Optimal. Leaf size=65 \[ \frac{2 e (a+b x)^5 (b d-a e)}{5 b^3}+\frac{(a+b x)^4 (b d-a e)^2}{4 b^3}+\frac{e^2 (a+b x)^6}{6 b^3} \]
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Rubi [A] time = 0.0666271, 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 = {27, 43} \[ \frac{2 e (a+b x)^5 (b d-a e)}{5 b^3}+\frac{(a+b x)^4 (b d-a e)^2}{4 b^3}+\frac{e^2 (a+b x)^6}{6 b^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^2 \, dx\\ &=\int \left (\frac{(b d-a e)^2 (a+b x)^3}{b^2}+\frac{2 e (b d-a e) (a+b x)^4}{b^2}+\frac{e^2 (a+b x)^5}{b^2}\right ) \, dx\\ &=\frac{(b d-a e)^2 (a+b x)^4}{4 b^3}+\frac{2 e (b d-a e) (a+b x)^5}{5 b^3}+\frac{e^2 (a+b x)^6}{6 b^3}\\ \end{align*}
Mathematica [A] time = 0.0197151, size = 122, normalized size = 1.88 \[ \frac{1}{4} b x^4 \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+\frac{1}{3} a x^3 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac{1}{2} a^2 d x^2 (2 a e+3 b d)+a^3 d^2 x+\frac{1}{5} b^2 e x^5 (3 a e+2 b d)+\frac{1}{6} b^3 e^2 x^6 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 169, normalized size = 2.6 \begin{align*}{\frac{{b}^{3}{e}^{2}{x}^{6}}{6}}+{\frac{ \left ( \left ( a{e}^{2}+2\,bde \right ){b}^{2}+2\,{b}^{2}{e}^{2}a \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 2\,ade+b{d}^{2} \right ){b}^{2}+2\, \left ( a{e}^{2}+2\,bde \right ) ab+b{e}^{2}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{2}{d}^{2}a+2\, \left ( 2\,ade+b{d}^{2} \right ) ab+ \left ( a{e}^{2}+2\,bde \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{d}^{2}b+ \left ( 2\,ade+b{d}^{2} \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.95951, size = 167, normalized size = 2.57 \begin{align*} \frac{1}{6} \, b^{3} e^{2} x^{6} + a^{3} d^{2} x + \frac{1}{5} \,{\left (2 \, b^{3} d e + 3 \, a b^{2} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{2} + 6 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{2} d^{2} + 6 \, a^{2} b d e + a^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.26756, size = 285, normalized size = 4.38 \begin{align*} \frac{1}{6} x^{6} e^{2} b^{3} + \frac{2}{5} x^{5} e d b^{3} + \frac{3}{5} x^{5} e^{2} b^{2} a + \frac{1}{4} x^{4} d^{2} b^{3} + \frac{3}{2} x^{4} e d b^{2} a + \frac{3}{4} x^{4} e^{2} b a^{2} + x^{3} d^{2} b^{2} a + 2 x^{3} e d b a^{2} + \frac{1}{3} x^{3} e^{2} a^{3} + \frac{3}{2} x^{2} d^{2} b a^{2} + x^{2} e d a^{3} + x d^{2} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.080874, size = 133, normalized size = 2.05 \begin{align*} a^{3} d^{2} x + \frac{b^{3} e^{2} x^{6}}{6} + x^{5} \left (\frac{3 a b^{2} e^{2}}{5} + \frac{2 b^{3} d e}{5}\right ) + x^{4} \left (\frac{3 a^{2} b e^{2}}{4} + \frac{3 a b^{2} d e}{2} + \frac{b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac{a^{3} e^{2}}{3} + 2 a^{2} b d e + a b^{2} d^{2}\right ) + x^{2} \left (a^{3} d e + \frac{3 a^{2} b d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12962, size = 176, normalized size = 2.71 \begin{align*} \frac{1}{6} \, b^{3} x^{6} e^{2} + \frac{2}{5} \, b^{3} d x^{5} e + \frac{1}{4} \, b^{3} d^{2} x^{4} + \frac{3}{5} \, a b^{2} x^{5} e^{2} + \frac{3}{2} \, a b^{2} d x^{4} e + a b^{2} d^{2} x^{3} + \frac{3}{4} \, a^{2} b x^{4} e^{2} + 2 \, a^{2} b d x^{3} e + \frac{3}{2} \, a^{2} b d^{2} x^{2} + \frac{1}{3} \, a^{3} x^{3} e^{2} + a^{3} d x^{2} e + a^{3} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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