Optimal. Leaf size=65 \[ -\frac {2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac {(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac {b^2 (d+e x)^6}{6 e^3} \]
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Rubi [A] time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 43} \[ -\frac {2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac {(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac {b^2 (d+e x)^6}{6 e^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (d+e x)^3 \, dx\\ &=\int \left (\frac {(-b d+a e)^2 (d+e x)^3}{e^2}-\frac {2 b (b d-a e) (d+e x)^4}{e^2}+\frac {b^2 (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac {(b d-a e)^2 (d+e x)^4}{4 e^3}-\frac {2 b (b d-a e) (d+e x)^5}{5 e^3}+\frac {b^2 (d+e x)^6}{6 e^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 122, normalized size = 1.88 \[ \frac {1}{4} e x^4 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac {1}{3} d x^3 \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+a^2 d^3 x+\frac {1}{2} a d^2 x^2 (3 a e+2 b d)+\frac {1}{5} b e^2 x^5 (2 a e+3 b d)+\frac {1}{6} b^2 e^3 x^6 \]
Antiderivative was successfully verified.
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fricas [B] time = 1.17, size = 130, normalized size = 2.00 \[ \frac {1}{6} x^{6} e^{3} b^{2} + \frac {3}{5} x^{5} e^{2} d b^{2} + \frac {2}{5} x^{5} e^{3} b a + \frac {3}{4} x^{4} e d^{2} b^{2} + \frac {3}{2} x^{4} e^{2} d b a + \frac {1}{4} x^{4} e^{3} a^{2} + \frac {1}{3} x^{3} d^{3} b^{2} + 2 x^{3} e d^{2} b a + x^{3} e^{2} d a^{2} + x^{2} d^{3} b a + \frac {3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 127, normalized size = 1.95 \[ \frac {1}{6} \, b^{2} x^{6} e^{3} + \frac {3}{5} \, b^{2} d x^{5} e^{2} + \frac {3}{4} \, b^{2} d^{2} x^{4} e + \frac {1}{3} \, b^{2} d^{3} x^{3} + \frac {2}{5} \, a b x^{5} e^{3} + \frac {3}{2} \, a b d x^{4} e^{2} + 2 \, a b d^{2} x^{3} e + a b d^{3} x^{2} + \frac {1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac {3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 125, normalized size = 1.92 \[ \frac {b^{2} e^{3} x^{6}}{6}+a^{2} d^{3} x +\frac {\left (2 a b \,e^{3}+3 b^{2} d \,e^{2}\right ) x^{5}}{5}+\frac {\left (a^{2} e^{3}+6 d \,e^{2} a b +3 d^{2} e \,b^{2}\right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a^{2}+6 d^{2} e a b +d^{3} b^{2}\right ) x^{3}}{3}+\frac {\left (3 d^{2} e \,a^{2}+2 d^{3} a b \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.37, size = 124, normalized size = 1.91 \[ \frac {1}{6} \, b^{2} e^{3} x^{6} + a^{2} d^{3} x + \frac {1}{5} \, {\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 115, normalized size = 1.77 \[ x^3\,\left (a^2\,d\,e^2+2\,a\,b\,d^2\,e+\frac {b^2\,d^3}{3}\right )+x^4\,\left (\frac {a^2\,e^3}{4}+\frac {3\,a\,b\,d\,e^2}{2}+\frac {3\,b^2\,d^2\,e}{4}\right )+a^2\,d^3\,x+\frac {b^2\,e^3\,x^6}{6}+\frac {a\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )}{2}+\frac {b\,e^2\,x^5\,\left (2\,a\,e+3\,b\,d\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 133, normalized size = 2.05 \[ a^{2} d^{3} x + \frac {b^{2} e^{3} x^{6}}{6} + x^{5} \left (\frac {2 a b e^{3}}{5} + \frac {3 b^{2} d e^{2}}{5}\right ) + x^{4} \left (\frac {a^{2} e^{3}}{4} + \frac {3 a b d e^{2}}{2} + \frac {3 b^{2} d^{2} e}{4}\right ) + x^{3} \left (a^{2} d e^{2} + 2 a b d^{2} e + \frac {b^{2} d^{3}}{3}\right ) + x^{2} \left (\frac {3 a^{2} d^{2} e}{2} + a b d^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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