Optimal. Leaf size=90 \[ \frac {\left (c d^2-a e^2\right )^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{4+m}}{e^3 (4+m)}+\frac {c^2 d^2 (d+e x)^{5+m}}{e^3 (5+m)} \]
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Rubi [A]
time = 0.04, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45}
\begin {gather*} \frac {\left (c d^2-a e^2\right )^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{m+4}}{e^3 (m+4)}+\frac {c^2 d^2 (d+e x)^{m+5}}{e^3 (m+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx &=\int (a e+c d x)^2 (d+e x)^{2+m} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2 (d+e x)^{2+m}}{e^2}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{3+m}}{e^2}+\frac {c^2 d^2 (d+e x)^{4+m}}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{4+m}}{e^3 (4+m)}+\frac {c^2 d^2 (d+e x)^{5+m}}{e^3 (5+m)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 79, normalized size = 0.88 \begin {gather*} \frac {(d+e x)^{3+m} \left (\frac {\left (c d^2-a e^2\right )^2}{3+m}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)}{4+m}+\frac {c^2 d^2 (d+e x)^2}{5+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. \(182\) vs.
\(2(90)=180\).
time = 0.81, size = 183, normalized size = 2.03
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{3+m} \left (c^{2} d^{2} e^{2} m^{2} x^{2}+2 a c d \,e^{3} m^{2} x +7 c^{2} d^{2} e^{2} m \,x^{2}+a^{2} e^{4} m^{2}+16 a c d \,e^{3} m x -2 c^{2} d^{3} e m x +12 c^{2} d^{2} x^{2} e^{2}+9 a^{2} e^{4} m -2 a c \,d^{2} e^{2} m +30 a c d \,e^{3} x -6 c^{2} d^{3} e x +20 a^{2} e^{4}-10 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right )}{e^{3} \left (m^{3}+12 m^{2}+47 m +60\right )}\) | \(183\) |
norman | \(\frac {\left (a^{2} e^{4} m^{2}+6 a c \,d^{2} e^{2} m^{2}+3 c^{2} d^{4} m^{2}+9 a^{2} e^{4} m +46 a c \,d^{2} e^{2} m +15 c^{2} d^{4} m +20 a^{2} e^{4}+80 a c \,d^{2} e^{2}+20 c^{2} d^{4}\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{3}+12 m^{2}+47 m +60}+\frac {d^{3} \left (a^{2} e^{4} m^{2}+9 a^{2} e^{4} m -2 a c \,d^{2} e^{2} m +20 a^{2} e^{4}-10 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {c^{2} d^{2} e^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {d^{2} \left (3 a^{2} e^{4} m^{2}+2 a c \,d^{2} e^{2} m^{2}+27 a^{2} e^{4} m +10 a c \,d^{2} e^{2} m -2 c^{2} d^{4} m +60 a^{2} e^{4}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {d \left (3 a^{2} e^{4} m^{2}+6 a c \,d^{2} e^{2} m^{2}+c^{2} d^{4} m^{2}+27 a^{2} e^{4} m +42 a c \,d^{2} e^{2} m +c^{2} d^{4} m +60 a^{2} e^{4}+60 a c \,d^{2} e^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {c d e \left (2 a \,e^{2} m +3 c \,d^{2} m +10 e^{2} a +10 c \,d^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{m^{2}+9 m +20}\) | \(495\) |
risch | \(\frac {\left (c^{2} d^{2} e^{5} m^{2} x^{5}+2 a c d \,e^{6} m^{2} x^{4}+3 c^{2} d^{3} e^{4} m^{2} x^{4}+7 c^{2} d^{2} e^{5} m \,x^{5}+a^{2} e^{7} m^{2} x^{3}+6 a c \,d^{2} e^{5} m^{2} x^{3}+16 a c d \,e^{6} m \,x^{4}+3 c^{2} d^{4} e^{3} m^{2} x^{3}+19 c^{2} d^{3} e^{4} m \,x^{4}+12 c^{2} d^{2} e^{5} x^{5}+3 a^{2} d \,e^{6} m^{2} x^{2}+9 a^{2} e^{7} m \,x^{3}+6 a c \,d^{3} e^{4} m^{2} x^{2}+46 a c \,d^{2} e^{5} m \,x^{3}+30 a c d \,e^{6} x^{4}+c^{2} d^{5} e^{2} m^{2} x^{2}+15 c^{2} d^{4} e^{3} m \,x^{3}+30 c^{2} d^{3} e^{4} x^{4}+3 a^{2} d^{2} e^{5} m^{2} x +27 a^{2} d \,e^{6} m \,x^{2}+20 a^{2} e^{7} x^{3}+2 a c \,d^{4} e^{3} m^{2} x +42 a c \,d^{3} e^{4} m \,x^{2}+80 a c \,d^{2} e^{5} x^{3}+c^{2} d^{5} e^{2} m \,x^{2}+20 c^{2} d^{4} e^{3} x^{3}+a^{2} d^{3} e^{4} m^{2}+27 a^{2} d^{2} e^{5} m x +60 a^{2} d \,e^{6} x^{2}+10 a c \,d^{4} e^{3} m x +60 a c \,d^{3} e^{4} x^{2}-2 c^{2} d^{6} e m x +9 a^{2} d^{3} e^{4} m +60 a^{2} d^{2} e^{5} x -2 a c \,d^{5} e^{2} m +20 a^{2} d^{3} e^{4}-10 a c \,d^{5} e^{2}+2 c^{2} d^{7}\right ) \left (e x +d \right )^{m}}{\left (4+m \right ) \left (5+m \right ) \left (3+m \right ) e^{3}}\) | \(536\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 703 vs.
