Optimal. Leaf size=65 \[ \frac{e (a+b x)^6 (b d-a e)}{3 b^3}+\frac{(a+b x)^5 (b d-a e)^2}{5 b^3}+\frac{e^2 (a+b x)^7}{7 b^3} \]
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Rubi [A] time = 0.086089, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac{e (a+b x)^6 (b d-a e)}{3 b^3}+\frac{(a+b x)^5 (b d-a e)^2}{5 b^3}+\frac{e^2 (a+b x)^7}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^2 \, dx\\ &=\int \left (\frac{(b d-a e)^2 (a+b x)^4}{b^2}+\frac{2 e (b d-a e) (a+b x)^5}{b^2}+\frac{e^2 (a+b x)^6}{b^2}\right ) \, dx\\ &=\frac{(b d-a e)^2 (a+b x)^5}{5 b^3}+\frac{e (b d-a e) (a+b x)^6}{3 b^3}+\frac{e^2 (a+b x)^7}{7 b^3}\\ \end{align*}
Mathematica [B] time = 0.0256762, size = 148, normalized size = 2.28 \[ \frac{1}{5} b^2 x^5 \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+a b x^4 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+\frac{1}{3} a^2 x^3 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+a^3 d x^2 (a e+2 b d)+a^4 d^2 x+\frac{1}{3} b^3 e x^6 (2 a e+b d)+\frac{1}{7} b^4 e^2 x^7 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 163, normalized size = 2.5 \begin{align*}{\frac{{e}^{2}{b}^{4}{x}^{7}}{7}}+{\frac{ \left ( 4\,{e}^{2}a{b}^{3}+2\,de{b}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,{e}^{2}{b}^{2}{a}^{2}+8\,dea{b}^{3}+{b}^{4}{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{e}^{2}{a}^{3}b+12\,de{b}^{2}{a}^{2}+4\,{d}^{2}a{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ({e}^{2}{a}^{4}+8\,de{a}^{3}b+6\,{d}^{2}{b}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{4}+4\,{d}^{2}{a}^{3}b \right ){x}^{2}}{2}}+{a}^{4}{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18639, size = 211, normalized size = 3.25 \begin{align*} \frac{1}{7} \, b^{4} e^{2} x^{7} + a^{4} d^{2} x + \frac{1}{3} \,{\left (b^{4} d e + 2 \, a b^{3} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{2} + 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x^{5} +{\left (a b^{3} d^{2} + 3 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, a^{2} b^{2} d^{2} + 8 \, a^{3} b d e + a^{4} e^{2}\right )} x^{3} +{\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50431, size = 363, normalized size = 5.58 \begin{align*} \frac{1}{7} x^{7} e^{2} b^{4} + \frac{1}{3} x^{6} e d b^{4} + \frac{2}{3} x^{6} e^{2} b^{3} a + \frac{1}{5} x^{5} d^{2} b^{4} + \frac{8}{5} x^{5} e d b^{3} a + \frac{6}{5} x^{5} e^{2} b^{2} a^{2} + x^{4} d^{2} b^{3} a + 3 x^{4} e d b^{2} a^{2} + x^{4} e^{2} b a^{3} + 2 x^{3} d^{2} b^{2} a^{2} + \frac{8}{3} x^{3} e d b a^{3} + \frac{1}{3} x^{3} e^{2} a^{4} + 2 x^{2} d^{2} b a^{3} + x^{2} e d a^{4} + x d^{2} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.097381, size = 168, normalized size = 2.58 \begin{align*} a^{4} d^{2} x + \frac{b^{4} e^{2} x^{7}}{7} + x^{6} \left (\frac{2 a b^{3} e^{2}}{3} + \frac{b^{4} d e}{3}\right ) + x^{5} \left (\frac{6 a^{2} b^{2} e^{2}}{5} + \frac{8 a b^{3} d e}{5} + \frac{b^{4} d^{2}}{5}\right ) + x^{4} \left (a^{3} b e^{2} + 3 a^{2} b^{2} d e + a b^{3} d^{2}\right ) + x^{3} \left (\frac{a^{4} e^{2}}{3} + \frac{8 a^{3} b d e}{3} + 2 a^{2} b^{2} d^{2}\right ) + x^{2} \left (a^{4} d e + 2 a^{3} b d^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12583, size = 230, normalized size = 3.54 \begin{align*} \frac{1}{7} \, b^{4} x^{7} e^{2} + \frac{1}{3} \, b^{4} d x^{6} e + \frac{1}{5} \, b^{4} d^{2} x^{5} + \frac{2}{3} \, a b^{3} x^{6} e^{2} + \frac{8}{5} \, a b^{3} d x^{5} e + a b^{3} d^{2} x^{4} + \frac{6}{5} \, a^{2} b^{2} x^{5} e^{2} + 3 \, a^{2} b^{2} d x^{4} e + 2 \, a^{2} b^{2} d^{2} x^{3} + a^{3} b x^{4} e^{2} + \frac{8}{3} \, a^{3} b d x^{3} e + 2 \, a^{3} b d^{2} x^{2} + \frac{1}{3} \, a^{4} x^{3} e^{2} + a^{4} d x^{2} e + a^{4} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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