Optimal. Leaf size=217 \[ \frac {e^4 \left (15 c^2 d^4-24 a c d^2 e^2+10 a^2 e^4\right ) x}{c^6 d^6}+\frac {e^5 \left (3 c d^2-2 a e^2\right ) x^2}{c^5 d^5}+\frac {e^6 x^3}{3 c^4 d^4}-\frac {\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}-\frac {3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}+\frac {20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7} \]
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Rubi [A]
time = 0.19, antiderivative size = 217, 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 {e^4 x \left (10 a^2 e^4-24 a c d^2 e^2+15 c^2 d^4\right )}{c^6 d^6}-\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}-\frac {3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}+\frac {20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}+\frac {e^5 x^2 \left (3 c d^2-2 a e^2\right )}{c^5 d^5}+\frac {e^6 x^3}{3 c^4 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac {(d+e x)^6}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac {15 c^2 d^4 e^4-24 a c d^2 e^6+10 a^2 e^8}{c^6 d^6}+\frac {2 e^5 \left (3 c d^2-2 a e^2\right ) x}{c^5 d^5}+\frac {e^6 x^2}{c^4 d^4}+\frac {\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^4}+\frac {6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)^3}+\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)^2}+\frac {20 \left (c d^2 e-a e^3\right )^3}{c^6 d^6 (a e+c d x)}\right ) \, dx\\ &=\frac {e^4 \left (15 c^2 d^4-24 a c d^2 e^2+10 a^2 e^4\right ) x}{c^6 d^6}+\frac {e^5 \left (3 c d^2-2 a e^2\right ) x^2}{c^5 d^5}+\frac {e^6 x^3}{3 c^4 d^4}-\frac {\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}-\frac {3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}+\frac {20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 335, normalized size = 1.54 \begin {gather*} \frac {-37 a^6 e^{12}+3 a^5 c d e^{10} (47 d-17 e x)+3 a^4 c^2 d^2 e^8 \left (-65 d^2+81 d e x+13 e^2 x^2\right )+a^3 c^3 d^3 e^6 \left (110 d^3-405 d^2 e x-27 d e^2 x^2+73 e^3 x^3\right )-3 a^2 c^4 d^4 e^4 \left (5 d^4-90 d^3 e x+45 d^2 e^2 x^2+63 d e^3 x^3-5 e^4 x^4\right )-3 a c^5 d^5 e^2 \left (d^5+15 d^4 e x-60 d^3 e^2 x^2-45 d^2 e^3 x^3+15 d e^4 x^4+e^5 x^5\right )+c^6 d^6 \left (-d^6-9 d^5 e x-45 d^4 e^2 x^2+45 d^2 e^4 x^4+9 d e^5 x^5+e^6 x^6\right )-60 e^3 \left (-c d^2+a e^2\right )^3 (a e+c d x)^3 \log (a e+c d x)}{3 c^7 d^7 (a e+c d x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.97, size = 399, normalized size = 1.84
method | result | size |
