Optimal. Leaf size=92 \[ -\frac{3 b^2 (d+e x)^7 (b d-a e)}{7 e^4}+\frac{b (d+e x)^6 (b d-a e)^2}{2 e^4}-\frac{(d+e x)^5 (b d-a e)^3}{5 e^4}+\frac{b^3 (d+e x)^8}{8 e^4} \]
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Rubi [A] time = 0.120025, antiderivative size = 92, 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 = {27, 43} \[ -\frac{3 b^2 (d+e x)^7 (b d-a e)}{7 e^4}+\frac{b (d+e x)^6 (b d-a e)^2}{2 e^4}-\frac{(d+e x)^5 (b d-a e)^3}{5 e^4}+\frac{b^3 (d+e x)^8}{8 e^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^4 \, dx\\ &=\int \left (\frac{(-b d+a e)^3 (d+e x)^4}{e^3}+\frac{3 b (b d-a e)^2 (d+e x)^5}{e^3}-\frac{3 b^2 (b d-a e) (d+e x)^6}{e^3}+\frac{b^3 (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac{(b d-a e)^3 (d+e x)^5}{5 e^4}+\frac{b (b d-a e)^2 (d+e x)^6}{2 e^4}-\frac{3 b^2 (b d-a e) (d+e x)^7}{7 e^4}+\frac{b^3 (d+e x)^8}{8 e^4}\\ \end{align*}
Mathematica [B] time = 0.0352289, size = 217, normalized size = 2.36 \[ \frac{1}{2} b e^2 x^6 \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+\frac{1}{5} e x^5 \left (12 a^2 b d e^2+a^3 e^3+18 a b^2 d^2 e+4 b^3 d^3\right )+\frac{1}{4} d x^4 \left (18 a^2 b d e^2+4 a^3 e^3+12 a b^2 d^2 e+b^3 d^3\right )+a d^2 x^3 \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+\frac{1}{2} a^2 d^3 x^2 (4 a e+3 b d)+a^3 d^4 x+\frac{1}{7} b^2 e^3 x^7 (3 a e+4 b d)+\frac{1}{8} b^3 e^4 x^8 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 319, normalized size = 3.5 \begin{align*}{\frac{{e}^{4}{b}^{3}{x}^{8}}{8}}+{\frac{ \left ( \left ( a{e}^{4}+4\,bd{e}^{3} \right ){b}^{2}+2\,a{b}^{2}{e}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ){b}^{2}+2\, \left ( a{e}^{4}+4\,bd{e}^{3} \right ) ab+b{e}^{4}{a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ){b}^{2}+2\, \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ) ab+ \left ( a{e}^{4}+4\,bd{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 4\,a{d}^{3}e+b{d}^{4} \right ){b}^{2}+2\, \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ) ab+ \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{2}{d}^{4}a+2\, \left ( 4\,a{d}^{3}e+b{d}^{4} \right ) ab+ \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{d}^{4}b+ \left ( 4\,a{d}^{3}e+b{d}^{4} \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.973833, size = 304, normalized size = 3.3 \begin{align*} \frac{1}{8} \, b^{3} e^{4} x^{8} + a^{3} d^{4} x + \frac{1}{7} \,{\left (4 \, b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, b^{3} d^{2} e^{2} + 4 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, b^{3} d^{3} e + 18 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{4} + 12 \, a b^{2} d^{3} e + 18 \, a^{2} b d^{2} e^{2} + 4 \, a^{3} d e^{3}\right )} x^{4} +{\left (a b^{2} d^{4} + 4 \, a^{2} b d^{3} e + 2 \, a^{3} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{4} + 4 \, a^{3} 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 = 1.38826, size = 520, normalized size = 5.65 \begin{align*} \frac{1}{8} x^{8} e^{4} b^{3} + \frac{4}{7} x^{7} e^{3} d b^{3} + \frac{3}{7} x^{7} e^{4} b^{2} a + x^{6} e^{2} d^{2} b^{3} + 2 x^{6} e^{3} d b^{2} a + \frac{1}{2} x^{6} e^{4} b a^{2} + \frac{4}{5} x^{5} e d^{3} b^{3} + \frac{18}{5} x^{5} e^{2} d^{2} b^{2} a + \frac{12}{5} x^{5} e^{3} d b a^{2} + \frac{1}{5} x^{5} e^{4} a^{3} + \frac{1}{4} x^{4} d^{4} b^{3} + 3 x^{4} e d^{3} b^{2} a + \frac{9}{2} x^{4} e^{2} d^{2} b a^{2} + x^{4} e^{3} d a^{3} + x^{3} d^{4} b^{2} a + 4 x^{3} e d^{3} b a^{2} + 2 x^{3} e^{2} d^{2} a^{3} + \frac{3}{2} x^{2} d^{4} b a^{2} + 2 x^{2} e d^{3} a^{3} + x d^{4} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.094974, size = 243, normalized size = 2.64 \begin{align*} a^{3} d^{4} x + \frac{b^{3} e^{4} x^{8}}{8} + x^{7} \left (\frac{3 a b^{2} e^{4}}{7} + \frac{4 b^{3} d e^{3}}{7}\right ) + x^{6} \left (\frac{a^{2} b e^{4}}{2} + 2 a b^{2} d e^{3} + b^{3} d^{2} e^{2}\right ) + x^{5} \left (\frac{a^{3} e^{4}}{5} + \frac{12 a^{2} b d e^{3}}{5} + \frac{18 a b^{2} d^{2} e^{2}}{5} + \frac{4 b^{3} d^{3} e}{5}\right ) + x^{4} \left (a^{3} d e^{3} + \frac{9 a^{2} b d^{2} e^{2}}{2} + 3 a b^{2} d^{3} e + \frac{b^{3} d^{4}}{4}\right ) + x^{3} \left (2 a^{3} d^{2} e^{2} + 4 a^{2} b d^{3} e + a b^{2} d^{4}\right ) + x^{2} \left (2 a^{3} d^{3} e + \frac{3 a^{2} b d^{4}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19078, size = 320, normalized size = 3.48 \begin{align*} \frac{1}{8} \, b^{3} x^{8} e^{4} + \frac{4}{7} \, b^{3} d x^{7} e^{3} + b^{3} d^{2} x^{6} e^{2} + \frac{4}{5} \, b^{3} d^{3} x^{5} e + \frac{1}{4} \, b^{3} d^{4} x^{4} + \frac{3}{7} \, a b^{2} x^{7} e^{4} + 2 \, a b^{2} d x^{6} e^{3} + \frac{18}{5} \, a b^{2} d^{2} x^{5} e^{2} + 3 \, a b^{2} d^{3} x^{4} e + a b^{2} d^{4} x^{3} + \frac{1}{2} \, a^{2} b x^{6} e^{4} + \frac{12}{5} \, a^{2} b d x^{5} e^{3} + \frac{9}{2} \, a^{2} b d^{2} x^{4} e^{2} + 4 \, a^{2} b d^{3} x^{3} e + \frac{3}{2} \, a^{2} b d^{4} x^{2} + \frac{1}{5} \, a^{3} x^{5} e^{4} + a^{3} d x^{4} e^{3} + 2 \, a^{3} d^{2} x^{3} e^{2} + 2 \, a^{3} d^{3} x^{2} e + a^{3} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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