Optimal. Leaf size=82 \[ \frac {(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^3 (m+1)}-\frac {(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac {c (d+e x)^{m+3}}{e^3 (m+3)} \]
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Rubi [A] time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {698} \begin {gather*} \frac {(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^3 (m+1)}-\frac {(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac {c (d+e x)^{m+3}}{e^3 (m+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int (d+e x)^m \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^m}{e^2}+\frac {(-2 c d+b e) (d+e x)^{1+m}}{e^2}+\frac {c (d+e x)^{2+m}}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^3 (1+m)}-\frac {(2 c d-b e) (d+e x)^{2+m}}{e^3 (2+m)}+\frac {c (d+e x)^{3+m}}{e^3 (3+m)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 1.01 \begin {gather*} \frac {(d+e x)^{m+1} \left (c d^2-e (b d-a e)\right )}{e^3 (m+1)}-\frac {(2 c d-b e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac {c (d+e x)^{m+3}}{e^3 (m+3)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (a+b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 201, normalized size = 2.45 \begin {gather*} \frac {{\left (a d e^{2} m^{2} + 2 \, c d^{3} - 3 \, b d^{2} e + 6 \, a d e^{2} + {\left (c e^{3} m^{2} + 3 \, c e^{3} m + 2 \, c e^{3}\right )} x^{3} + {\left (3 \, b e^{3} + {\left (c d e^{2} + b e^{3}\right )} m^{2} + {\left (c d e^{2} + 4 \, b e^{3}\right )} m\right )} x^{2} - {\left (b d^{2} e - 5 \, a d e^{2}\right )} m + {\left (6 \, a e^{3} + {\left (b d e^{2} + a e^{3}\right )} m^{2} - {\left (2 \, c d^{2} e - 3 \, b d e^{2} - 5 \, a e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 353, normalized size = 4.30 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c m^{2} x^{3} e^{3} + {\left (x e + d\right )}^{m} c d m^{2} x^{2} e^{2} + {\left (x e + d\right )}^{m} b m^{2} x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} c m x^{3} e^{3} + {\left (x e + d\right )}^{m} b d m^{2} x e^{2} + {\left (x e + d\right )}^{m} c d m x^{2} e^{2} - 2 \, {\left (x e + d\right )}^{m} c d^{2} m x e + {\left (x e + d\right )}^{m} a m^{2} x e^{3} + 4 \, {\left (x e + d\right )}^{m} b m x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} c x^{3} e^{3} + {\left (x e + d\right )}^{m} a d m^{2} e^{2} + 3 \, {\left (x e + d\right )}^{m} b d m x e^{2} - {\left (x e + d\right )}^{m} b d^{2} m e + 2 \, {\left (x e + d\right )}^{m} c d^{3} + 5 \, {\left (x e + d\right )}^{m} a m x e^{3} + 3 \, {\left (x e + d\right )}^{m} b x^{2} e^{3} + 5 \, {\left (x e + d\right )}^{m} a d m e^{2} - 3 \, {\left (x e + d\right )}^{m} b d^{2} e + 6 \, {\left (x e + d\right )}^{m} a x e^{3} + 6 \, {\left (x e + d\right )}^{m} a d e^{2}}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 135, normalized size = 1.65 \begin {gather*} \frac {\left (c \,e^{2} m^{2} x^{2}+b \,e^{2} m^{2} x +3 c \,e^{2} m \,x^{2}+a \,e^{2} m^{2}+4 b \,e^{2} m x -2 c d e m x +2 c \,e^{2} x^{2}+5 a \,e^{2} m -b d e m +3 b \,e^{2} x -2 c d e x +6 a \,e^{2}-3 b d e +2 c \,d^{2}\right ) \left (e x +d \right )^{m +1}}{\left (m^{3}+6 m^{2}+11 m +6\right ) e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 132, normalized size = 1.61 \begin {gather*} \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 201, normalized size = 2.45 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {c\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}+\frac {d\,\left (2\,c\,d^2-b\,d\,e\,m-3\,b\,d\,e+a\,e^2\,m^2+5\,a\,e^2\,m+6\,a\,e^2\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {x\,\left (-2\,c\,d^2\,e\,m+b\,d\,e^2\,m^2+3\,b\,d\,e^2\,m+a\,e^3\,m^2+5\,a\,e^3\,m+6\,a\,e^3\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {x^2\,\left (m+1\right )\,\left (3\,b\,e+b\,e\,m+c\,d\,m\right )}{e\,\left (m^3+6\,m^2+11\,m+6\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.14, size = 1416, normalized size = 17.27
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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