Optimal. Leaf size=38 \[ \frac {(a+b x)^4 (b d-a e)}{4 b^2}+\frac {e (a+b x)^5}{5 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 43} \begin {gather*} \frac {(a+b x)^4 (b d-a e)}{4 b^2}+\frac {e (a+b x)^5}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x) \, dx\\ &=\int \left (\frac {(b d-a e) (a+b x)^3}{b}+\frac {e (a+b x)^4}{b}\right ) \, dx\\ &=\frac {(b d-a e) (a+b x)^4}{4 b^2}+\frac {e (a+b x)^5}{5 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 67, normalized size = 1.76 \begin {gather*} a^3 d x+\frac {1}{2} a^2 x^2 (a e+3 b d)+\frac {1}{4} b^2 x^4 (3 a e+b d)+a b x^3 (a e+b d)+\frac {1}{5} b^3 e x^5 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.35, size = 72, normalized size = 1.89 \begin {gather*} \frac {1}{5} x^{5} e b^{3} + \frac {1}{4} x^{4} d b^{3} + \frac {3}{4} x^{4} e b^{2} a + x^{3} d b^{2} a + x^{3} e b a^{2} + \frac {3}{2} x^{2} d b a^{2} + \frac {1}{2} x^{2} e a^{3} + x d a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 76, normalized size = 2.00 \begin {gather*} \frac {1}{5} \, b^{3} x^{5} e + \frac {1}{4} \, b^{3} d x^{4} + \frac {3}{4} \, a b^{2} x^{4} e + a b^{2} d x^{3} + a^{2} b x^{3} e + \frac {3}{2} \, a^{2} b d x^{2} + \frac {1}{2} \, a^{3} x^{2} e + a^{3} d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 94, normalized size = 2.47 \begin {gather*} \frac {b^{3} e \,x^{5}}{5}+a^{3} d x +\frac {\left (2 a \,b^{2} e +\left (a e +b d \right ) b^{2}\right ) x^{4}}{4}+\frac {\left (a^{2} b e +a \,b^{2} d +2 \left (a e +b d \right ) a b \right ) x^{3}}{3}+\frac {\left (2 a^{2} b d +\left (a e +b d \right ) a^{2}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 69, normalized size = 1.82 \begin {gather*} \frac {1}{5} \, b^{3} e x^{5} + a^{3} d x + \frac {1}{4} \, {\left (b^{3} d + 3 \, a b^{2} e\right )} x^{4} + {\left (a b^{2} d + a^{2} b e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 65, normalized size = 1.71 \begin {gather*} x^2\,\left (\frac {e\,a^3}{2}+\frac {3\,b\,d\,a^2}{2}\right )+x^4\,\left (\frac {d\,b^3}{4}+\frac {3\,a\,e\,b^2}{4}\right )+\frac {b^3\,e\,x^5}{5}+a^3\,d\,x+a\,b\,x^3\,\left (a\,e+b\,d\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.08, size = 73, normalized size = 1.92 \begin {gather*} a^{3} d x + \frac {b^{3} e x^{5}}{5} + x^{4} \left (\frac {3 a b^{2} e}{4} + \frac {b^{3} d}{4}\right ) + x^{3} \left (a^{2} b e + a b^{2} d\right ) + x^{2} \left (\frac {a^{3} e}{2} + \frac {3 a^{2} b d}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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