default | \(\frac {e^{4} \left (\frac {1}{3} c^{2} d^{2} e^{2} x^{3}-2 a c d \,e^{3} x^{2}+3 c^{2} d^{3} e \,x^{2}+10 a^{2} e^{4} x -24 a c \,d^{2} e^{2} x +15 c^{2} d^{4} x \right )}{c^{6} d^{6}}-\frac {15 e^{2} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}{c^{7} d^{7} \left (c d x +a e \right )}-\frac {a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}{3 c^{7} d^{7} \left (c d x +a e \right )^{3}}-\frac {20 e^{3} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \ln \left (c d x +a e \right )}{c^{7} d^{7}}+\frac {3 e \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right )}{c^{7} d^{7} \left (c d x +a e \right )^{2}}\) | \(399\) |
risch | \(\frac {e^{6} x^{3}}{3 c^{4} d^{4}}-\frac {2 e^{7} a \,x^{2}}{c^{5} d^{5}}+\frac {3 e^{5} x^{2}}{c^{4} d^{3}}+\frac {10 e^{8} a^{2} x}{c^{6} d^{6}}-\frac {24 e^{6} a x}{c^{5} d^{4}}+\frac {15 e^{4} x}{c^{4} d^{2}}+\frac {\left (-15 a^{4} e^{10} d c +60 e^{8} a^{3} d^{3} c^{2}-90 a^{2} e^{6} d^{5} c^{3}+60 a \,c^{4} d^{7} e^{4}-15 e^{2} d^{9} c^{5}\right ) x^{2}-3 e \left (9 a^{5} e^{10}-35 a^{4} c \,d^{2} e^{8}+50 a^{3} c^{2} d^{4} e^{6}-30 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}+c^{5} d^{10}\right ) x -\frac {37 a^{6} e^{12}-141 a^{5} c \,d^{2} e^{10}+195 a^{4} c^{2} d^{4} e^{8}-110 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}+3 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}{3 c d}}{c^{6} d^{6} \left (c d x +a e \right )^{3}}-\frac {20 e^{9} \ln \left (c d x +a e \right ) a^{3}}{c^{7} d^{7}}+\frac {60 e^{7} \ln \left (c d x +a e \right ) a^{2}}{c^{6} d^{5}}-\frac {60 e^{5} \ln \left (c d x +a e \right ) a}{c^{5} d^{3}}+\frac {20 e^{3} \ln \left (c d x +a e \right )}{c^{4} d}\) | \(418\) |
norman | \(\frac {\frac {e^{7} \left (5 a^{2} e^{4}-18 a c \,d^{2} e^{2}+25 c^{2} d^{4}\right ) x^{7}}{c^{3} d^{3}}-\frac {110 a^{6} e^{12}-285 a^{5} c \,d^{2} e^{10}+186 a^{4} c^{2} d^{4} e^{8}+53 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}+3 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}{3 d^{4} c^{7}}+\frac {e^{9} x^{9}}{3 c d}-\frac {\left (110 a^{6} e^{18}+525 c \,d^{2} a^{5} e^{16}-1299 a^{4} c^{2} d^{4} e^{14}-28 a^{3} c^{3} d^{6} e^{12}+1041 a^{2} c^{4} d^{8} e^{10}+921 a \,c^{5} d^{10} e^{8}+326 c^{6} d^{12} e^{6}\right ) x^{3}}{3 c^{7} d^{7} e^{3}}-\frac {\left (110 a^{6} e^{16}-15 c \,d^{2} a^{5} e^{14}-429 a^{4} c^{2} d^{4} e^{12}+296 a^{3} c^{3} d^{6} e^{10}+270 a^{2} c^{4} d^{8} e^{8}+151 a \,c^{5} d^{10} e^{6}+25 c^{6} d^{12} e^{4}\right ) x^{2}}{c^{7} d^{6} e^{2}}-\frac {\left (110 a^{6} e^{14}-195 a^{5} c \,d^{2} e^{12}-39 a^{4} c^{2} d^{4} e^{10}+179 a^{3} c^{3} d^{6} e^{8}+88 a^{2} c^{4} d^{8} e^{6}+18 a \,c^{5} d^{10} e^{4}+4 c^{6} d^{12} e^{2}\right ) x}{c^{7} d^{5} e}-\frac {\left (90 a^{5} e^{16}-45 a^{4} c \,d^{2} e^{14}-279 a^{3} c^{2} d^{4} e^{12}+221 