\(2 (88) = 176\).
time = 0.32, size = 703, normalized size = 7.81 \begin {gather*} \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 )} c^{2} d^{4} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {2 \, {\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} a c d^{3} e^{\left (m \log \left (x e + d\right ) - 1\right )}}{m^{2} + 3 \, m + 2} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} e^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d x^{3} e^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} x^{2} e^{2} + 6 \, d^{3} m x e - 6 \, d^{4}\right )} c^{2} d^{3} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left (x e + d\right )}^{m + 1} a^{2} d^{2} e}{m + 1} + \frac {4 \, {\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 )} a c d^{2} e^{\left (m \log \left (x e + d\right ) - 1\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} e^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d x^{4} e^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} x^{3} e^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} x^{2} e^{2} - 24 \, d^{4} m x e + 24 \, d^{5}\right )} c^{2} d^{2} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} + \frac {2 \, {\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} a^{2} d e^{\left (m \log \left (x e + d\right ) + 1\right )}}{m^{2} + 3 \, m + 2} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} e^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d x^{3} e^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} x^{2} e^{2} + 6 \, d^{3} m x e - 6 \, d^{4}\right )} a c d e^{\left (m \log \left (x e + d\right ) - 1\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \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 )} a^{2} e^{\left (m \log \left (x e + d\right ) + 1\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. 426 vs.
\(2 (88) = 176\).
time = 3.07, size = 426, normalized size = 4.73 \begin {gather*} -\frac {{\left (2 \, c^{2} d^{6} m x e - 2 \, c^{2} d^{7} - {\left (a^{2} m^{2} + 9 \, a^{2} m + 20 \, a^{2}\right )} x^{3} e^{7} - {\left (2 \, {\left (a c d m^{2} + 8 \, a c d m + 15 \, a c d\right )} x^{4} + 3 \, {\left (a^{2} d m^{2} + 9 \, a^{2} d m + 20 \, a^{2} d\right )} x^{2}\right )} e^{6} - {\left ({\left (c^{2} d^{2} m^{2} + 7 \, c^{2} d^{2} m + 12 \, c^{2} d^{2}\right )} x^{5} + 2 \, {\left (3 \, a c d^{2} m^{2} + 23 \, a c d^{2} m + 40 \, a c d^{2}\right )} x^{3} + 3 \, {\left (a^{2} d^{2} m^{2} + 9 \, a^{2} d^{2} m + 20 \, a^{2} d^{2}\right )} x\right )} e^{5} - {\left (a^{2} d^{3} m^{2} + 9 \, a^{2} d^{3} m + 20 \, a^{2} d^{3} + {\left (3 \, c^{2} d^{3} m^{2} + 19 \, c^{2} d^{3} m + 30 \, c^{2} d^{3}\right )} x^{4} + 6 \, {\left (a c d^{3} m^{2} + 7 \, a c d^{3} m + 10 \, a c d^{3}\right )} x^{2}\right )} e^{4} - {\left ({\left (3 \, c^{2} d^{4} m^{2} + 15 \, c^{2} d^{4} m + 20 \, c^{2} d^{4}\right )} x^{3} + 2 \, {\left (a c d^{4} m^{2} + 5 \, a c d^{4} m\right )} x\right )} e^{3} + {\left (2 \, a c d^{5} m + 10 \, a c d^{5} - {\left (c^{2} d^{5} m^{2} + c^{2} d^{5} m\right )} x^{2}\right )} e^{2}\right )} {\left (x e + d\right )}^{m} e^{\left (-3\right )}}{m^{3} + 12 \, m^{2} + 47 \, m + 60} \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. 2494 vs.
\(2 (78) = 156\).
time = 1.11, size = 2494, normalized size = 27.71 \begin {gather*} \text {Too large to display} \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. 804 vs.
