Optimal. Leaf size=96 \[ a^2 \text {b1} x+\frac {1}{2} a (4 b \text {b1}+a \text {c1}) x^2+\frac {2}{3} \left (2 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+\frac {1}{2} \left (2 b \text {b1} c+2 b^2 \text {c1}+a c \text {c1}\right ) x^4+\frac {1}{5} c (\text {b1} c+4 b \text {c1}) x^5+\frac {1}{6} c^2 \text {c1} x^6 \]
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Rubi [A]
time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {645}
\begin {gather*} a^2 \text {b1} x+\frac {1}{2} x^4 \left (a c \text {c1}+2 b^2 \text {c1}+2 b \text {b1} c\right )+\frac {2}{3} x^3 \left (2 a b \text {c1}+a \text {b1} c+2 b^2 \text {b1}\right )+\frac {1}{2} a x^2 (a \text {c1}+4 b \text {b1})+\frac {1}{5} c x^5 (4 b \text {c1}+\text {b1} c)+\frac {1}{6} c^2 \text {c1} x^6 \end {gather*}
Antiderivative was successfully verified.
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Rule 645
Rubi steps
\begin {align*} \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^2 \, dx &=\int \left (a^2 \text {b1}+a (4 b \text {b1}+a \text {c1}) x+2 \left (2 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^2+2 \left (2 b \text {b1} c+2 b^2 \text {c1}+a c \text {c1}\right ) x^3+c (\text {b1} c+4 b \text {c1}) x^4+c^2 \text {c1} x^5\right ) \, dx\\ &=a^2 \text {b1} x+\frac {1}{2} a (4 b \text {b1}+a \text {c1}) x^2+\frac {2}{3} \left (2 b^2 \text {b1}+a \text {b1} c+2 a b \text {c1}\right ) x^3+\frac {1}{2} \left (2 b \text {b1} c+2 b^2 \text {c1}+a c \text {c1}\right ) x^4+\frac {1}{5} c (\text {b1} c+4 b \text {c1}) x^5+\frac {1}{6} c^2 \text {c1} x^6\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 91, normalized size = 0.95 \begin {gather*} \frac {1}{30} x \left (15 a^2 (2 \text {b1}+\text {c1} x)+5 a x (4 b (3 \text {b1}+2 \text {c1} x)+c x (4 \text {b1}+3 \text {c1} x))+x^2 \left (10 b^2 (4 \text {b1}+3 \text {c1} x)+6 b c x (5 \text {b1}+4 \text {c1} x)+c^2 x^2 (6 \text {b1}+5 \text {c1} x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 95, normalized size = 0.99
method | result | size |
norman | \(\frac {c^{2} \mathit {c1} \,x^{6}}{6}+\left (\frac {4}{5} \mathit {c1} b c +\frac {1}{5} \mathit {b1} \,c^{2}\right ) x^{5}+\left (\frac {1}{2} a c \mathit {c1} +b^{2} \mathit {c1} +b \mathit {b1} c \right ) x^{4}+\left (\frac {4}{3} a b \mathit {c1} +\frac {2}{3} a \mathit {b1} c +\frac {4}{3} b^{2} \mathit {b1} \right ) x^{3}+\left (\frac {1}{2} \mathit {c1} \,a^{2}+2 \mathit {b1} a b \right ) x^{2}+a^{2} \mathit {b1} x\) | \(89\) |
default | \(\frac {c^{2} \mathit {c1} \,x^{6}}{6}+\frac {\left (4 \mathit {c1} b c +\mathit {b1} \,c^{2}\right ) x^{5}}{5}+\frac {\left (4 b \mathit {b1} c +\mathit {c1} \left (2 a c +4 b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\mathit {b1} \left (2 a c +4 b^{2}\right )+4 a b \mathit {c1} \right ) x^{3}}{3}+\frac {\left (\mathit {c1} \,a^{2}+4 \mathit {b1} a b \right ) x^{2}}{2}+a^{2} \mathit {b1} x\) | \(95\) |
