Optimal. Leaf size=38 \[ -\frac {(b c-a d) (c+d x)^3}{3 d^2}+\frac {b (c+d x)^4}{4 d^2} \]
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Rubi [A]
time = 0.02, 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 {b (c+d x)^4}{4 d^2}-\frac {(c+d x)^3 (b c-a d)}{3 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int (a+b x) (c+d x)^2 \, dx &=\int \left (\frac {(-b c+a d) (c+d x)^2}{d}+\frac {b (c+d x)^3}{d}\right ) \, dx\\ &=-\frac {(b c-a d) (c+d x)^3}{3 d^2}+\frac {b (c+d x)^4}{4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 1.24 \begin {gather*} \frac {1}{12} x \left (12 a c^2+6 c (b c+2 a d) x+4 d (2 b c+a d) x^2+3 b d^2 x^3\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 | \(\frac {b \,d^{2} x^{4}}{4}+\left (\frac {1}{3} a \,d^{2}+\frac {2}{3} b d c \right ) x^{3}+\left (a c d +\frac {1}{2} b \,c^{2}\right ) x^{2}+a \,c^{2} x\) | \(48\) |
default | \(\frac {b \,d^{2} x^{4}}{4}+\frac {\left (a \,d^{2}+2 b d c \right ) x^{3}}{3}+\frac {\left (2 a c d +b \,c^{2}\right ) x^{2}}{2}+a \,c^{2} x\) | \(49\) |
gosper | \(\frac {1}{4} b \,d^{2} x^{4}+\frac {1}{3} x^{3} a \,d^{2}+\frac {2}{3} x^{3} b d c +x^{2} a c d +\frac {1}{2} b \,c^{2} x^{2}+a \,c^{2} x\) | \(50\) |
risch | \(\frac {1}{4} b \,d^{2} x^{4}+\frac {1}{3} x^{3} a \,d^{2}+\frac {2}{3} x^{3} b d c +x^{2} a c d +\frac {1}{2} b \,c^{2} x^{2}+a \,c^{2} x\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, b d^{2} x^{4} + a c^{2} x + \frac {1}{3} \, {\left (2 \, b c d + a d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{2} + 2 \, a c d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 48, normalized size = 1.26 \begin {gather*} \frac {1}{4} \, b d^{2} x^{4} + a c^{2} x + \frac {1}{3} \, {\left (2 \, b c d + a d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{2} + 2 \, a c 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 c^{2} x + \frac {b d^{2} x^{4}}{4} + x^{3} \left (\frac {a d^{2}}{3} + \frac {2 b c d}{3}\right ) + x^{2} \left (a c d + \frac {b c^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 49, normalized size = 1.29 \begin {gather*} \frac {1}{4} \, b d^{2} x^{4} + \frac {2}{3} \, b c d x^{3} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{2} \, b c^{2} x^{2} + a c d x^{2} + a c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 47, normalized size = 1.24 \begin {gather*} x^2\,\left (\frac {b\,c^2}{2}+a\,d\,c\right )+x^3\,\left (\frac {a\,d^2}{3}+\frac {2\,b\,c\,d}{3}\right )+\frac {b\,d^2\,x^4}{4}+a\,c^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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