Optimal. Leaf size=78 \[ \frac {(b d-a e)^2 (d+e x)^{1+m}}{e^3 (1+m)}-\frac {2 b (b d-a e) (d+e x)^{2+m}}{e^3 (2+m)}+\frac {b^2 (d+e x)^{3+m}}{e^3 (3+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 78, 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, 45}
\begin {gather*} \frac {(b d-a e)^2 (d+e x)^{m+1}}{e^3 (m+1)}-\frac {2 b (b d-a e) (d+e x)^{m+2}}{e^3 (m+2)}+\frac {b^2 (d+e x)^{m+3}}{e^3 (m+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (d+e x)^m \, dx\\ &=\int \left (\frac {(-b d+a e)^2 (d+e x)^m}{e^2}-\frac {2 b (b d-a e) (d+e x)^{1+m}}{e^2}+\frac {b^2 (d+e x)^{2+m}}{e^2}\right ) \, dx\\ &=\frac {(b d-a e)^2 (d+e x)^{1+m}}{e^3 (1+m)}-\frac {2 b (b d-a e) (d+e x)^{2+m}}{e^3 (2+m)}+\frac {b^2 (d+e x)^{3+m}}{e^3 (3+m)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 67, normalized size = 0.86 \begin {gather*} \frac {(d+e x)^{1+m} \left (\frac {(b d-a e)^2}{1+m}-\frac {2 b (b d-a e) (d+e x)}{2+m}+\frac {b^2 (d+e x)^2}{3+m}\right )}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs.
\(2(78)=156\).
time = 0.49, size = 159, normalized size = 2.04
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{1+m} \left (b^{2} e^{2} m^{2} x^{2}+2 a b \,e^{2} m^{2} x +3 b^{2} e^{2} m \,x^{2}+a^{2} e^{2} m^{2}+8 a b \,e^{2} m x -2 b^{2} d e m x +2 b^{2} x^{2} e^{2}+5 a^{2} e^{2} m -2 a b d e m +6 a b \,e^{2} x -2 b^{2} d e x +6 a^{2} e^{2}-6 a b d e +2 b^{2} d^{2}\right )}{e^{3} \left (m^{3}+6 m^{2}+11 m +6\right )}\) | \(159\) |
norman | \(\frac {b^{2} x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}+\frac {d \left (a^{2} e^{2} m^{2}+5 a^{2} e^{2} m -2 a b d e m +6 a^{2} e^{2}-6 a b d e +2 b^{2} d^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{3}+6 m^{2}+11 m +6\right )}+\frac {\left (a^{2} e^{2} m^{2}+2 a b d e \,m^{2}+5 a^{2} e^{2} m +6 a b d e m -2 b^{2} d^{2} m +6 a^{2} e^{2}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+6 m^{2}+11 m +6\right )}+\frac {\left (2 a e m +b d m +6 a e \right ) b \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+5 m +6\right )}\) | \(224\) |
risch | \(\frac {\left (b^{2} e^{3} m^{2} x^{3}+2 a b \,e^{3} m^{2} x^{2}+b^{2} d \,e^{2} m^{2} x^{2}+3 b^{2} e^{3} m \,x^{3}+a^{2} e^{3} m^{2} x +2 a b d \,e^{2} m^{2} x +8 a b \,e^{3} m \,x^{2}+b^{2} d \,e^{2} m \,x^{2}+2 b^{2} x^{3} e^{3}+a^{2} d \,e^{2} m^{2}+5 a^{2} e^{3} m x +6 a b d \,e^{2} m x +6 a b \,e^{3} x^{2}-2 b^{2} d^{2} e m x +5 a^{2} d \,e^{2} m +6 a^{2} e^{3} x -2 a b \,d^{2} e m +6 a^{2} d \,e^{2}-6 a b \,d^{2} e +2 b^{2} d^{3}\right ) \left (e x +d \right )^{m}}{\left (2+m \right ) \left (3+m \right ) \left (1+m \right ) e^{3}}\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 141, normalized size = 1.81 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} a^{2} e^{\left (-1\right )}}{m + 1} + \frac {2 \, {\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} a b e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} x^{3} e^{3} + {\left (m^{2} + m\right )} d x^{2} e^{2} - 2 \, d^{2} m x e + 2 \, d^{3}\right )} b^{2} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 193 vs.
\(2 (80) = 160\).
time = 3.04, size = 193, normalized size = 2.47 \begin {gather*} \frac {{\left (2 \, b^{2} d^{3} + {\left ({\left (b^{2} m^{2} + 3 \, b^{2} m + 2 \, b^{2}\right )} x^{3} + 2 \, {\left (a b m^{2} + 4 \, a b m + 3 \, a b\right )} x^{2} + {\left (a^{2} m^{2} + 5 \, a^{2} m + 6 \, a^{2}\right )} x\right )} e^{3} + {\left (a^{2} d m^{2} + 5 \, a^{2} d m + 6 \, a^{2} d + {\left (b^{2} d m^{2} + b^{2} d m\right )} x^{2} + 2 \, {\left (a b d m^{2} + 3 \, a b d m\right )} x\right )} e^{2} - 2 \, {\left (b^{2} d^{2} m x + a b d^{2} m + 3 \, a b d^{2}\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1506 vs.
