Optimal. Leaf size=44 \[ \frac {1}{2} x^2 (a \text {c1}+2 b \text {b1})+a \text {b1} x+\frac {1}{3} x^3 (2 b \text {c1}+\text {b1} c)+\frac {1}{4} c \text {c1} x^4 \]
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Rubi [A] time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {631} \[ \frac {1}{2} x^2 (a \text {c1}+2 b \text {b1})+a \text {b1} x+\frac {1}{3} x^3 (2 b \text {c1}+\text {b1} c)+\frac {1}{4} c \text {c1} x^4 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin {align*} \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right ) \, dx &=\int \left (a \text {b1}+(2 b \text {b1}+a \text {c1}) x+(\text {b1} c+2 b \text {c1}) x^2+c \text {c1} x^3\right ) \, dx\\ &=a \text {b1} x+\frac {1}{2} (2 b \text {b1}+a \text {c1}) x^2+\frac {1}{3} (\text {b1} c+2 b \text {c1}) x^3+\frac {1}{4} c \text {c1} x^4\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.93 \[ \frac {1}{12} x (6 a (2 \text {b1}+\text {c1} x)+x (4 b (3 \text {b1}+2 \text {c1} x)+c x (4 \text {b1}+3 \text {c1} x))) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 41, normalized size = 0.93 \[ \frac {1}{12} x \left (12 a \text {b1}+6 a \text {c1} x+12 b \text {b1} x+8 b \text {c1} x^2+4 \text {b1} c x^2+3 c \text {c1} x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 39, normalized size = 0.89 \[ \frac {1}{4} x^{4} c_{1} c + \frac {1}{3} x^{3} c b_{1} + \frac {2}{3} x^{3} c_{1} b + x^{2} b_{1} b + \frac {1}{2} x^{2} c_{1} a + x b_{1} a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 39, normalized size = 0.89 \[ \frac {1}{4} \, c c_{1} x^{4} + \frac {1}{3} \, b_{1} c x^{3} + \frac {2}{3} \, b c_{1} x^{3} + b b_{1} x^{2} + \frac {1}{2} \, a c_{1} x^{2} + a b_{1} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 38, normalized size = 0.86
method | result | size |
norman | \(\frac {c \mathit {c1} \,x^{4}}{4}+\left (\frac {2 b \mathit {c1}}{3}+\frac {\mathit {b1} c}{3}\right ) x^{3}+\left (\frac {a \mathit {c1}}{2}+b \mathit {b1} \right ) x^{2}+a \mathit {b1} x\) | \(38\) |
default | \(a \mathit {b1} x +\frac {\left (a \mathit {c1} +2 b \mathit {b1} \right ) x^{2}}{2}+\frac {\left (2 b \mathit {c1} +\mathit {b1} c \right ) x^{3}}{3}+\frac {c \mathit {c1} \,x^{4}}{4}\) | \(39\) |
gosper | \(\frac {1}{4} c \mathit {c1} \,x^{4}+\frac {2}{3} x^{3} b \mathit {c1} +\frac {1}{3} x^{3} \mathit {b1} c +\frac {1}{2} x^{2} a \mathit {c1} +x^{2} b \mathit {b1} +a \mathit {b1} x\) | \(40\) |
risch | \(\frac {1}{4} c \mathit {c1} \,x^{4}+\frac {2}{3} x^{3} b \mathit {c1} +\frac {1}{3} x^{3} \mathit {b1} c +\frac {1}{2} x^{2} a \mathit {c1} +x^{2} b \mathit {b1} +a \mathit {b1} x\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 38, normalized size = 0.86 \[ \frac {1}{4} \, c c_{1} x^{4} + \frac {1}{3} \, {\left (b_{1} c + 2 \, b c_{1}\right )} x^{3} + a b_{1} x + \frac {1}{2} \, {\left (2 \, b b_{1} + a c_{1}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 37, normalized size = 0.84 \[ \frac {c\,c_{1}\,x^4}{4}+\left (\frac {2\,b\,c_{1}}{3}+\frac {b_{1}\,c}{3}\right )\,x^3+\left (\frac {a\,c_{1}}{2}+b\,b_{1}\right )\,x^2+a\,b_{1}\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 39, normalized size = 0.89 \[ a b_{1} x + \frac {c c_{1} x^{4}}{4} + x^{3} \left (\frac {2 b c_{1}}{3} + \frac {b_{1} c}{3}\right ) + x^{2} \left (\frac {a c_{1}}{2} + b b_{1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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