Optimal. Leaf size=65 \[ -\frac {b (d+e x)^6 (b d-a e)}{3 e^3}+\frac {(d+e x)^5 (b d-a e)^2}{5 e^3}+\frac {b^2 (d+e x)^7}{7 e^3} \]
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Rubi [A] time = 0.10, 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} \begin {gather*} -\frac {b (d+e x)^6 (b d-a e)}{3 e^3}+\frac {(d+e x)^5 (b d-a e)^2}{5 e^3}+\frac {b^2 (d+e x)^7}{7 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (d+e x)^4 \, dx\\ &=\int \left (\frac {(-b d+a e)^2 (d+e x)^4}{e^2}-\frac {2 b (b d-a e) (d+e x)^5}{e^2}+\frac {b^2 (d+e x)^6}{e^2}\right ) \, dx\\ &=\frac {(b d-a e)^2 (d+e x)^5}{5 e^3}-\frac {b (b d-a e) (d+e x)^6}{3 e^3}+\frac {b^2 (d+e x)^7}{7 e^3}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 148, normalized size = 2.28 \begin {gather*} \frac {1}{5} e^2 x^5 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+d e x^4 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+\frac {1}{3} d^2 x^3 \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+a^2 d^4 x+a d^3 x^2 (2 a e+b d)+\frac {1}{3} b e^3 x^6 (a e+2 b d)+\frac {1}{7} b^2 e^4 x^7 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^4 \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 = 170, normalized size = 2.62 \begin {gather*} \frac {1}{7} x^{7} e^{4} b^{2} + \frac {2}{3} x^{6} e^{3} d b^{2} + \frac {1}{3} x^{6} e^{4} b a + \frac {6}{5} x^{5} e^{2} d^{2} b^{2} + \frac {8}{5} x^{5} e^{3} d b a + \frac {1}{5} x^{5} e^{4} a^{2} + x^{4} e d^{3} b^{2} + 3 x^{4} e^{2} d^{2} b a + x^{4} e^{3} d a^{2} + \frac {1}{3} x^{3} d^{4} b^{2} + \frac {8}{3} x^{3} e d^{3} b a + 2 x^{3} e^{2} d^{2} a^{2} + x^{2} d^{4} b a + 2 x^{2} e d^{3} a^{2} + x d^{4} a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 164, normalized size = 2.52 \begin {gather*} \frac {1}{7} \, b^{2} x^{7} e^{4} + \frac {2}{3} \, b^{2} d x^{6} e^{3} + \frac {6}{5} \, b^{2} d^{2} x^{5} e^{2} + b^{2} d^{3} x^{4} e + \frac {1}{3} \, b^{2} d^{4} x^{3} + \frac {1}{3} \, a b x^{6} e^{4} + \frac {8}{5} \, a b d x^{5} e^{3} + 3 \, a b d^{2} x^{4} e^{2} + \frac {8}{3} \, a b d^{3} x^{3} e + a b d^{4} x^{2} + \frac {1}{5} \, a^{2} x^{5} e^{4} + a^{2} d x^{4} e^{3} + 2 \, a^{2} d^{2} x^{3} e^{2} + 2 \, a^{2} d^{3} x^{2} e + a^{2} d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 163, normalized size = 2.51 \begin {gather*} \frac {b^{2} e^{4} x^{7}}{7}+a^{2} d^{4} x +\frac {\left (2 e^{4} a b +4 d \,e^{3} b^{2}\right ) x^{6}}{6}+\frac {\left (a^{2} e^{4}+8 d \,e^{3} a b +6 d^{2} e^{2} b^{2}\right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a^{2}+12 d^{2} e^{2} a b +4 d^{3} e \,b^{2}\right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a^{2}+8 d^{3} e a b +d^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e \,a^{2}+2 d^{4} a b \right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.37, size = 156, normalized size = 2.40 \begin {gather*} \frac {1}{7} \, b^{2} e^{4} x^{7} + a^{2} d^{4} x + \frac {1}{3} \, {\left (2 \, b^{2} d e^{3} + a b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, b^{2} d^{2} e^{2} + 8 \, a b d e^{3} + a^{2} e^{4}\right )} x^{5} + {\left (b^{2} d^{3} e + 3 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} d^{4} + 8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2}\right )} x^{3} + {\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 144, normalized size = 2.22 \begin {gather*} x^3\,\left (2\,a^2\,d^2\,e^2+\frac {8\,a\,b\,d^3\,e}{3}+\frac {b^2\,d^4}{3}\right )+x^5\,\left (\frac {a^2\,e^4}{5}+\frac {8\,a\,b\,d\,e^3}{5}+\frac {6\,b^2\,d^2\,e^2}{5}\right )+a^2\,d^4\,x+\frac {b^2\,e^4\,x^7}{7}+a\,d^3\,x^2\,\left (2\,a\,e+b\,d\right )+\frac {b\,e^3\,x^6\,\left (a\,e+2\,b\,d\right )}{3}+d\,e\,x^4\,\left (a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 168, normalized size = 2.58 \begin {gather*} a^{2} d^{4} x + \frac {b^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac {a b e^{4}}{3} + \frac {2 b^{2} d e^{3}}{3}\right ) + x^{5} \left (\frac {a^{2} e^{4}}{5} + \frac {8 a b d e^{3}}{5} + \frac {6 b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + b^{2} d^{3} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac {8 a b d^{3} e}{3} + \frac {b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e + a b d^{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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