Optimal. Leaf size=92 \[ \frac{3 e^2 (a+b x)^7 (b d-a e)}{7 b^4}+\frac{e (a+b x)^6 (b d-a e)^2}{2 b^4}+\frac{(a+b x)^5 (b d-a e)^3}{5 b^4}+\frac{e^3 (a+b x)^8}{8 b^4} \]
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Rubi [A] time = 0.130013, antiderivative size = 92, 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} \[ \frac{3 e^2 (a+b x)^7 (b d-a e)}{7 b^4}+\frac{e (a+b x)^6 (b d-a e)^2}{2 b^4}+\frac{(a+b x)^5 (b d-a e)^3}{5 b^4}+\frac{e^3 (a+b x)^8}{8 b^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^3 \, dx\\ &=\int \left (\frac{(b d-a e)^3 (a+b x)^4}{b^3}+\frac{3 e (b d-a e)^2 (a+b x)^5}{b^3}+\frac{3 e^2 (b d-a e) (a+b x)^6}{b^3}+\frac{e^3 (a+b x)^7}{b^3}\right ) \, dx\\ &=\frac{(b d-a e)^3 (a+b x)^5}{5 b^4}+\frac{e (b d-a e)^2 (a+b x)^6}{2 b^4}+\frac{3 e^2 (b d-a e) (a+b x)^7}{7 b^4}+\frac{e^3 (a+b x)^8}{8 b^4}\\ \end{align*}
Mathematica [B] time = 0.0311339, size = 217, normalized size = 2.36 \[ \frac{1}{2} b^2 e x^6 \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+\frac{1}{5} b x^5 \left (18 a^2 b d e^2+4 a^3 e^3+12 a b^2 d^2 e+b^3 d^3\right )+\frac{1}{4} a x^4 \left (12 a^2 b d e^2+a^3 e^3+18 a b^2 d^2 e+4 b^3 d^3\right )+a^2 d x^3 \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+\frac{1}{2} a^3 d^2 x^2 (3 a e+4 b d)+a^4 d^3 x+\frac{1}{7} b^3 e^2 x^7 (4 a e+3 b d)+\frac{1}{8} b^4 e^3 x^8 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 229, normalized size = 2.5 \begin{align*}{\frac{{e}^{3}{b}^{4}{x}^{8}}{8}}+{\frac{ \left ( 4\,{e}^{3}a{b}^{3}+3\,d{e}^{2}{b}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{e}^{3}{b}^{2}{a}^{2}+12\,d{e}^{2}a{b}^{3}+3\,{d}^{2}e{b}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,{e}^{3}{a}^{3}b+18\,d{e}^{2}{b}^{2}{a}^{2}+12\,{d}^{2}ea{b}^{3}+{d}^{3}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ({e}^{3}{a}^{4}+12\,d{e}^{2}{a}^{3}b+18\,{d}^{2}e{b}^{2}{a}^{2}+4\,{d}^{3}a{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{4}+12\,{d}^{2}e{a}^{3}b+6\,{d}^{3}{b}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{4}+4\,{d}^{3}{a}^{3}b \right ){x}^{2}}{2}}+{a}^{4}{d}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18706, size = 304, normalized size = 3.3 \begin{align*} \frac{1}{8} \, b^{4} e^{3} x^{8} + a^{4} d^{3} x + \frac{1}{7} \,{\left (3 \, b^{4} d e^{2} + 4 \, a b^{3} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (b^{4} d^{2} e + 4 \, a b^{3} d e^{2} + 2 \, a^{2} b^{2} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{3} + 12 \, a b^{3} d^{2} e + 18 \, a^{2} b^{2} d e^{2} + 4 \, a^{3} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, a b^{3} d^{3} + 18 \, a^{2} b^{2} d^{2} e + 12 \, a^{3} b d e^{2} + a^{4} e^{3}\right )} x^{4} +{\left (2 \, a^{2} b^{2} d^{3} + 4 \, a^{3} b d^{2} e + a^{4} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d^{3} + 3 \, a^{4} d^{2} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.51139, size = 520, normalized size = 5.65 \begin{align*} \frac{1}{8} x^{8} e^{3} b^{4} + \frac{3}{7} x^{7} e^{2} d b^{4} + \frac{4}{7} x^{7} e^{3} b^{3} a + \frac{1}{2} x^{6} e d^{2} b^{4} + 2 x^{6} e^{2} d b^{3} a + x^{6} e^{3} b^{2} a^{2} + \frac{1}{5} x^{5} d^{3} b^{4} + \frac{12}{5} x^{5} e d^{2} b^{3} a + \frac{18}{5} x^{5} e^{2} d b^{2} a^{2} + \frac{4}{5} x^{5} e^{3} b a^{3} + x^{4} d^{3} b^{3} a + \frac{9}{2} x^{4} e d^{2} b^{2} a^{2} + 3 x^{4} e^{2} d b a^{3} + \frac{1}{4} x^{4} e^{3} a^{4} + 2 x^{3} d^{3} b^{2} a^{2} + 4 x^{3} e d^{2} b a^{3} + x^{3} e^{2} d a^{4} + 2 x^{2} d^{3} b a^{3} + \frac{3}{2} x^{2} e d^{2} a^{4} + x d^{3} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.109806, size = 243, normalized size = 2.64 \begin{align*} a^{4} d^{3} x + \frac{b^{4} e^{3} x^{8}}{8} + x^{7} \left (\frac{4 a b^{3} e^{3}}{7} + \frac{3 b^{4} d e^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} e^{3} + 2 a b^{3} d e^{2} + \frac{b^{4} d^{2} e}{2}\right ) + x^{5} \left (\frac{4 a^{3} b e^{3}}{5} + \frac{18 a^{2} b^{2} d e^{2}}{5} + \frac{12 a b^{3} d^{2} e}{5} + \frac{b^{4} d^{3}}{5}\right ) + x^{4} \left (\frac{a^{4} e^{3}}{4} + 3 a^{3} b d e^{2} + \frac{9 a^{2} b^{2} d^{2} e}{2} + a b^{3} d^{3}\right ) + x^{3} \left (a^{4} d e^{2} + 4 a^{3} b d^{2} e + 2 a^{2} b^{2} d^{3}\right ) + x^{2} \left (\frac{3 a^{4} d^{2} e}{2} + 2 a^{3} b d^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16648, size = 324, normalized size = 3.52 \begin{align*} \frac{1}{8} \, b^{4} x^{8} e^{3} + \frac{3}{7} \, b^{4} d x^{7} e^{2} + \frac{1}{2} \, b^{4} d^{2} x^{6} e + \frac{1}{5} \, b^{4} d^{3} x^{5} + \frac{4}{7} \, a b^{3} x^{7} e^{3} + 2 \, a b^{3} d x^{6} e^{2} + \frac{12}{5} \, a b^{3} d^{2} x^{5} e + a b^{3} d^{3} x^{4} + a^{2} b^{2} x^{6} e^{3} + \frac{18}{5} \, a^{2} b^{2} d x^{5} e^{2} + \frac{9}{2} \, a^{2} b^{2} d^{2} x^{4} e + 2 \, a^{2} b^{2} d^{3} x^{3} + \frac{4}{5} \, a^{3} b x^{5} e^{3} + 3 \, a^{3} b d x^{4} e^{2} + 4 \, a^{3} b d^{2} x^{3} e + 2 \, a^{3} b d^{3} x^{2} + \frac{1}{4} \, a^{4} x^{4} e^{3} + a^{4} d x^{3} e^{2} + \frac{3}{2} \, a^{4} d^{2} x^{2} e + a^{4} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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