Optimal. Leaf size=65 \[ \frac{d (a+b x)^6 (b c-a d)}{3 b^3}+\frac{(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac{d^2 (a+b x)^7}{7 b^3} \]
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Rubi [A] time = 0.0897019, 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} \[ \frac{d (a+b x)^6 (b c-a d)}{3 b^3}+\frac{(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac{d^2 (a+b x)^7}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx &=\int (a+b x)^4 (c+d x)^2 \, dx\\ &=\int \left (\frac{(b c-a d)^2 (a+b x)^4}{b^2}+\frac{2 d (b c-a d) (a+b x)^5}{b^2}+\frac{d^2 (a+b x)^6}{b^2}\right ) \, dx\\ &=\frac{(b c-a d)^2 (a+b x)^5}{5 b^3}+\frac{d (b c-a d) (a+b x)^6}{3 b^3}+\frac{d^2 (a+b x)^7}{7 b^3}\\ \end{align*}
Mathematica [B] time = 0.0200078, size = 148, normalized size = 2.28 \[ \frac{1}{5} b^2 x^5 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a b x^4 \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{3} a^2 x^3 \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+a^3 c x^2 (a d+2 b c)+a^4 c^2 x+\frac{1}{3} b^3 d x^6 (2 a d+b c)+\frac{1}{7} b^4 d^2 x^7 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 231, normalized size = 3.6 \begin{align*}{\frac{{b}^{4}{d}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{d}^{2}a{b}^{3}+2\,{b}^{3} \left ( ad+bc \right ) d \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}{b}^{2}{d}^{2}+4\,a{b}^{2} \left ( ad+bc \right ) d+{b}^{2} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{2} \left ( ad+bc \right ) bd+2\,ab \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +2\,{b}^{2}ac \left ( ad+bc \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{2} \left ( 2\,cabd+ \left ( ad+bc \right ) ^{2} \right ) +4\,{a}^{2}bc \left ( ad+bc \right ) +{a}^{2}{b}^{2}{c}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{3}c \left ( ad+bc \right ) +2\,{a}^{3}b{c}^{2} \right ){x}^{2}}{2}}+{a}^{4}{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04526, size = 211, normalized size = 3.25 \begin{align*} \frac{1}{7} \, b^{4} d^{2} x^{7} + a^{4} c^{2} x + \frac{1}{3} \,{\left (b^{4} c d + 2 \, a b^{3} d^{2}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} c^{2} + 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x^{5} +{\left (a b^{3} c^{2} + 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{3} +{\left (2 \, a^{3} b c^{2} + a^{4} c d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68828, size = 363, normalized size = 5.58 \begin{align*} \frac{1}{7} x^{7} d^{2} b^{4} + \frac{1}{3} x^{6} d c b^{4} + \frac{2}{3} x^{6} d^{2} b^{3} a + \frac{1}{5} x^{5} c^{2} b^{4} + \frac{8}{5} x^{5} d c b^{3} a + \frac{6}{5} x^{5} d^{2} b^{2} a^{2} + x^{4} c^{2} b^{3} a + 3 x^{4} d c b^{2} a^{2} + x^{4} d^{2} b a^{3} + 2 x^{3} c^{2} b^{2} a^{2} + \frac{8}{3} x^{3} d c b a^{3} + \frac{1}{3} x^{3} d^{2} a^{4} + 2 x^{2} c^{2} b a^{3} + x^{2} d c a^{4} + x c^{2} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.126889, size = 168, normalized size = 2.58 \begin{align*} a^{4} c^{2} x + \frac{b^{4} d^{2} x^{7}}{7} + x^{6} \left (\frac{2 a b^{3} d^{2}}{3} + \frac{b^{4} c d}{3}\right ) + x^{5} \left (\frac{6 a^{2} b^{2} d^{2}}{5} + \frac{8 a b^{3} c d}{5} + \frac{b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac{a^{4} d^{2}}{3} + \frac{8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.191, size = 230, normalized size = 3.54 \begin{align*} \frac{1}{7} \, b^{4} d^{2} x^{7} + \frac{1}{3} \, b^{4} c d x^{6} + \frac{2}{3} \, a b^{3} d^{2} x^{6} + \frac{1}{5} \, b^{4} c^{2} x^{5} + \frac{8}{5} \, a b^{3} c d x^{5} + \frac{6}{5} \, a^{2} b^{2} d^{2} x^{5} + a b^{3} c^{2} x^{4} + 3 \, a^{2} b^{2} c d x^{4} + a^{3} b d^{2} x^{4} + 2 \, a^{2} b^{2} c^{2} x^{3} + \frac{8}{3} \, a^{3} b c d x^{3} + \frac{1}{3} \, a^{4} d^{2} x^{3} + 2 \, a^{3} b c^{2} x^{2} + a^{4} c d x^{2} + a^{4} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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