\(2 (88) = 176\).
time = 1.01, size = 804, normalized size = 8.93 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c^{2} d^{2} m^{2} x^{5} e^{5} + 3 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m^{2} x^{4} e^{4} + 3 \, {\left (x e + d\right )}^{m} c^{2} d^{4} m^{2} x^{3} e^{3} + {\left (x e + d\right )}^{m} c^{2} d^{5} m^{2} x^{2} e^{2} + 7 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m x^{5} e^{5} + 19 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m x^{4} e^{4} + 15 \, {\left (x e + d\right )}^{m} c^{2} d^{4} m x^{3} e^{3} + {\left (x e + d\right )}^{m} c^{2} d^{5} m x^{2} e^{2} - 2 \, {\left (x e + d\right )}^{m} c^{2} d^{6} m x e + 2 \, {\left (x e + d\right )}^{m} a c d m^{2} x^{4} e^{6} + 6 \, {\left (x e + d\right )}^{m} a c d^{2} m^{2} x^{3} e^{5} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{2} x^{5} e^{5} + 6 \, {\left (x e + d\right )}^{m} a c d^{3} m^{2} x^{2} e^{4} + 30 \, {\left (x e + d\right )}^{m} c^{2} d^{3} x^{4} e^{4} + 2 \, {\left (x e + d\right )}^{m} a c d^{4} m^{2} x e^{3} + 20 \, {\left (x e + d\right )}^{m} c^{2} d^{4} x^{3} e^{3} + 2 \, {\left (x e + d\right )}^{m} c^{2} d^{7} + 16 \, {\left (x e + d\right )}^{m} a c d m x^{4} e^{6} + 46 \, {\left (x e + d\right )}^{m} a c d^{2} m x^{3} e^{5} + 42 \, {\left (x e + d\right )}^{m} a c d^{3} m x^{2} e^{4} + 10 \, {\left (x e + d\right )}^{m} a c d^{4} m x e^{3} - 2 \, {\left (x e + d\right )}^{m} a c d^{5} m e^{2} + {\left (x e + d\right )}^{m} a^{2} m^{2} x^{3} e^{7} + 3 \, {\left (x e + d\right )}^{m} a^{2} d m^{2} x^{2} e^{6} + 30 \, {\left (x e + d\right )}^{m} a c d x^{4} e^{6} + 3 \, {\left (x e + d\right )}^{m} a^{2} d^{2} m^{2} x e^{5} + 80 \, {\left (x e + d\right )}^{m} a c d^{2} x^{3} e^{5} + {\left (x e + d\right )}^{m} a^{2} d^{3} m^{2} e^{4} + 60 \, {\left (x e + d\right )}^{m} a c d^{3} x^{2} e^{4} - 10 \, {\left (x e + d\right )}^{m} a c d^{5} e^{2} + 9 \, {\left (x e + d\right )}^{m} a^{2} m x^{3} e^{7} + 27 \, {\left (x e + d\right )}^{m} a^{2} d m x^{2} e^{6} + 27 \, {\left (x e + d\right )}^{m} a^{2} d^{2} m x e^{5} + 9 \, {\left (x e + d\right )}^{m} a^{2} d^{3} m e^{4} + 20 \, {\left (x e + d\right )}^{m} a^{2} x^{3} e^{7} + 60 \, {\left (x e + d\right )}^{m} a^{2} d x^{2} e^{6} + 60 \, {\left (x e + d\right )}^{m} a^{2} d^{2} x e^{5} + 20 \, {\left (x e + d\right )}^{m} a^{2} d^{3} e^{4}}{m^{3} e^{3} + 12 \, m^{2} e^{3} + 47 \, m e^{3} + 60 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.05, size = 486, normalized size = 5.40 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {x^3\,\left (a^2\,e^7\,m^2+9\,a^2\,e^7\,m+20\,a^2\,e^7+6\,a\,c\,d^2\,e^5\,m^2+46\,a\,c\,d^2\,e^5\,m+80\,a\,c\,d^2\,e^5+3\,c^2\,d^4\,e^3\,m^2+15\,c^2\,d^4\,e^3\,m+20\,c^2\,d^4\,e^3\right )}{e^3\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d^3\,\left (a^2\,e^4\,m^2+9\,a^2\,e^4\,m+20\,a^2\,e^4-2\,a\,c\,d^2\,e^2\,m-10\,a\,c\,d^2\,e^2+2\,c^2\,d^4\right )}{e^3\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d^2\,x\,\left (3\,a^2\,e^4\,m^2+27\,a^2\,e^4\,m+60\,a^2\,e^4+2\,a\,c\,d^2\,e^2\,m^2+10\,a\,c\,d^2\,e^2\,m-2\,c^2\,d^4\,m\right )}{e^2\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d\,x^2\,\left (3\,a^2\,e^4\,m^2+27\,a^2\,e^4\,m+60\,a^2\,e^4+6\,a\,c\,d^2\,e^2\,m^2+42\,a\,c\,d^2\,e^2\,m+60\,a\,c\,d^2\,e^2+c^2\,d^4\,m^2+c^2\,d^4\,m\right )}{e\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {c^2\,d^2\,e^2\,x^5\,\left (m^2+7\,m+12\right )}{m^3+12\,m^2+47\,m+60}+\frac {c\,d\,e\,x^4\,\left (m+3\right )\,\left (10\,a\,e^2+10\,c\,d^2+2\,a\,e^2\,m+3\,c\,d^2\,m\right )}{m^3+12\,m^2+47\,m+60}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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