Optimal. Leaf size=65 \[ \frac{2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac{(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac{d^2 (a+b x)^8}{8 b^3} \]
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Rubi [A] time = 0.129975, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {626, 43} \[ \frac{2 d (a+b x)^7 (b c-a d)}{7 b^3}+\frac{(a+b x)^6 (b c-a d)^2}{6 b^3}+\frac{d^2 (a+b x)^8}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx &=\int (a+b x)^5 (c+d x)^2 \, dx\\ &=\int \left (\frac{(b c-a d)^2 (a+b x)^5}{b^2}+\frac{2 d (b c-a d) (a+b x)^6}{b^2}+\frac{d^2 (a+b x)^7}{b^2}\right ) \, dx\\ &=\frac{(b c-a d)^2 (a+b x)^6}{6 b^3}+\frac{2 d (b c-a d) (a+b x)^7}{7 b^3}+\frac{d^2 (a+b x)^8}{8 b^3}\\ \end{align*}
Mathematica [B] time = 0.0313715, size = 189, normalized size = 2.91 \[ \frac{1}{6} b^3 x^6 \left (10 a^2 d^2+10 a b c d+b^2 c^2\right )+a b^2 x^5 \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{5}{4} a^2 b x^4 \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )+\frac{1}{3} a^3 x^3 \left (a^2 d^2+10 a b c d+10 b^2 c^2\right )+\frac{1}{2} a^4 c x^2 (2 a d+5 b c)+a^5 c^2 x+\frac{1}{7} b^4 d x^7 (5 a d+2 b c)+\frac{1}{8} b^5 d^2 x^8 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 315, normalized size = 4.9 \begin{align*}{\frac{{b}^{5}{d}^{2}{x}^{8}}{8}}+{\frac{ \left ( 3\,{b}^{4}a{d}^{2}+2\,{b}^{4} \left ( ad+bc \right ) d \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{b}^{3}{a}^{2}{d}^{2}+6\,{b}^{3}a \left ( ad+bc \right ) d+{b}^{3} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{3}{b}^{2}{d}^{2}+6\,{b}^{2}{a}^{2} \left ( ad+bc \right ) d+3\,{b}^{2}a \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\,{b}^{3}ac \left ( ad+bc \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{3} \left ( ad+bc \right ) bd+3\,b{a}^{2} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +6\,{b}^{2}{a}^{2}c \left ( ad+bc \right ) +{a}^{2}{b}^{3}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +6\,b{a}^{3}c \left ( ad+bc \right ) +3\,{b}^{2}{a}^{3}{c}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{4}c \left ( ad+bc \right ) +3\,b{a}^{4}{c}^{2} \right ){x}^{2}}{2}}+{a}^{5}{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0531, size = 266, normalized size = 4.09 \begin{align*} \frac{1}{8} \, b^{5} d^{2} x^{8} + a^{5} c^{2} x + \frac{1}{7} \,{\left (2 \, b^{5} c d + 5 \, a b^{4} d^{2}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} c^{2} + 10 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x^{6} +{\left (a b^{4} c^{2} + 4 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, a^{2} b^{3} c^{2} + 4 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} c^{2} + 10 \, a^{4} b c d + a^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.57015, size = 460, normalized size = 7.08 \begin{align*} \frac{1}{8} x^{8} d^{2} b^{5} + \frac{2}{7} x^{7} d c b^{5} + \frac{5}{7} x^{7} d^{2} b^{4} a + \frac{1}{6} x^{6} c^{2} b^{5} + \frac{5}{3} x^{6} d c b^{4} a + \frac{5}{3} x^{6} d^{2} b^{3} a^{2} + x^{5} c^{2} b^{4} a + 4 x^{5} d c b^{3} a^{2} + 2 x^{5} d^{2} b^{2} a^{3} + \frac{5}{2} x^{4} c^{2} b^{3} a^{2} + 5 x^{4} d c b^{2} a^{3} + \frac{5}{4} x^{4} d^{2} b a^{4} + \frac{10}{3} x^{3} c^{2} b^{2} a^{3} + \frac{10}{3} x^{3} d c b a^{4} + \frac{1}{3} x^{3} d^{2} a^{5} + \frac{5}{2} x^{2} c^{2} b a^{4} + x^{2} d c a^{5} + x c^{2} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.219368, size = 218, normalized size = 3.35 \begin{align*} a^{5} c^{2} x + \frac{b^{5} d^{2} x^{8}}{8} + x^{7} \left (\frac{5 a b^{4} d^{2}}{7} + \frac{2 b^{5} c d}{7}\right ) + x^{6} \left (\frac{5 a^{2} b^{3} d^{2}}{3} + \frac{5 a b^{4} c d}{3} + \frac{b^{5} c^{2}}{6}\right ) + x^{5} \left (2 a^{3} b^{2} d^{2} + 4 a^{2} b^{3} c d + a b^{4} c^{2}\right ) + x^{4} \left (\frac{5 a^{4} b d^{2}}{4} + 5 a^{3} b^{2} c d + \frac{5 a^{2} b^{3} c^{2}}{2}\right ) + x^{3} \left (\frac{a^{5} d^{2}}{3} + \frac{10 a^{4} b c d}{3} + \frac{10 a^{3} b^{2} c^{2}}{3}\right ) + x^{2} \left (a^{5} c d + \frac{5 a^{4} b c^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15711, size = 286, normalized size = 4.4 \begin{align*} \frac{1}{8} \, b^{5} d^{2} x^{8} + \frac{2}{7} \, b^{5} c d x^{7} + \frac{5}{7} \, a b^{4} d^{2} x^{7} + \frac{1}{6} \, b^{5} c^{2} x^{6} + \frac{5}{3} \, a b^{4} c d x^{6} + \frac{5}{3} \, a^{2} b^{3} d^{2} x^{6} + a b^{4} c^{2} x^{5} + 4 \, a^{2} b^{3} c d x^{5} + 2 \, a^{3} b^{2} d^{2} x^{5} + \frac{5}{2} \, a^{2} b^{3} c^{2} x^{4} + 5 \, a^{3} b^{2} c d x^{4} + \frac{5}{4} \, a^{4} b d^{2} x^{4} + \frac{10}{3} \, a^{3} b^{2} c^{2} x^{3} + \frac{10}{3} \, a^{4} b c d x^{3} + \frac{1}{3} \, a^{5} d^{2} x^{3} + \frac{5}{2} \, a^{4} b c^{2} x^{2} + a^{5} c d x^{2} + a^{5} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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