Optimal. Leaf size=131 \[ \frac {\left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^5 d^5}+\frac {e x \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac {(d+e x)^2 \left (c d^2-a e^2\right )^2}{2 c^3 d^3}+\frac {(d+e x)^3 \left (c d^2-a e^2\right )}{3 c^2 d^2}+\frac {(d+e x)^4}{4 c d} \]
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Rubi [A] time = 0.07, antiderivative size = 131, 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 x \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac {(d+e x)^2 \left (c d^2-a e^2\right )^2}{2 c^3 d^3}+\frac {(d+e x)^3 \left (c d^2-a e^2\right )}{3 c^2 d^2}+\frac {\left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^5 d^5}+\frac {(d+e x)^4}{4 c d} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac {(d+e x)^4}{a e+c d x} \, dx\\ &=\int \left (\frac {e \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac {\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)}+\frac {e \left (c d^2-a e^2\right )^2 (d+e x)}{c^3 d^3}+\frac {e \left (c d^2-a e^2\right ) (d+e x)^2}{c^2 d^2}+\frac {e (d+e x)^3}{c d}\right ) \, dx\\ &=\frac {e \left (c d^2-a e^2\right )^3 x}{c^4 d^4}+\frac {\left (c d^2-a e^2\right )^2 (d+e x)^2}{2 c^3 d^3}+\frac {\left (c d^2-a e^2\right ) (d+e x)^3}{3 c^2 d^2}+\frac {(d+e x)^4}{4 c d}+\frac {\left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^5 d^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 134, normalized size = 1.02 \begin {gather*} \frac {c d e x \left (-12 a^3 e^6+6 a^2 c d e^4 (8 d+e x)-4 a c^2 d^2 e^2 \left (18 d^2+6 d e x+e^2 x^2\right )+c^3 d^3 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{12 c^5 d^5} \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)^5}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 207, normalized size = 1.58 \begin {gather*} \frac {3 \, c^{4} d^{4} e^{4} x^{4} + 4 \, {\left (4 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (6 \, c^{4} d^{6} e^{2} - 4 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 12 \, {\left (4 \, c^{4} d^{7} e - 6 \, a c^{3} d^{5} e^{3} + 4 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x + 12 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (c d x + a e\right )}{12 \, c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 360, normalized size = 2.75 \begin {gather*} \frac {{\left (3 \, c^{3} d^{3} x^{4} e^{8} + 16 \, c^{3} d^{4} x^{3} e^{7} + 36 \, c^{3} d^{5} x^{2} e^{6} + 48 \, c^{3} d^{6} x e^{5} - 4 \, a c^{2} d^{2} x^{3} e^{9} - 24 \, a c^{2} d^{3} x^{2} e^{8} - 72 \, a c^{2} d^{4} x e^{7} + 6 \, a^{2} c d x^{2} e^{10} + 48 \, a^{2} c d^{2} x e^{9} - 12 \, a^{3} x e^{11}\right )} e^{\left (-4\right )}}{12 \, c^{4} d^{4}} + \frac {{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{5} d^{5}} + \frac {{\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 239, normalized size = 1.82 \begin {gather*} \frac {e^{4} x^{4}}{4 c d}-\frac {a \,e^{5} x^{3}}{3 c^{2} d^{2}}+\frac {4 e^{3} x^{3}}{3 c}+\frac {a^{2} e^{6} x^{2}}{2 c^{3} d^{3}}-\frac {2 a \,e^{4} x^{2}}{c^{2} d}+\frac {3 d \,e^{2} x^{2}}{c}+\frac {a^{4} e^{8} \ln \left (c d x +a e \right )}{c^{5} d^{5}}-\frac {4 a^{3} e^{6} \ln \left (c d x +a e \right )}{c^{4} d^{3}}-\frac {a^{3} e^{7} x}{c^{4} d^{4}}+\frac {6 a^{2} e^{4} \ln \left (c d x +a e \right )}{c^{3} d}+\frac {4 a^{2} e^{5} x}{c^{3} d^{2}}-\frac {4 a d \,e^{2} \ln \left (c d x +a e \right )}{c^{2}}-\frac {6 a \,e^{3} x}{c^{2}}+\frac {d^{3} \ln \left (c d x +a e \right )}{c}+\frac {4 d^{2} e x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 205, normalized size = 1.56 \begin {gather*} \frac {3 \, c^{3} d^{3} e^{4} x^{4} + 4 \, {\left (4 \, c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{3} + 6 \, {\left (6 \, c^{3} d^{5} e^{2} - 4 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{2} + 12 \, {\left (4 \, c^{3} d^{6} e - 6 \, a c^{2} d^{4} e^{3} + 4 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} x}{12 \, c^{4} d^{4}} + \frac {{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 217, normalized size = 1.66 \begin {gather*} x^3\,\left (\frac {4\,e^3}{3\,c}-\frac {a\,e^5}{3\,c^2\,d^2}\right )+x\,\left (\frac {4\,d^2\,e}{c}-\frac {a\,e\,\left (\frac {6\,d\,e^2}{c}-\frac {a\,e\,\left (\frac {4\,e^3}{c}-\frac {a\,e^5}{c^2\,d^2}\right )}{c\,d}\right )}{c\,d}\right )+x^2\,\left (\frac {3\,d\,e^2}{c}-\frac {a\,e\,\left (\frac {4\,e^3}{c}-\frac {a\,e^5}{c^2\,d^2}\right )}{2\,c\,d}\right )+\frac {\ln \left (a\,e+c\,d\,x\right )\,\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^5\,d^5}+\frac {e^4\,x^4}{4\,c\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 153, normalized size = 1.17 \begin {gather*} x^{3} \left (- \frac {a e^{5}}{3 c^{2} d^{2}} + \frac {4 e^{3}}{3 c}\right ) + x^{2} \left (\frac {a^{2} e^{6}}{2 c^{3} d^{3}} - \frac {2 a e^{4}}{c^{2} d} + \frac {3 d e^{2}}{c}\right ) + x \left (- \frac {a^{3} e^{7}}{c^{4} d^{4}} + \frac {4 a^{2} e^{5}}{c^{3} d^{2}} - \frac {6 a e^{3}}{c^{2}} + \frac {4 d^{2} e}{c}\right ) + \frac {e^{4} x^{4}}{4 c d} + \frac {\left (a e^{2} - c d^{2}\right )^{4} \log {\left (a e + c d x \right )}}{c^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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