a^{2} c^{3} d^{6} e^{10}+195 a \,c^{4} d^{8} e^{8}+196 c^{5} d^{10} e^{6}\right ) x^{4}}{c^{6} d^{6} e^{2}}-\frac {\left (60 a^{4} e^{14}-135 a^{3} c \,d^{2} e^{12}+66 d^{4} a^{2} c^{2} e^{10}+5 a \,c^{3} d^{6} e^{8}+130 c^{4} d^{8} e^{6}\right ) x^{5}}{c^{5} d^{5} e}-\frac {e^{8} \left (e^{2} a -4 c \,d^{2}\right ) x^{8}}{c^{2} d^{2}}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}-\frac {20 e^{3} \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right ) \ln \left (c d x +a e \right )}{c^{7} d^{7}}\) | \(707\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 399, normalized size = 1.84 \begin {gather*} -\frac {c^{6} d^{12} + 3 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 110 \, a^{3} c^{3} d^{6} e^{6} + 195 \, a^{4} c^{2} d^{4} e^{8} - 141 \, a^{5} c d^{2} e^{10} + 37 \, a^{6} e^{12} + 45 \, {\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 9 \, {\left (c^{6} d^{11} e + 5 \, a c^{5} d^{9} e^{3} - 30 \, a^{2} c^{4} d^{7} e^{5} + 50 \, a^{3} c^{3} d^{5} e^{7} - 35 \, a^{4} c^{2} d^{3} e^{9} + 9 \, a^{5} c d e^{11}\right )} x}{3 \, {\left (c^{10} d^{10} x^{3} + 3 \, a c^{9} d^{9} x^{2} e + 3 \, a^{2} c^{8} d^{8} x e^{2} + a^{3} c^{7} d^{7} e^{3}\right )}} + \frac {c^{2} d^{2} x^{3} e^{6} + 3 \, {\left (3 \, c^{2} d^{3} e^{5} - 2 \, a c d e^{7}\right )} x^{2} + 3 \, {\left (15 \, c^{2} d^{4} e^{4} - 24 \, a c d^{2} e^{6} + 10 \, a^{2} e^{8}\right )} x}{3 \, c^{6} d^{6}} + \frac {20 \, {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \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. 633 vs.
\(2 (206) = 412\).
time = 3.43, size = 633, normalized size = 2.92 \begin {gather*} -\frac {9 \, c^{6} d^{11} x e + c^{6} d^{12} + 45 \, a c^{5} d^{9} x e^{3} + 51 \, a^{5} c d x e^{11} + 37 \, a^{6} e^{12} - 3 \, {\left (13 \, a^{4} c^{2} d^{2} x^{2} + 47 \, a^{5} c d^{2}\right )} e^{10} - {\left (73 \, a^{3} c^{3} d^{3} x^{3} + 243 \, a^{4} c^{2} d^{3} x\right )} e^{9} - 3 \, {\left (5 \, a^{2} c^{4} d^{4} x^{4} - 9 \, a^{3} c^{3} d^{4} x^{2} - 65 \, a^{4} c^{2} d^{4}\right )} e^{8} + 3 \, {\left (a c^{5} d^{5} x^{5} + 63 \, a^{2} c^{4} d^{5} x^{3} + 135 \, a^{3} c^{3} d^{5} x\right )} e^{7} - {\left (c^{6} d^{6} x^{6} - 45 \, a c^{5} d^{6} x^{4} - 135 \, a^{2} c^{4} d^{6} x^{2} + 110 \, a^{3} c^{3} d^{6}\right )} e^{6} - 9 \, {\left (c^{6} d^{7} x^{5} + 15 \, a c^{5} d^{7} x^{3} + 30 \, a^{2} c^{4} d^{7} x\right )} e^{5} - 15 \, {\left (3 \, c^{6} d^{8} x^{4} + 12 \, a c^{5} d^{8} x^{2} - a^{2} c^{4} d^{8}\right )} e^{4} + 3 \, {\left (15 \, c^{6} d^{10} x^{2} + a c^{5} d^{10}\right )} e^{2} - 60 \, {\left (c^{6} d^{9} x^{3} e^{3} + 3 \, a c^{5} d^{8} x^{2} e^{4} - 3 \, a^{5} c d x