Optimal. Leaf size=38 \[ \frac {(a+b x)^3 (b c-a d)}{3 b^2}+\frac {d (a+b x)^4}{4 b^2} \]
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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 = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {626, 43} \[ \frac {(a+b x)^3 (b c-a d)}{3 b^2}+\frac {d (a+b x)^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int (a+b x) \left (a c+(b c+a d) x+b d x^2\right ) \, dx &=\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 \[ \frac {1}{12} x \left (6 a^2 (2 c+d x)+4 a b x (3 c+2 d x)+b^2 x^2 (4 c+3 d x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 49, normalized size = 1.29 \[ \frac {1}{4} x^{4} d b^{2} + \frac {1}{3} x^{3} c b^{2} + \frac {2}{3} x^{3} d b a + x^{2} c b a + \frac {1}{2} x^{2} d a^{2} + x c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 49, normalized size = 1.29 \[ \frac {1}{4} \, b^{2} d x^{4} + \frac {1}{3} \, b^{2} c x^{3} + \frac {2}{3} \, a b d x^{3} + a b c x^{2} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 55, normalized size = 1.45 \[ \frac {b^{2} d \,x^{4}}{4}+a^{2} c x +\frac {\left (a b d +\left (a d +b c \right ) b \right ) x^{3}}{3}+\frac {\left (a b c +\left (a d +b c \right ) a \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 48, normalized size = 1.26 \[ \frac {1}{4} \, b^{2} d x^{4} + a^{2} c x + \frac {1}{3} \, {\left (b^{2} c + 2 \, a b d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b c + a^{2} d\right )} x^{2} \]
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 \[ x^2\,\left (\frac {d\,a^2}{2}+b\,c\,a\right )+x^3\,\left (\frac {c\,b^2}{3}+\frac {2\,a\,d\,b}{3}\right )+\frac {b^2\,d\,x^4}{4}+a^2\,c\,x \]
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 \[ a^{2} c x + \frac {b^{2} d x^{4}}{4} + x^{3} \left (\frac {2 a b d}{3} + \frac {b^{2} c}{3}\right ) + x^{2} \left (\frac {a^{2} d}{2} + a b c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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