Optimal. Leaf size=219 \[ \frac {15 e^2 \left (c d^2-a e^2\right )^4 x}{c^6 d^6}-\frac {\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac {10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac {5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac {3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac {e^6 (a e+c d x)^5}{5 c^7 d^7}+\frac {6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7} \]
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Rubi [A]
time = 0.20, antiderivative size = 219, 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^6 (a e+c d x)^5}{5 c^7 d^7}-\frac {\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac {6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7}+\frac {3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac {5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac {10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac {15 e^2 x \left (c d^2-a e^2\right )^4}{c^6 d^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {(d+e x)^6}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6}+\frac {\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^2}+\frac {6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {20 \left (c d^2 e-a e^3\right )^3 (a e+c d x)}{c^6 d^6}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {6 \left (c d^2 e^5-a e^7\right ) (a e+c d x)^3}{c^6 d^6}+\frac {e^6 (a e+c d x)^4}{c^6 d^6}\right ) \, dx\\ &=\frac {15 e^2 \left (c d^2-a e^2\right )^4 x}{c^6 d^6}-\frac {\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac {10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac {5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac {3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac {e^6 (a e+c d x)^5}{5 c^7 d^7}+\frac {6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 339, normalized size = 1.55 \begin {gather*} \frac {-10 a^6 e^{12}+10 a^5 c d e^{10} (6 d+5 e x)-30 a^4 c^2 d^2 e^8 \left (5 d^2+8 d e x-e^2 x^2\right )+10 a^3 c^3 d^3 e^6 \left (20 d^3+45 d^2 e x-15 d e^2 x^2-e^3 x^3\right )-5 a^2 c^4 d^4 e^4 \left (30 d^4+80 d^3 e x-60 d^2 e^2 x^2-10 d e^3 x^3-e^4 x^4\right )+a c^5 d^5 e^2 \left (60 d^5+150 d^4 e x-300 d^3 e^2 x^2-100 d^2 e^3 x^3-25 d e^4 x^4-3 e^5 x^5\right )+c^6 d^6 \left (-10 d^6+150 d^4 e^2 x^2+100 d^3 e^3 x^3+50 d^2 e^4 x^4+15 d e^5 x^5+2 e^6 x^6\right )-60 e \left (-c d^2+a e^2\right )^5 (a e+c d x) \log (a e+c d x)}{10 c^7 d^7 (a e+c d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.85, size = 402, normalized size = 1.84
method | result | size |
default | \(\frac {e^{2} \left (\frac {1}{5} e^{4} x^{5} c^{4} d^{4}-\frac {1}{2} a \,c^{3} d^{3} e^{5} x^{4}+\frac {3}{2} c^{4} d^{5} e^{3} x^{4}+a^{2} c^{2} d^{2} e^{6} x^{3}-4 a \,c^{3} d^{4} e^{4} x^{3}+5 c^{4} d^{6} e^{2} x^{3}-2 a^{3} c d \,e^{7} x^{2}+9 a^{2} c^{2} d^{3} e^{5} x^{2}-15 a \,c^{3} d^{5} e^{3} x^{2}+10 c^{4} d^{7} e \,x^{2}+5 a^{4} e^{8} x -24 a^{3} c \,d^{2} e^{6} x +45 a^{2} c^{2} d^{4} e^{4} x -40 a \,c^{3} d^{6} e^{2} x +15 c^{4} d^{8} x \right )}{c^{6} d^{6}}-\frac {a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}{c^{7} d^{7} \left (c d x +a e \right )}-\frac {6 e \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \ln \left (c d x +a e \right )}{c^{7} d^{7}}\) | \(402\) |
