Optimal. Leaf size=38 \[ \frac{(a+b x)^5 (b d-a e)}{5 b^2}+\frac{e (a+b x)^6}{6 b^2} \]
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Rubi [A] time = 0.0158858, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 43} \[ \frac{(a+b x)^5 (b d-a e)}{5 b^2}+\frac{e (a+b x)^6}{6 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x) \, dx\\ &=\int \left (\frac{(b d-a e) (a+b x)^4}{b}+\frac{e (a+b x)^5}{b}\right ) \, dx\\ &=\frac{(b d-a e) (a+b x)^5}{5 b^2}+\frac{e (a+b x)^6}{6 b^2}\\ \end{align*}
Mathematica [B] time = 0.0161652, size = 84, normalized size = 2.21 \[ \frac{1}{30} x \left (15 a^2 b^2 x^2 (4 d+3 e x)+20 a^3 b x (3 d+2 e x)+15 a^4 (2 d+e x)+6 a b^3 x^3 (5 d+4 e x)+b^4 x^4 (6 d+5 e x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 97, normalized size = 2.6 \begin{align*}{\frac{e{b}^{4}{x}^{6}}{6}}+{\frac{ \left ( 4\,ea{b}^{3}+d{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,{a}^{2}e{b}^{2}+4\,ad{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 4\,e{a}^{3}b+6\,d{b}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{4}+4\,d{a}^{3}b \right ){x}^{2}}{2}}+{a}^{4}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1702, size = 130, normalized size = 3.42 \begin{align*} \frac{1}{6} \, b^{4} e x^{6} + a^{4} d x + \frac{1}{5} \,{\left (b^{4} d + 4 \, a b^{3} e\right )} x^{5} + \frac{1}{2} \,{\left (2 \, a b^{3} d + 3 \, a^{2} b^{2} e\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} d + 2 \, a^{3} b e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52879, size = 217, normalized size = 5.71 \begin{align*} \frac{1}{6} x^{6} e b^{4} + \frac{1}{5} x^{5} d b^{4} + \frac{4}{5} x^{5} e b^{3} a + x^{4} d b^{3} a + \frac{3}{2} x^{4} e b^{2} a^{2} + 2 x^{3} d b^{2} a^{2} + \frac{4}{3} x^{3} e b a^{3} + 2 x^{2} d b a^{3} + \frac{1}{2} x^{2} e a^{4} + x d a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.076732, size = 100, normalized size = 2.63 \begin{align*} a^{4} d x + \frac{b^{4} e x^{6}}{6} + x^{5} \left (\frac{4 a b^{3} e}{5} + \frac{b^{4} d}{5}\right ) + x^{4} \left (\frac{3 a^{2} b^{2} e}{2} + a b^{3} d\right ) + x^{3} \left (\frac{4 a^{3} b e}{3} + 2 a^{2} b^{2} d\right ) + x^{2} \left (\frac{a^{4} e}{2} + 2 a^{3} b d\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15537, size = 138, normalized size = 3.63 \begin{align*} \frac{1}{6} \, b^{4} x^{6} e + \frac{1}{5} \, b^{4} d x^{5} + \frac{4}{5} \, a b^{3} x^{5} e + a b^{3} d x^{4} + \frac{3}{2} \, a^{2} b^{2} x^{4} e + 2 \, a^{2} b^{2} d x^{3} + \frac{4}{3} \, a^{3} b x^{3} e + 2 \, a^{3} b d x^{2} + \frac{1}{2} \, a^{4} x^{2} e + a^{4} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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