x)^5 \left (6 a^2 d^2 f^2-6 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )}{5 b^5}+\frac{d f (a+b x)^6 (-2 a d f+b c f+b d e)}{3 b^5}+\frac{(a+b x)^4 (b c-a d) (b e-a f) (-2 a d f+b c f+b d e)}{2 b^5}+\frac{(a+b x)^3 (b c-a d)^2 (b e-a f)^2}{3 b^5}+\frac{d^2 f^2 (a+b x)^7}{7 b^5} \]
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Rubi [A] time = 0.256423, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ \frac{(a+b x)^5 \left (6 a^2 d^2 f^2-6 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )}{5 b^5}+\frac{d f (a+b x)^6 (-2 a d f+b c f+b d e)}{3 b^5}+\frac{(a+b x)^4 (b c-a d) (b e-a f) (-2 a d f+b c f+b d e)}{2 b^5}+\frac{(a+b x)^3 (b c-a d)^2 (b e-a f)^2}{3 b^5}+\frac{d^2 f^2 (a+b x)^7}{7 b^5} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin{align*} \int (a+b x)^2 (c+d x)^2 (e+f x)^2 \, dx &=\int \left (\frac{(b c-a d)^2 (b e-a f)^2 (a+b x)^2}{b^4}+\frac{2 (b c-a d) (b e-a f) (b d e+b c f-2 a d f) (a+b x)^3}{b^4}+\frac{\left (6 a^2 d^2 f^2-6 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) (a+b x)^4}{b^4}+\frac{2 d f (b d e+b c f-2 a d f) (a+b x)^5}{b^4}+\frac{d^2 f^2 (a+b x)^6}{b^4}\right ) \, dx\\ &=\frac{(b c-a d)^2 (b e-a f)^2 (a+b x)^3}{3 b^5}+\frac{(b c-a d) (b e-a f) (b d e+b c f-2 a d f) (a+b x)^4}{2 b^5}+\frac{\left (6 a^2 d^2 f^2-6 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) (a+b x)^5}{5 b^5}+\frac{d f (b d e+b c f-2 a d f) (a+b x)^6}{3 b^5}+\frac{d^2 f^2 (a+b x)^7}{7 b^5}\\ \end{align*}
Mathematica [A] time = 0.0839336, size = 241, normalized size = 1.25 \[ \frac{1}{5} x^5 \left (a^2 d^2 f^2+4 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+\frac{1}{2} x^4 \left (a^2 d f (c f+d e)+a b \left (c^2 f^2+4 c d e f+d^2 e^2\right )+b^2 c e (c f+d e)\right )+\frac{1}{3} x^3 \left (a^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )+4 a b c e (c f+d e)+b^2 c^2 e^2\right )+a^2 c^2 e^2 x+\frac{1}{3} b d f x^6 (a d f+b c f+b d e)+a c e x^2 (a c f+a d e+b c e)+\frac{1}{7} b^2 d^2 f^2 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 286, normalized size = 1.5 \begin{align*}{\frac{{b}^{2}{d}^{2}{f}^{2}{x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){f}^{2}+2\,{b}^{2}{d}^{2}ef \right ){x}^{6}}{6}}+{\frac{ \left ( \left ({a}^{2}{d}^{2}+4\,abcd+{b}^{2}{c}^{2} \right ){f}^{2}+2\, \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ) ef+{b}^{2}{d}^{2}{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){f}^{2}+2\, \left ({a}^{2}{d}^{2}+4\,abcd+{b}^{2}{c}^{2} \right ) ef+ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{2}{c}^{2}{f}^{2}+2\, \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ) ef+ \left ({a}^{2}{d}^{2}+4\,abcd+{b}^{2}{c}^{2} \right ){e}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{c}^{2}ef+ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){e}^{2} \right ){x}^{2}}{2}}+{a}^{2}{c}^{2}{e}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07493, size = 363, normalized size = 1.88 \begin{align*} \frac{1}{7} \, b^{2} d^{2} f^{2} x^{7} + a^{2} c^{2} e^{2} x + \frac{1}{3} \,{\left (b^{2} d^{2} e f +{\left (b^{2} c d + a b d^{2}\right )} f^{2}\right )} x^{6} + \frac{1}{5} \,{\left (b^{2} d^{2} e^{2} + 4 \,{\left (b^{2} c d + a b d^{2}\right )} e f +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2}\right )} x^{5} + \frac{1}{2} \,{\left ({\left (b^{2} c d + a b d^{2}\right )} e^{2} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e f +{\left (a b