Optimal. Leaf size=179 \[ -\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^6 d^6 (a e+c d x)}-\frac {5 e \left (c d^2-a e^2\right )^4}{2 c^6 d^6 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^5}{3 c^6 d^6 (a e+c d x)^3}+\frac {10 e^3 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^6 d^6}+\frac {e^4 x \left (5 c d^2-4 a e^2\right )}{c^5 d^5}+\frac {e^5 x^2}{2 c^4 d^4} \]
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Rubi [A] time = 0.17, antiderivative size = 179, 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 = {626, 43} \begin {gather*} \frac {e^4 x \left (5 c d^2-4 a e^2\right )}{c^5 d^5}-\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^6 d^6 (a e+c d x)}-\frac {5 e \left (c d^2-a e^2\right )^4}{2 c^6 d^6 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^5}{3 c^6 d^6 (a e+c d x)^3}+\frac {10 e^3 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^6 d^6}+\frac {e^5 x^2}{2 c^4 d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^9}{\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)^5}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac {5 c d^2 e^4-4 a e^6}{c^5 d^5}+\frac {e^5 x}{c^4 d^4}+\frac {\left (c d^2-a e^2\right )^5}{c^5 d^5 (a e+c d x)^4}+\frac {5 e \left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)^3}+\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}+\frac {10 e^3 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}\right ) \, dx\\ &=\frac {e^4 \left (5 c d^2-4 a e^2\right ) x}{c^5 d^5}+\frac {e^5 x^2}{2 c^4 d^4}-\frac {\left (c d^2-a e^2\right )^5}{3 c^6 d^6 (a e+c d x)^3}-\frac {5 e \left (c d^2-a e^2\right )^4}{2 c^6 d^6 (a e+c d x)^2}-\frac {10 e^2 \left (c d^2-a e^2\right )^3}{c^6 d^6 (a e+c d x)}+\frac {10 e^3 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^6 d^6}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 259, normalized size = 1.45 \begin {gather*} \frac {47 a^5 e^{10}+a^4 c d e^8 (81 e x-130 d)+a^3 c^2 d^2 e^6 \left (110 d^2-270 d e x-9 e^2 x^2\right )-a^2 c^3 d^3 e^4 \left (20 d^3-270 d^2 e x+90 d e^2 x^2+63 e^3 x^3\right )-5 a c^4 d^4 e^2 \left (d^4+12 d^3 e x-36 d^2 e^2 x^2-18 d e^3 x^3+3 e^4 x^4\right )+60 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^3 \log (a e+c d x)+c^5 d^5 \left (-2 d^5-15 d^4 e x-60 d^3 e^2 x^2+30 d e^4 x^4+3 e^5 x^5\right )}{6 c^6 d^6 (a e+c d x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^9}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 488, normalized size = 2.73 \begin {gather*} \frac {3 \, c^{5} d^{5} e^{5} x^{5} - 2 \, c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} - 20 \, a^{2} c^{3} d^{6} e^{4} + 110 \, a^{3} c^{2} d^{4} e^{6} - 130 \, a^{4} c d^{2} e^{8} + 47 \, a^{5} e^{10} + 15 \, {\left (2 \, c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 9 \, {\left (10 \, a c^{4} d^{5} e^{5} - 7 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{3} - 3 \, {\left (20 \, c^{5} d^{8} e^{2} - 60 \, a c^{4} d^{6} e^{4} + 30 \, a^{2} c^{3} d^{4} e^{6} + 3 \, a^{3} c^{2} d^{2} e^{8}\right )} x^{2} - 3 \, {\left (5 \, c^{5} d^{9} e + 20 \, a c^{4} d^{7} e^{3} - 90 \, a^{2} c^{3} d^{5} e^{5} + 90 \, a^{3} c^{2} d^{3} e^{7} - 27 \, a^{4} c d e^{9}\right )} x + 60 \, {\left (a^{3} c^{2} d^{4} e^{6} - 2 \, a^{4} c d^{2} e^{8} + a^{5} e^{10} + {\left (c^{5} d^{7} e^{3} - 2 \, a c^{4} d^{5} e^{5} + a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 3 \, {\left (a c^{4} d^{6} e^{4} - 2 \, a^{2} c^{3} d^{4} e^{6} + a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 3 \, {\left (a^{2} c^{3} d^{5} e^{5} - 2 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x\right )} \log \left (c d x + a e\right )}{6 \, {\left (c^{9} d^{9} x^{3} + 3 \, a c^{8} d^{8} e x^{2} + 3 \, a^{2} c^{7} d^{7} e^{2} x + a^{3} c^{6} d^{6} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] 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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maple [B] time = 0.05, size = 436, normalized size = 2.44 \begin {gather*} \frac {a^{5} e^{10}}{3 \left (c d x +a e \right )^{3} c^{6} d^{6}}-\frac {5 a^{4} e^{8}}{3 \left (c d x +a e \right )^{3} c^{5} d^{4}}+\frac {10 a^{3} e^{6}}{3 \left (c d x +a e \right )^{3} c^{4} d^{2}}-\frac {10 a^{2} e^{4}}{3 \left (c d x +a e \right )^{3} c^{3}}+\frac {5 a \,d^{2} e^{2}}{3 \left (c d x +a e \right )^{3} c^{2}}-\frac {d^{4}}{3 \left (c d x +a e \right )^{3} c}-\frac {5 a^{4} e^{9}}{2 \left (c d x +a e \right )^{2} c^{6} d^{6}}+\frac {10 a^{3} e^{7}}{\left (c d x +a e \right )^{2} c^{5} d^{4}}-\frac {15 a^{2} e^{5}}{\left (c d x +a e \right )^{2} c^{4} d^{2}}+\frac {10 a \,e^{3}}{\left (c d x +a e \right )^{2} c^{3}}-\frac {5 d^{2} e}{2 \left (c d x +a e \right )^{2} c^{2}}+\frac {10 a^{3} e^{8}}{\left (c d x +a e \right ) c^{6} d^{6}}-\frac {30 a^{2} e^{6}}{\left (c d x +a e \right ) c^{5} d^{4}}+\frac {30 a \,e^{4}}{\left (c d x +a e \right ) c^{4} d^{2}}-\frac {10 e^{2}}{\left (c d x +a e \right ) c^{3}}+\frac {e^{5} x^{2}}{2 c^{4} d^{4}}+\frac {10 a^{2} e^{7} \ln \left (c d x +a e \right )}{c^{6} d^{6}}-\frac {20 a \,e^{5} \ln \left (c d x +a e \right )}{c^{5} d^{4}}-\frac {4 a \,e^{6} x}{c^{5} d^{5}}+\frac {10 e^{3} \ln \left (c d x +a e \right )}{c^{4} d^{2}}+\frac {5 e^{4} x}{c^{4} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 326, normalized size = 1.82 \begin {gather*} -\frac {2 \, c^{5} d^{10} + 5 \, a c^{4} d^{8} e^{2} + 20 \, a^{2} c^{3} d^{6} e^{4} - 110 \, a^{3} c^{2} d^{4} e^{6} + 130 \, a^{4} c d^{2} e^{8} - 47 \, a^{5} e^{10} + 60 \, {\left (c^{5} d^{8} e^{2} - 3 \, a c^{4} d^{6} e^{4} + 3 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 15 \, {\left (c^{5} d^{9} e + 4 \, a c^{4} d^{7} e^{3} - 18 \, a^{2} c^{3} d^{5} e^{5} + 20 \, a^{3} c^{2} d^{3} e^{7} - 7 \, a^{4} c d e^{9}\right )} x}{6 \, {\left (c^{9} d^{9} x^{3} + 3 \, a c^{8} d^{8} e x^{2} + 3 \, a^{2} c^{7} d^{7} e^{2} x + a^{3} c^{6} d^{6} e^{3}\right )}} + \frac {c d e^{5} x^{2} + 2 \, {\left (5 \, c d^{2} e^{4} - 4 \, a e^{6}\right )} x}{2 \, c^{5} d^{5}} + \frac {10 \, {\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 331, normalized size = 1.85 \begin {gather*} x\,\left (\frac {5\,e^4}{c^4\,d^3}-\frac {4\,a\,e^6}{c^5\,d^5}\right )-\frac {x^2\,\left (-10\,a^3\,c\,d\,e^8+30\,a^2\,c^2\,d^3\,e^6-30\,a\,c^3\,d^5\,e^4+10\,c^4\,d^7\,e^2\right )+x\,\left (-\frac {35\,a^4\,e^9}{2}+50\,a^3\,c\,d^2\,e^7-45\,a^2\,c^2\,d^4\,e^5+10\,a\,c^3\,d^6\,e^3+\frac {5\,c^4\,d^8\,e}{2}\right )+\frac {-47\,a^5\,e^{10}+130\,a^4\,c\,d^2\,e^8-110\,a^3\,c^2\,d^4\,e^6+20\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2+2\,c^5\,d^{10}}{6\,c\,d}}{a^3\,c^5\,d^5\,e^3+3\,a^2\,c^6\,d^6\,e^2\,x+3\,a\,c^7\,d^7\,e\,x^2+c^8\,d^8\,x^3}+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (10\,a^2\,e^7-20\,a\,c\,d^2\,e^5+10\,c^2\,d^4\,e^3\right )}{c^6\,d^6}+\frac {e^5\,x^2}{2\,c^4\,d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.65, size = 337, normalized size = 1.88 \begin {gather*} x \left (- \frac {4 a e^{6}}{c^{5} d^{5}} + \frac {5 e^{4}}{c^{4} d^{3}}\right ) + \frac {47 a^{5} e^{10} - 130 a^{4} c d^{2} e^{8} + 110 a^{3} c^{2} d^{4} e^{6} - 20 a^{2} c^{3} d^{6} e^{4} - 5 a c^{4} d^{8} e^{2} - 2 c^{5} d^{10} + x^{2} \left (60 a^{3} c^{2} d^{2} e^{8} - 180 a^{2} c^{3} d^{4} e^{6} + 180 a c^{4} d^{6} e^{4} - 60 c^{5} d^{8} e^{2}\right ) + x \left (105 a^{4} c d e^{9} - 300 a^{3} c^{2} d^{3} e^{7} + 270 a^{2} c^{3} d^{5} e^{5} - 60 a c^{4} d^{7} e^{3} - 15 c^{5} d^{9} e\right )}{6 a^{3} c^{6} d^{6} e^{3} + 18 a^{2} c^{7} d^{7} e^{2} x + 18 a c^{8} d^{8} e x^{2} + 6 c^{9} d^{9} x^{3}} + \frac {e^{5} x^{2}}{2 c^{4} d^{4}} + \frac {10 e^{3} \left (a e^{2} - c d^{2}\right )^{2} \log {\left (a e + c d x \right )}}{c^{6} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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