Optimal. Leaf size=38 \[ \frac {(a+b x)^3 (b d-a e)}{3 b^2}+\frac {e (a+b x)^4}{4 b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {27, 43} \begin {gather*} \frac {(a+b x)^3 (b d-a e)}{3 b^2}+\frac {e (a+b x)^4}{4 b^2} \end {gather*}
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 ) \, dx &=\int (a+b x)^2 (d+e x) \, dx\\ &=\int \left (\frac {(b d-a e) (a+b x)^2}{b}+\frac {e (a+b x)^3}{b}\right ) \, dx\\ &=\frac {(b d-a e) (a+b x)^3}{3 b^2}+\frac {e (a+b x)^4}{4 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.21 \begin {gather*} \frac {1}{12} x \left (6 a^2 (2 d+e x)+4 a b x (3 d+2 e x)+b^2 x^2 (4 d+3 e x)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.35, size = 49, normalized size = 1.29 \begin {gather*} \frac {1}{4} x^{4} e b^{2} + \frac {1}{3} x^{3} d b^{2} + \frac {2}{3} x^{3} e b a + x^{2} d b a + \frac {1}{2} x^{2} e a^{2} + x d a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 52, normalized size = 1.37 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} e + \frac {1}{3} \, b^{2} d x^{3} + \frac {2}{3} \, a b x^{3} e + a b d x^{2} + \frac {1}{2} \, a^{2} x^{2} e + a^{2} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 49, normalized size = 1.29 \begin {gather*} \frac {b^{2} e \,x^{4}}{4}+a^{2} d x +\frac {\left (2 b e a +b^{2} d \right ) x^{3}}{3}+\frac {\left (e \,a^{2}+2 b d a \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, b^{2} e x^{4} + a^{2} d x + \frac {1}{3} \, {\left (b^{2} d + 2 \, a b e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 47, normalized size = 1.24 \begin {gather*} x^2\,\left (\frac {e\,a^2}{2}+b\,d\,a\right )+x^3\,\left (\frac {d\,b^2}{3}+\frac {2\,a\,e\,b}{3}\right )+\frac {b^2\,e\,x^4}{4}+a^2\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 49, normalized size = 1.29 \begin {gather*} a^{2} d x + \frac {b^{2} e x^{4}}{4} + x^{3} \left (\frac {2 a b e}{3} + \frac {b^{2} d}{3}\right ) + x^{2} \left (\frac {a^{2} e}{2} + a b d\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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