c^{2} + a^{2} c d\right )} f^{2}\right )} x^{4} + \frac{1}{3} \,{\left (a^{2} c^{2} f^{2} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e^{2} + 4 \,{\left (a b c^{2} + a^{2} c d\right )} e f\right )} x^{3} +{\left (a^{2} c^{2} e f +{\left (a b c^{2} + a^{2} c d\right )} e^{2}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37963, size = 774, normalized size = 4.01 \begin{align*} \frac{1}{7} x^{7} f^{2} d^{2} b^{2} + \frac{1}{3} x^{6} f e d^{2} b^{2} + \frac{1}{3} x^{6} f^{2} d c b^{2} + \frac{1}{3} x^{6} f^{2} d^{2} b a + \frac{1}{5} x^{5} e^{2} d^{2} b^{2} + \frac{4}{5} x^{5} f e d c b^{2} + \frac{1}{5} x^{5} f^{2} c^{2} b^{2} + \frac{4}{5} x^{5} f e d^{2} b a + \frac{4}{5} x^{5} f^{2} d c b a + \frac{1}{5} x^{5} f^{2} d^{2} a^{2} + \frac{1}{2} x^{4} e^{2} d c b^{2} + \frac{1}{2} x^{4} f e c^{2} b^{2} + \frac{1}{2} x^{4} e^{2} d^{2} b a + 2 x^{4} f e d c b a + \frac{1}{2} x^{4} f^{2} c^{2} b a + \frac{1}{2} x^{4} f e d^{2} a^{2} + \frac{1}{2} x^{4} f^{2} d c a^{2} + \frac{1}{3} x^{3} e^{2} c^{2} b^{2} + \frac{4}{3} x^{3} e^{2} d c b a + \frac{4}{3} x^{3} f e c^{2} b a + \frac{1}{3} x^{3} e^{2} d^{2} a^{2} + \frac{4}{3} x^{3} f e d c a^{2} + \frac{1}{3} x^{3} f^{2} c^{2} a^{2} + x^{2} e^{2} c^{2} b a + x^{2} e^{2} d c a^{2} + x^{2} f e c^{2} a^{2} + x e^{2} c^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.100924, size = 345, normalized size = 1.79 \begin{align*} a^{2} c^{2} e^{2} x + \frac{b^{2} d^{2} f^{2} x^{7}}{7} + x^{6} \left (\frac{a b d^{2} f^{2}}{3} + \frac{b^{2} c d f^{2}}{3} + \frac{b^{2} d^{2} e f}{3}\right ) + x^{5} \left (\frac{a^{2} d^{2} f^{2}}{5} + \frac{4 a b c d f^{2}}{5} + \frac{4 a b d^{2} e f}{5} + \frac{b^{2} c^{2} f^{2}}{5} + \frac{4 b^{2} c d e f}{5} + \frac{b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} c d f^{2}}{2} + \frac{a^{2} d^{2} e f}{2} + \frac{a b c^{2} f^{2}}{2} + 2 a b c d e f + \frac{a b d^{2} e^{2}}{2} + \frac{b^{2} c^{2} e f}{2} + \frac{b^{2} c d e^{2}}{2}\right ) + x^{3} \left (\frac{a^{2} c^{2} f^{2}}{3} + \frac{4 a^{2} c d e f}{3} + \frac{a^{2} d^{2} e^{2}}{3} + \frac{4 a b c^{2} e f}{3} + \frac{4 a b c d e^{2}}{3} + \frac{b^{2} c^{2} e^{2}}{3}\right ) + x^{2} \left (a^{2} c^{2} e f + a^{2} c d e^{2} + a b c^{2} e^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.90346, size = 467, normalized size = 2.42 \begin{align*} \frac{1}{7} \, b^{2} d^{2} f^{2} x^{7} + \frac{1}{3} \, b^{2} c d f^{2} x^{6} + \frac{1}{3} \, a b d^{2} f^{2} x^{6} + \frac{1}{3} \, b^{2} d^{2} f x^{6} e + \frac{1}{5} \, b^{2} c^{2} f^{2} x^{5} + \frac{4}{5} \, a b c d f^{2} x^{5} + \frac{1}{5} \, a^{2} d^{2} f^{2} x^{5} + \frac{4}{5} \, b^{2} c d f x^{5} e + \frac{4}{5} \, a b d^{2} f x^{5} e + \frac{1}{2} \, a b c^{2} f^{2} x^{4} + \frac{1}{2} \, a^{2} c d f^{2} x^{4} + \frac{1}{5} \, b^{2} d^{2} x^{5} e^{2} + \frac{1}{2} \, b^{2} c^{2} f x^{4} e + 2 \, a b c d f x^{4} e + \frac{1}{2} \, a^{2} d^{2} f x^{4} e + \frac{1}{3} \, a^{2} c^{2} f^{2} x^{3} + \frac{1}{2} \, b^{2} c d x^{4} e^{2} + \frac{1}{2} \, a b d^{2} x^{4} e^{2} + \frac{4}{3} \, a b c^{2} f x^{3} e + \frac{4}{3} \, a^{2} c d f x^{3} e + \frac{1}{3} \, b^{2} c^{2} x^{3} e^{2} + \frac{4}{3} \, a b c d x^{3} e^{2} + \frac{1}{3} \, a^{2} d^{2} x^{3} e^{2} + a^{2} c^{2} f x^{2} e + a b c^{2} x^{2} e^{2} + a^{2} c d x^{2} e^{2} + a^{2} c^{2} x e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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