Optimal. Leaf size=119 \[ \frac{e^3 (a+b x)^8 (b d-a e)}{2 b^5}+\frac{6 e^2 (a+b x)^7 (b d-a e)^2}{7 b^5}+\frac{2 e (a+b x)^6 (b d-a e)^3}{3 b^5}+\frac{(a+b x)^5 (b d-a e)^4}{5 b^5}+\frac{e^4 (a+b x)^9}{9 b^5} \]
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Rubi [A] time = 0.164931, antiderivative size = 119, 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} \[ \frac{e^3 (a+b x)^8 (b d-a e)}{2 b^5}+\frac{6 e^2 (a+b x)^7 (b d-a e)^2}{7 b^5}+\frac{2 e (a+b x)^6 (b d-a e)^3}{3 b^5}+\frac{(a+b x)^5 (b d-a e)^4}{5 b^5}+\frac{e^4 (a+b x)^9}{9 b^5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^4 \, dx\\ &=\int \left (\frac{(b d-a e)^4 (a+b x)^4}{b^4}+\frac{4 e (b d-a e)^3 (a+b x)^5}{b^4}+\frac{6 e^2 (b d-a e)^2 (a+b x)^6}{b^4}+\frac{4 e^3 (b d-a e) (a+b x)^7}{b^4}+\frac{e^4 (a+b x)^8}{b^4}\right ) \, dx\\ &=\frac{(b d-a e)^4 (a+b x)^5}{5 b^5}+\frac{2 e (b d-a e)^3 (a+b x)^6}{3 b^5}+\frac{6 e^2 (b d-a e)^2 (a+b x)^7}{7 b^5}+\frac{e^3 (b d-a e) (a+b x)^8}{2 b^5}+\frac{e^4 (a+b x)^9}{9 b^5}\\ \end{align*}
Mathematica [B] time = 0.0385267, size = 273, normalized size = 2.29 \[ \frac{2}{7} b^2 e^2 x^7 \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+\frac{2}{3} b e x^6 \left (6 a^2 b d e^2+a^3 e^3+6 a b^2 d^2 e+b^3 d^3\right )+\frac{1}{5} x^5 \left (36 a^2 b^2 d^2 e^2+16 a^3 b d e^3+a^4 e^4+16 a b^3 d^3 e+b^4 d^4\right )+a d x^4 \left (6 a^2 b d e^2+a^3 e^3+6 a b^2 d^2 e+b^3 d^3\right )+\frac{2}{3} a^2 d^2 x^3 \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+2 a^3 d^3 x^2 (a e+b d)+a^4 d^4 x+\frac{1}{2} b^3 e^3 x^8 (a e+b d)+\frac{1}{9} b^4 e^4 x^9 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 295, normalized size = 2.5 \begin{align*}{\frac{{b}^{4}{e}^{4}{x}^{9}}{9}}+{\frac{ \left ( 4\,{e}^{4}a{b}^{3}+4\,{b}^{4}d{e}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 6\,{e}^{4}{b}^{2}{a}^{2}+16\,d{e}^{3}a{b}^{3}+6\,{d}^{2}{e}^{2}{b}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,{e}^{4}{a}^{3}b+24\,d{e}^{3}{b}^{2}{a}^{2}+24\,{d}^{2}{e}^{2}a{b}^{3}+4\,{d}^{3}e{b}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ({e}^{4}{a}^{4}+16\,d{e}^{3}{a}^{3}b+36\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}+16\,{d}^{3}ea{b}^{3}+{d}^{4}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,d{e}^{3}{a}^{4}+24\,{d}^{2}{e}^{2}{a}^{3}b+24\,{d}^{3}e{b}^{2}{a}^{2}+4\,{d}^{4}a{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{a}^{4}+16\,{d}^{3}e{a}^{3}b+6\,{d}^{4}{b}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{d}^{3}e{a}^{4}+4\,{d}^{4}{a}^{3}b \right ){x}^{2}}{2}}+{a}^{4}{d}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09669, size = 385, normalized size = 3.24 \begin{align*} \frac{1}{9} \, b^{4} e^{4} x^{9} + a^{4} d^{4} x + \frac{1}{2} \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{8} + \frac{2}{7} \,{\left (3 \, b^{4} d^{2} e^{2} + 8 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{7} + \frac{2}{3} \,{\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{4} + 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} + 16 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} x^{5} +{\left (a b^{3} d^{4} + 6 \, a^{2} b^{2} d^{3} e + 6 \, a^{3} b d^{2} e^{2} + a^{4} d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} d^{4} + 8 \, a^{3} b d^{3} e + 3 \, a^{4} d^{2} e^{2}\right )} x^{3} + 2 \,{\left (a^{3} b d^{4} + a^{4} d^{3} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.51175, size = 680, normalized size = 5.71 \begin{align*} \frac{1}{9} x^{9} e^{4} b^{4} + \frac{1}{2} x^{8} e^{3} d b^{4} + \frac{1}{2} x^{8} e^{4} b^{3} a + \frac{6}{7} x^{7} e^{2} d^{2} b^{4} + \frac{16}{7} x^{7} e^{3} d b^{3} a + \frac{6}{7} x^{7} e^{4} b^{2} a^{2} + \frac{2}{3} x^{6} e d^{3} b^{4} + 4 x^{6} e^{2} d^{2} b^{3} a + 4 x^{6} e^{3} d b^{2} a^{2} + \frac{2}{3} x^{6} e^{4} b a^{3} + \frac{1}{5} x^{5} d^{4} b^{4} + \frac{16}{5} x^{5} e d^{3} b^{3} a + \frac{36}{5} x^{5} e^{2} d^{2} b^{2} a^{2} + \frac{16}{5} x^{5} e^{3} d b a^{3} + \frac{1}{5} x^{5} e^{4} a^{4} + x^{4} d^{4} b^{3} a + 6 x^{4} e d^{3} b^{2} a^{2} + 6 x^{4} e^{2} d^{2} b a^{3} + x^{4} e^{3} d a^{4} + 2 x^{3} d^{4} b^{2} a^{2} + \frac{16}{3} x^{3} e d^{3} b a^{3} + 2 x^{3} e^{2} d^{2} a^{4} + 2 x^{2} d^{4} b a^{3} + 2 x^{2} e d^{3} a^{4} + x d^{4} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.115406, size = 318, normalized size = 2.67 \begin{align*} a^{4} d^{4} x + \frac{b^{4} e^{4} x^{9}}{9} + x^{8} \left (\frac{a b^{3} e^{4}}{2} + \frac{b^{4} d e^{3}}{2}\right ) + x^{7} \left (\frac{6 a^{2} b^{2} e^{4}}{7} + \frac{16 a b^{3} d e^{3}}{7} + \frac{6 b^{4} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac{2 a^{3} b e^{4}}{3} + 4 a^{2} b^{2} d e^{3} + 4 a b^{3} d^{2} e^{2} + \frac{2 b^{4} d^{3} e}{3}\right ) + x^{5} \left (\frac{a^{4} e^{4}}{5} + \frac{16 a^{3} b d e^{3}}{5} + \frac{36 a^{2} b^{2} d^{2} e^{2}}{5} + \frac{16 a b^{3} d^{3} e}{5} + \frac{b^{4} d^{4}}{5}\right ) + x^{4} \left (a^{4} d e^{3} + 6 a^{3} b d^{2} e^{2} + 6 a^{2} b^{2} d^{3} e + a b^{3} d^{4}\right ) + x^{3} \left (2 a^{4} d^{2} e^{2} + \frac{16 a^{3} b d^{3} e}{3} + 2 a^{2} b^{2} d^{4}\right ) + x^{2} \left (2 a^{4} d^{3} e + 2 a^{3} b d^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1317, size = 420, normalized size = 3.53 \begin{align*} \frac{1}{9} \, b^{4} x^{9} e^{4} + \frac{1}{2} \, b^{4} d x^{8} e^{3} + \frac{6}{7} \, b^{4} d^{2} x^{7} e^{2} + \frac{2}{3} \, b^{4} d^{3} x^{6} e + \frac{1}{5} \, b^{4} d^{4} x^{5} + \frac{1}{2} \, a b^{3} x^{8} e^{4} + \frac{16}{7} \, a b^{3} d x^{7} e^{3} + 4 \, a b^{3} d^{2} x^{6} e^{2} + \frac{16}{5} \, a b^{3} d^{3} x^{5} e + a b^{3} d^{4} x^{4} + \frac{6}{7} \, a^{2} b^{2} x^{7} e^{4} + 4 \, a^{2} b^{2} d x^{6} e^{3} + \frac{36}{5} \, a^{2} b^{2} d^{2} x^{5} e^{2} + 6 \, a^{2} b^{2} d^{3} x^{4} e + 2 \, a^{2} b^{2} d^{4} x^{3} + \frac{2}{3} \, a^{3} b x^{6} e^{4} + \frac{16}{5} \, a^{3} b d x^{5} e^{3} + 6 \, a^{3} b d^{2} x^{4} e^{2} + \frac{16}{3} \, a^{3} b d^{3} x^{3} e + 2 \, a^{3} b d^{4} x^{2} + \frac{1}{5} \, a^{4} x^{5} e^{4} + a^{4} d x^{4} e^{3} + 2 \, a^{4} d^{2} x^{3} e^{2} + 2 \, a^{4} d^{3} x^{2} e + a^{4} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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