norman | \(\frac {\frac {e^{3} \left (3 a^{4} e^{8}-16 a^{3} c \,d^{2} e^{6}+35 a^{2} c^{2} d^{4} e^{4}-40 a \,c^{3} d^{6} e^{2}+25 c^{4} d^{8}\right ) x^{3}}{c^{5} d^{5}}-\frac {6 a^{6} e^{12}-27 a^{5} c \,d^{2} e^{10}+45 a^{4} c^{2} d^{4} e^{8}-30 a^{3} c^{3} d^{6} e^{6}+9 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}}{c^{7} d^{6}}+\frac {e^{7} x^{7}}{5 c d}-\frac {e^{4} \left (2 e^{6} a^{3}-11 e^{4} d^{2} a^{2} c +25 d^{4} e^{2} c^{2} a -30 d^{6} c^{3}\right ) x^{4}}{2 c^{4} d^{4}}+\frac {e^{5} \left (5 a^{2} e^{4}-28 a c \,d^{2} e^{2}+65 c^{2} d^{4}\right ) x^{5}}{10 c^{3} d^{3}}-\frac {e^{6} \left (3 e^{2} a -17 c \,d^{2}\right ) x^{6}}{10 c^{2} d^{2}}-\frac {\left (6 a^{6} e^{14}-27 a^{5} c \,d^{2} e^{12}+48 a^{4} c^{2} d^{4} e^{10}-45 a^{3} c^{3} d^{6} e^{8}+30 a^{2} c^{4} d^{8} e^{6}-21 a \,c^{5} d^{10} e^{4}+16 c^{6} d^{12} e^{2}\right ) x}{c^{7} d^{7} e}}{\left (c d x +a e \right ) \left (e x +d \right )}-\frac {6 e \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \ln \left (c d x +a e \right )}{c^{7} d^{7}}\) | \(486\) |
risch | \(-\frac {e^{7} a \,x^{4}}{2 c^{3} d^{3}}+\frac {e^{8} a^{2} x^{3}}{c^{4} d^{4}}+\frac {6 d^{3} e \ln \left (c d x +a e \right )}{c^{2}}-\frac {40 e^{4} a x}{c^{3}}-\frac {a^{6} e^{12}}{c^{7} d^{7} \left (c d x +a e \right )}+\frac {6 a^{5} e^{10}}{c^{6} d^{5} \left (c d x +a e \right )}-\frac {15 a^{4} e^{8}}{c^{5} d^{3} \left (c d x +a e \right )}+\frac {20 a^{3} e^{6}}{c^{4} d \left (c d x +a e \right )}-\frac {15 d \,a^{2} e^{4}}{c^{3} \left (c d x +a e \right )}+\frac {6 d^{3} a \,e^{2}}{c^{2} \left (c d x +a e \right )}+\frac {5 e^{10} a^{4} x}{c^{6} d^{6}}-\frac {24 e^{8} a^{3} x}{c^{5} d^{4}}+\frac {45 e^{6} a^{2} x}{c^{4} d^{2}}-\frac {4 e^{6} a \,x^{3}}{c^{3} d^{2}}-\frac {2 e^{9} a^{3} x^{2}}{c^{5} d^{5}}+\frac {9 e^{7} a^{2} x^{2}}{c^{4} d^{3}}-\frac {15 e^{5} a \,x^{2}}{c^{3} d}-\frac {d^{5}}{c \left (c d x +a e \right )}+\frac {5 e^{4} x^{3}}{c^{2}}-\frac {6 e^{11} \ln \left (c d x +a e \right ) a^{5}}{c^{7} d^{7}}+\frac {30 e^{9} \ln \left (c d x +a e \right ) a^{4}}{c^{6} d^{5}}-\frac {60 e^{7} \ln \left (c d x +a e \right ) a^{3}}{c^{5} d^{3}}+\frac {60 e^{5} \ln \left (c d x +a e \right ) a^{2}}{c^{4} d}-\frac {30 d \,e^{3} \ln \left (c d x +a e \right ) a}{c^{3}}+\frac {e^{6} x^{5}}{5 c^{2} d^{2}}+\frac {3 e^{5} x^{4}}{2 c^{2} d}+\frac {10 e^{3} d \,x^{2}}{c^{2}}+\frac {15 e^{2} d^{2} x}{c^{2}}\) | \(502\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 375, normalized size = 1.71 \begin {gather*} -\frac {c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{c^{8} d^{8} x + a c^{7} d^{7} e} + \frac {2 \, c^{4} d^{4} x^{5} e^{6} + 5 \, {\left (3 \, c^{4} d^{5} e^{5} - a c^{3} d^{3} e^{7}\right )} x^{4} + 10 \, {\left (5 \, c^{4} d^{6} e^{4} - 4 \, a c^{3} d^{4} e^{6} + a^{2} c^{2} d^{2} e^{8}\right )} x^{3} + 10 \, {\left (10 \, c^{4} d^{7} e^{3} - 15 \, a c^{3} d^{5} e^{5} + 9 \, a^{2} c^{2} d^{3} e^{7} - 2 \, a^{3} c d e^{9}\right )} x^{2} + 10 \, {\left (15 \, c^{4} d^{8} e^{2} - 40 \, a c^{3} d^{6} e^{4} + 45 \, a^{2} c^{2} d^{4} e^{6} - 24 \, a^{3} c d^{2} e^{8} + 5 \, a^{4} e^{10}\right )} x}{10 \, c^{6} d^{6}} + \frac {6 \, {\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\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. 527 vs.