e^{11} - a^{6} e^{12} - 3 \, {\left (a^{4} c^{2} d^{2} x^{2} - a^{5} c d^{2}\right )} e^{10} - {\left (a^{3} c^{3} d^{3} x^{3} - 9 \, a^{4} c^{2} d^{3} x\right )} e^{9} + 3 \, {\left (3 \, a^{3} c^{3} d^{4} x^{2} - a^{4} c^{2} d^{4}\right )} e^{8} + 3 \, {\left (a^{2} c^{4} d^{5} x^{3} - 3 \, a^{3} c^{3} d^{5} x\right )} e^{7} - {\left (9 \, a^{2} c^{4} d^{6} x^{2} - a^{3} c^{3} d^{6}\right )} e^{6} - 3 \, {\left (a c^{5} d^{7} x^{3} - a^{2} c^{4} d^{7} x\right )} e^{5}\right )} \log \left (c d x + a e\right )}{3 \, {\left (c^{10} d^{10} x^{3} + 3 \, a c^{9} d^{9} x^{2} e + 3 \, a^{2} c^{8} d^{8} x e^{2} + a^{3} c^{7} d^{7} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 375, normalized size = 1.73 \begin {gather*} \frac {20 \, {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{7}} - \frac {c^{6} d^{12} + 3 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 110 \, a^{3} c^{3} d^{6} e^{6} + 195 \, a^{4} c^{2} d^{4} e^{8} - 141 \, a^{5} c d^{2} e^{10} + 37 \, a^{6} e^{12} + 45 \, {\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 9 \, {\left (c^{6} d^{11} e + 5 \, a c^{5} d^{9} e^{3} - 30 \, a^{2} c^{4} d^{7} e^{5} + 50 \, a^{3} c^{3} d^{5} e^{7} - 35 \, a^{4} c^{2} d^{3} e^{9} + 9 \, a^{5} c d e^{11}\right )} x}{3 \, {\left (c d x + a e\right )}^{3} c^{7} d^{7}} + \frac {c^{8} d^{8} x^{3} e^{6} + 9 \, c^{8} d^{9} x^{2} e^{5} + 45 \, c^{8} d^{10} x e^{4} - 6 \, a c^{7} d^{7} x^{2} e^{7} - 72 \, a c^{7} d^{8} x e^{6} + 30 \, a^{2} c^{6} d^{6} x e^{8}}{3 \, c^{12} d^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 452, normalized size = 2.08 \begin {gather*} x^2\,\left (\frac {3\,e^5}{c^4\,d^3}-\frac {2\,a\,e^7}{c^5\,d^5}\right )-x\,\left (\frac {6\,a^2\,e^8}{c^6\,d^6}-\frac {15\,e^4}{c^4\,d^2}+\frac {4\,a\,e\,\left (\frac {6\,e^5}{c^4\,d^3}-\frac {4\,a\,e^7}{c^5\,d^5}\right )}{c\,d}\right )-\frac {x\,\left (27\,a^5\,e^{11}-105\,a^4\,c\,d^2\,e^9+150\,a^3\,c^2\,d^4\,e^7-90\,a^2\,c^3\,d^6\,e^5+15\,a\,c^4\,d^8\,e^3+3\,c^5\,d^{10}\,e\right )+x^2\,\left (15\,a^4\,c\,d\,e^{10}-60\,a^3\,c^2\,d^3\,e^8+90\,a^2\,c^3\,d^5\,e^6-60\,a\,c^4\,d^7\,e^4+15\,c^5\,d^9\,e^2\right )+\frac {37\,a^6\,e^{12}-141\,a^5\,c\,d^2\,e^{10}+195\,a^4\,c^2\,d^4\,e^8-110\,a^3\,c^3\,d^6\,e^6+15\,a^2\,c^4\,d^8\,e^4+3\,a\,c^5\,d^{10}\,e^2+c^6\,d^{12}}{3\,c\,d}}{a^3\,c^6\,d^6\,e^3+3\,a^2\,c^7\,d^7\,e^2\,x+3\,a\,c^8\,d^8\,e\,x^2+c^9\,d^9\,x^3}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (20\,a^3\,e^9-60\,a^2\,c\,d^2\,e^7+60\,a\,c^2\,d^4\,e^5-20\,c^3\,d^6\,e^3\right )}{c^7\,d^7}+\frac {e^6\,x^3}{3\,c^4\,d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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