Optimal. Leaf size=38 \[ \frac {(b c-a d) (a+b x)^3}{3 b^2}+\frac {d (a+b x)^4}{4 b^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} \frac {(a+b x)^3 (b c-a d)}{3 b^2}+\frac {d (a+b x)^4}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x)^2 (c+d x) \, dx &=\int \left (\frac {(b c-a d) (a+b x)^2}{b}+\frac {d (a+b x)^3}{b}\right ) \, dx\\ &=\frac {(b c-a d) (a+b x)^3}{3 b^2}+\frac {d (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 c+d x)+4 a b x (3 c+2 d x)+b^2 x^2 (4 c+3 d x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 49, normalized size = 1.29
method | result | size |
norman | \(\frac {b^{2} d \,x^{4}}{4}+\left (\frac {2}{3} a b d +\frac {1}{3} b^{2} c \right ) x^{3}+\left (\frac {1}{2} a^{2} d +a b c \right ) x^{2}+a^{2} c x\) | \(48\) |
default | \(\frac {b^{2} d \,x^{4}}{4}+\frac {\left (2 a b d +b^{2} c \right ) x^{3}}{3}+\frac {\left (a^{2} d +2 a b c \right ) x^{2}}{2}+a^{2} c x\) | \(49\) |
gosper | \(\frac {1}{4} b^{2} d \,x^{4}+\frac {2}{3} x^{3} a b d +\frac {1}{3} b^{2} c \,x^{3}+\frac {1}{2} x^{2} a^{2} d +x^{2} a b c +a^{2} c x\) | \(50\) |
risch | \(\frac {1}{4} b^{2} d \,x^{4}+\frac {2}{3} x^{3} a b d +\frac {1}{3} b^{2} c \,x^{3}+\frac {1}{2} x^{2} a^{2} d +x^{2} a b c +a^{2} c x\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, b^{2} d x^{4} + a^{2} c x + \frac {1}{3} \, {\left (b^{2} c + 2 \, a b d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c + a^{2} d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, b^{2} d x^{4} + a^{2} c x + \frac {1}{3} \, {\left (b^{2} c + 2 \, a b d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c + a^{2} d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 49, normalized size = 1.29 \begin {gather*} a^{2} c x + \frac {b^{2} d x^{4}}{4} + x^{3} \cdot \left (\frac {2 a b d}{3} + \frac {b^{2} c}{3}\right ) + x^{2} \left (\frac {a^{2} d}{2} + a b c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 49, normalized size = 1.29 \begin {gather*} \frac {1}{4} \, b^{2} d x^{4} + \frac {1}{3} \, b^{2} c x^{3} + \frac {2}{3} \, a b d x^{3} + a b c x^{2} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \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 {d\,a^2}{2}+b\,c\,a\right )+x^3\,\left (\frac {c\,b^2}{3}+\frac {2\,a\,d\,b}{3}\right )+\frac {b^2\,d\,x^4}{4}+a^2\,c\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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