gosper | \(\frac {1}{6} c^{2} \mathit {c1} \,x^{6}+\frac {4}{5} x^{5} \mathit {c1} b c +\frac {1}{5} x^{5} \mathit {b1} \,c^{2}+\frac {1}{2} x^{4} a c \mathit {c1} +x^{4} b^{2} \mathit {c1} +x^{4} b \mathit {b1} c +\frac {4}{3} x^{3} a b \mathit {c1} +\frac {2}{3} x^{3} a \mathit {b1} c +\frac {4}{3} x^{3} b^{2} \mathit {b1} +\frac {1}{2} x^{2} \mathit {c1} \,a^{2}+2 x^{2} \mathit {b1} a b +a^{2} \mathit {b1} x\) | \(99\) |
risch | \(\frac {1}{6} c^{2} \mathit {c1} \,x^{6}+\frac {4}{5} x^{5} \mathit {c1} b c +\frac {1}{5} x^{5} \mathit {b1} \,c^{2}+\frac {1}{2} x^{4} a c \mathit {c1} +x^{4} b^{2} \mathit {c1} +x^{4} b \mathit {b1} c +\frac {4}{3} x^{3} a b \mathit {c1} +\frac {2}{3} x^{3} a \mathit {b1} c +\frac {4}{3} x^{3} b^{2} \mathit {b1} +\frac {1}{2} x^{2} \mathit {c1} \,a^{2}+2 x^{2} \mathit {b1} a b +a^{2} \mathit {b1} x\) | \(99\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.52, size = 91, normalized size = 0.95 \begin {gather*} \frac {1}{6} \, c^{2} c_{1} x^{6} + \frac {1}{5} \, {\left (b_{1} c^{2} + 4 \, b c c_{1}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, b b_{1} c + {\left (2 \, b^{2} + a c\right )} c_{1}\right )} x^{4} + a^{2} b_{1} x + \frac {2}{3} \, {\left (2 \, b^{2} b_{1} + a b_{1} c + 2 \, a b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a b b_{1} + a^{2} c_{1}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 91, normalized size = 0.95 \begin {gather*} \frac {1}{6} \, c^{2} c_{1} x^{6} + \frac {1}{5} \, {\left (b_{1} c^{2} + 4 \, b c c_{1}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, b b_{1} c + {\left (2 \, b^{2} + a c\right )} c_{1}\right )} x^{4} + a^{2} b_{1} x + \frac {2}{3} \, {\left (2 \, b^{2} b_{1} + a b_{1} c + 2 \, a b c_{1}\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a b b_{1} + a^{2} c_{1}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 100, normalized size = 1.04 \begin {gather*} a^{2} b_{1} x + \frac {c^{2} c_{1} x^{6}}{6} + x^{5} \cdot \left (\frac {4 b c c_{1}}{5} + \frac {b_{1} c^{2}}{5}\right ) + x^{4} \left (\frac {a c c_{1}}{2} + b^{2} c_{1} + b b_{1} c\right ) + x^{3} \cdot \left (\frac {4 a b c_{1}}{3} + \frac {2 a b_{1} c}{3} + \frac {4 b^{2} b_{1}}{3}\right ) + x^{2} \left (\frac {a^{2} c_{1}}{2} + 2 a b b_{1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.83, size = 98, normalized size = 1.02 \begin {gather*} \frac {1}{6} \, c^{2} c_{1} x^{6} + \frac {1}{5} \, b_{1} c^{2} x^{5} + \frac {4}{5} \, b c c_{1} x^{5} + b b_{1} c x^{4} + b^{2} c_{1} x^{4} + \frac {1}{2} \, a c c_{1} x^{4} + \frac {4}{3} \, b^{2} b_{1} x^{3} + \frac {2}{3} \, a b_{1} c x^{3} + \frac {4}{3} \, a b c_{1} x^{3} + 2 \, a b b_{1} x^{2} + \frac {1}{2} \, a^{2} c_{1} x^{2} + a^{2} b_{1} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 88, normalized size = 0.92 \begin {gather*} x^3\,\left (\frac {4\,b_{1}\,b^2}{3}+\frac {4\,a\,c_{1}\,b}{3}+\frac {2\,a\,b_{1}\,c}{3}\right )+x^4\,\left (c_{1}\,b^2+b_{1}\,c\,b+\frac {a\,c\,c_{1}}{2}\right )+x^2\,\left (\frac {c_{1}\,a^2}{2}+2\,b\,b_{1}\,a\right )+x^5\,\left (\frac {b_{1}\,c^2}{5}+\frac {4\,b\,c_{1}\,c}{5}\right )+\frac {c^2\,c_{1}\,x^6}{6}+a^2\,b_{1}\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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