\(2 (66) = 132\).
time = 0.55, size = 1506, normalized size = 19.31 \begin {gather*} \begin {cases} d^{m} \left (a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}\right ) & \text {for}\: e = 0 \\- \frac {a^{2} e^{2}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} - \frac {2 a b d e}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} - \frac {4 a b e^{2} x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {2 b^{2} d^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {3 b^{2} d^{2}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {4 b^{2} d e x \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {4 b^{2} d e x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} + \frac {2 b^{2} e^{2} x^{2} \log {\left (\frac {d}{e} + x \right )}}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} & \text {for}\: m = -3 \\- \frac {a^{2} e^{2}}{d e^{3} + e^{4} x} + \frac {2 a b d e \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} + \frac {2 a b d e}{d e^{3} + e^{4} x} + \frac {2 a b e^{2} x \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} - \frac {2 b^{2} d^{2} \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} - \frac {2 b^{2} d^{2}}{d e^{3} + e^{4} x} - \frac {2 b^{2} d e x \log {\left (\frac {d}{e} + x \right )}}{d e^{3} + e^{4} x} + \frac {b^{2} e^{2} x^{2}}{d e^{3} + e^{4} x} & \text {for}\: m = -2 \\\frac {a^{2} \log {\left (\frac {d}{e} + x \right )}}{e} - \frac {2 a b d \log {\left (\frac {d}{e} + x \right )}}{e^{2}} + \frac {2 a b x}{e} + \frac {b^{2} d^{2} \log {\left (\frac {d}{e} + x \right )}}{e^{3}} - \frac {b^{2} d x}{e^{2}} + \frac {b^{2} x^{2}}{2 e} & \text {for}\: m = -1 \\\frac {a^{2} d e^{2} m^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {5 a^{2} d e^{2} m \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {6 a^{2} d e^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {a^{2} e^{3} m^{2} x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {5 a^{2} e^{3} m x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {6 a^{2} e^{3} x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} - \frac {2 a b d^{2} e m \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} - \frac {6 a b d^{2} e \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {2 a b d e^{2} m^{2} x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {6 a b d e^{2} m x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {2 a b e^{3} m^{2} x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {8 a b e^{3} m x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {6 a b e^{3} x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {2 b^{2} d^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} - \frac {2 b^{2} d^{2} e m x \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {b^{2} d e^{2} m^{2} x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {b^{2} d e^{2} m x^{2} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {b^{2} e^{3} m^{2} x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {3 b^{2} e^{3} m x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} + \frac {2 b^{2} e^{3} x^{3} \left (d + e x\right )^{m}}{e^{3} m^{3} + 6 e^{3} m^{2} + 11 e^{3} m + 6 e^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 388 vs.
\(2 (80) = 160\).
time = 1.27, size = 388, normalized size = 4.97 \begin {gather*} \frac {{\left (x e + d\right )}^{m} b^{2} m^{2} x^{3} e^{3} + {\left (x e + d\right )}^{m} b^{2} d m^{2} x^{2} e^{2} + 2 \, {\left (x e + d\right )}^{m} a b m^{2} x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} b^{2} m x^{3} e^{3} + 2 \, {\left (x e + d\right )}^{m} a b d m^{2} x e^{2} + {\left (x e + d\right )}^{m} b^{2} d m x^{2} e^{2} - 2 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m x e + {\left (x e + d\right )}^{m} a^{2} m^{2} x e^{3} + 8 \, {\left (x e + d\right )}^{m} a b m x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} b^{2} x^{3} e^{3} + {\left (x e + d\right )}^{m} a^{2} d m^{2} e^{2} + 6 \, {\left (x e + d\right )}^{m} a b d m x e^{2} - 2 \, {\left (x e + d\right )}^{m} a b d^{2} m e + 2 \, {\left (x e + d\right )}^{m} b^{2} d^{3} + 5 \, {\left (x e + d\right )}^{m} a^{2} m x e^{3} + 6 \, {\left (x e + d\right )}^{m} a b x^{2} e^{3} + 5 \, {\left (x e + d\right )}^{m} a^{2} d m e^{2} - 6 \, {\left (x e + d\right )}^{m} a b d^{2} e + 6 \, {\left (x e + d\right )}^{m} a^{2} x e^{3} + 6 \, {\left (x e + d\right )}^{m} a^{2} 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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Mupad [B]
time = 0.72, size = 226, normalized size = 2.90 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {b^2\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}+\frac {d\,\left (a^2\,e^2\,m^2+5\,a^2\,e^2\,m+6\,a^2\,e^2-2\,a\,b\,d\,e\,m-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {x\,\left (a^2\,e^3\,m^2+5\,a^2\,e^3\,m+6\,a^2\,e^3+2\,a\,b\,d\,e^2\,m^2+6\,a\,b\,d\,e^2\,m-2\,b^2\,d^2\,e\,m\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {b\,x^2\,\left (m+1\right )\,\left (6\,a\,e+2\,a\,e\,m+b\,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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