\(2 (211) = 422\).
time = 3.37, size = 527, normalized size = 2.41 \begin {gather*} -\frac {10 \, c^{6} d^{12} - 50 \, a^{5} c d x e^{11} + 10 \, a^{6} e^{12} - 30 \, {\left (a^{4} c^{2} d^{2} x^{2} + 2 \, a^{5} c d^{2}\right )} e^{10} + 10 \, {\left (a^{3} c^{3} d^{3} x^{3} + 24 \, a^{4} c^{2} d^{3} x\right )} e^{9} - 5 \, {\left (a^{2} c^{4} d^{4} x^{4} - 30 \, a^{3} c^{3} d^{4} x^{2} - 30 \, a^{4} c^{2} d^{4}\right )} e^{8} + {\left (3 \, a c^{5} d^{5} x^{5} - 50 \, a^{2} c^{4} d^{5} x^{3} - 450 \, a^{3} c^{3} d^{5} x\right )} e^{7} - {\left (2 \, c^{6} d^{6} x^{6} - 25 \, a c^{5} d^{6} x^{4} + 300 \, a^{2} c^{4} d^{6} x^{2} + 200 \, a^{3} c^{3} d^{6}\right )} e^{6} - 5 \, {\left (3 \, c^{6} d^{7} x^{5} - 20 \, a c^{5} d^{7} x^{3} - 80 \, a^{2} c^{4} d^{7} x\right )} e^{5} - 50 \, {\left (c^{6} d^{8} x^{4} - 6 \, a c^{5} d^{8} x^{2} - 3 \, a^{2} c^{4} d^{8}\right )} e^{4} - 50 \, {\left (2 \, c^{6} d^{9} x^{3} + 3 \, a c^{5} d^{9} x\right )} e^{3} - 30 \, {\left (5 \, c^{6} d^{10} x^{2} + 2 \, a c^{5} d^{10}\right )} e^{2} - 60 \, {\left (c^{6} d^{11} x e - 5 \, a c^{5} d^{9} x e^{3} + a c^{5} d^{10} e^{2} + 10 \, a^{2} c^{4} d^{7} x e^{5} - 5 \, a^{2} c^{4} d^{8} e^{4} - 10 \, a^{3} c^{3} d^{5} x e^{7} + 10 \, a^{3} c^{3} d^{6} e^{6} + 5 \, a^{4} c^{2} d^{3} x e^{9} - 10 \, a^{4} c^{2} d^{4} e^{8} - a^{5} c d x e^{11} + 5 \, a^{5} c d^{2} e^{10} - a^{6} e^{12}\right )} \log \left (c d x + a e\right )}{10 \, {\left (c^{8} d^{8} x + a c^{7} d^{7} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.66, size = 345, normalized size = 1.58 \begin {gather*} x^{4} \left (- \frac {a e^{7}}{2 c^{3} d^{3}} + \frac {3 e^{5}}{2 c^{2} d}\right ) + x^{3} \left (\frac {a^{2} e^{8}}{c^{4} d^{4}} - \frac {4 a e^{6}}{c^{3} d^{2}} + \frac {5 e^{4}}{c^{2}}\right ) + x^{2} \left (- \frac {2 a^{3} e^{9}}{c^{5} d^{5}} + \frac {9 a^{2} e^{7}}{c^{4} d^{3}} - \frac {15 a e^{5}}{c^{3} d} + \frac {10 d e^{3}}{c^{2}}\right ) + x \left (\frac {5 a^{4} e^{10}}{c^{6} d^{6}} - \frac {24 a^{3} e^{8}}{c^{5} d^{4}} + \frac {45 a^{2} e^{6}}{c^{4} d^{2}} - \frac {40 a e^{4}}{c^{3}} + \frac {15 d^{2} e^{2}}{c^{2}}\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}}{a c^{7} d^{7} e + c^{8} d^{8} x} + \frac {e^{6} x^{5}}{5 c^{2} d^{2}} - \frac {6 e \left (a e^{2} - c d^{2}\right )^{5} \log {\left (a e + c d x \right )}}{c^{7} d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.78, size = 395, normalized size = 1.80 \begin {gather*} \frac {6 \, {\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{7} d^{7}} - \frac {c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{{\left (c d x + a e\right )} c^{7} d^{7}} + \frac {2 \, c^{8} d^{8} x^{5} e^{6} + 15 \, c^{8} d^{9} x^{4} e^{5} + 50 \, c^{8} d^{10} x^{3} e^{4} + 100 \, c^{8} d^{11} x^{2} e^{3} + 150 \, c^{8} d^{12} x e^{2} - 5 \, a c^{7} d^{7} x^{4} e^{7} - 40 \, a c^{7} d^{8} x^{3} e^{6} - 150 \, a c^{7} d^{9} x^{2} e^{5} - 400 \, a c^{7} d^{10} x e^{4} + 10 \, a^{2} c^{6} d^{6} x^{3} e^{8} + 90 \, a^{2} c^{6} d^{7} x^{2} e^{7} + 450 \, a^{2} c^{6} d^{8} x e^{6} - 20 \, a^{3} c^{5} d^{5} x^{2} e^{9} - 240 \, a^{3} c^{5} d^{6} x e^{8} + 50 \, a^{4} c^{4} d^{4} x e^{10}}{10 \, c^{10} d^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 625, normalized size = 2.85 \begin {gather*} x^4\,\left (\frac {3\,e^5}{2\,c^2\,d}-\frac {a\,e^7}{2\,c^3\,d^3}\right )+x^2\,\left (\frac {10\,d\,e^3}{c^2}+\frac {a\,e\,\left (\frac {a^2\,e^8}{c^4\,d^4}-\frac {15\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c\,d}\right )}{c\,d}-\frac {a^2\,e^2\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{2\,c^2\,d^2}\right )-x^3\,\left (\frac {a^2\,e^8}{3\,c^4\,d^4}-\frac {5\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{3\,c\,d}\right )+x\,\left (\frac {15\,d^2\,e^2}{c^2}+\frac {a^2\,e^2\,\left (\frac {a^2\,e^8}{c^4\,d^4}-\frac {15\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c\,d}\right )}{c^2\,d^2}-\frac {2\,a\,e\,\left (\frac {20\,d\,e^3}{c^2}+\frac {2\,a\,e\,\left (\frac {a^2\,e^8}{c^4\,d^4}-\frac {15\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c\,d}\right )}{c\,d}-\frac {a^2\,e^2\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c^2\,d^2}\right )}{c\,d}\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}}{c\,d\,\left (x\,c^7\,d^7+a\,e\,c^6\,d^6\right )}+\frac {e^6\,x^5}{5\,c^2\,d^2}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (6\,a^5\,e^{11}-30\,a^4\,c\,d^2\,e^9+60\,a^3\,c^2\,d^4\,e^7-60\,a^2\,c^3\,d^6\,e^5+30\,a\,c^4\,d^8\,e^3-6\,c^5\,d^{10}\,e\right )}{c^7\,d^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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