Optimal. Leaf size=54 \[ -\frac {c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac {e}{2 c^2 d^2 (a e+c d x)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 54, 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 {c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac {e}{2 c^2 d^2 (a e+c d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{\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}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac {c d^2-a e^2}{c d (a e+c d x)^4}+\frac {e}{c d (a e+c d x)^3}\right ) \, dx\\ &=-\frac {c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac {e}{2 c^2 d^2 (a e+c d x)^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.69 \begin {gather*} -\frac {a e^2+c d (2 d+3 e x)}{6 c^2 d^2 (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)^5}{\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 [A] time = 0.39, size = 74, normalized size = 1.37 \begin {gather*} -\frac {3 \, c d e x + 2 \, c d^{2} + a e^{2}}{6 \, {\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} e x^{2} + 3 \, a^{2} c^{3} d^{3} e^{2} x + a^{3} c^{2} d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 641, normalized size = 11.87 \begin {gather*} -\frac {3 \, c^{7} d^{13} x^{4} e^{4} + 11 \, c^{7} d^{14} x^{3} e^{3} + 15 \, c^{7} d^{15} x^{2} e^{2} + 9 \, c^{7} d^{16} x e + 2 \, c^{7} d^{17} - 18 \, a c^{6} d^{11} x^{4} e^{6} - 65 \, a c^{6} d^{12} x^{3} e^{5} - 87 \, a c^{6} d^{13} x^{2} e^{4} - 51 \, a c^{6} d^{14} x e^{3} - 11 \, a c^{6} d^{15} e^{2} + 45 \, a^{2} c^{5} d^{9} x^{4} e^{8} + 159 \, a^{2} c^{5} d^{10} x^{3} e^{7} + 207 \, a^{2} c^{5} d^{11} x^{2} e^{6} + 117 \, a^{2} c^{5} d^{12} x e^{5} + 24 \, a^{2} c^{5} d^{13} e^{4} - 60 \, a^{3} c^{4} d^{7} x^{4} e^{10} - 205 \, a^{3} c^{4} d^{8} x^{3} e^{9} - 255 \, a^{3} c^{4} d^{9} x^{2} e^{8} - 135 \, a^{3} c^{4} d^{10} x e^{7} - 25 \, a^{3} c^{4} d^{11} e^{6} + 45 \, a^{4} c^{3} d^{5} x^{4} e^{12} + 145 \, a^{4} c^{3} d^{6} x^{3} e^{11} + 165 \, a^{4} c^{3} d^{7} x^{2} e^{10} + 75 \, a^{4} c^{3} d^{8} x e^{9} + 10 \, a^{4} c^{3} d^{9} e^{8} - 18 \, a^{5} c^{2} d^{3} x^{4} e^{14} - 51 \, a^{5} c^{2} d^{4} x^{3} e^{13} - 45 \, a^{5} c^{2} d^{5} x^{2} e^{12} - 9 \, a^{5} c^{2} d^{6} x e^{11} + 3 \, a^{5} c^{2} d^{7} e^{10} + 3 \, a^{6} c d x^{4} e^{16} + 5 \, a^{6} c d^{2} x^{3} e^{15} - 3 \, a^{6} c d^{3} x^{2} e^{14} - 9 \, a^{6} c d^{4} x e^{13} - 4 \, a^{6} c d^{5} e^{12} + a^{7} x^{3} e^{17} + 3 \, a^{7} d x^{2} e^{16} + 3 \, a^{7} d^{2} x e^{15} + a^{7} d^{3} e^{14}}{6 \, {\left (c^{8} d^{14} - 6 \, a c^{7} d^{12} e^{2} + 15 \, a^{2} c^{6} d^{10} e^{4} - 20 \, a^{3} c^{5} d^{8} e^{6} + 15 \, a^{4} c^{4} d^{6} e^{8} - 6 \, a^{5} c^{3} d^{4} e^{10} + a^{6} c^{2} d^{2} e^{12}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 51, normalized size = 0.94 \begin {gather*} -\frac {e}{2 \left (c d x +a e \right )^{2} c^{2} d^{2}}-\frac {-a \,e^{2}+c \,d^{2}}{3 \left (c d x +a e \right )^{3} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 74, normalized size = 1.37 \begin {gather*} -\frac {3 \, c d e x + 2 \, c d^{2} + a e^{2}}{6 \, {\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} e x^{2} + 3 \, a^{2} c^{3} d^{3} e^{2} x + a^{3} c^{2} d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 77, normalized size = 1.43 \begin {gather*} -\frac {\frac {2\,c\,d^2+a\,e^2}{6\,c^2\,d^2}+\frac {e\,x}{2\,c\,d}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 80, normalized size = 1.48 \begin {gather*} \frac {- a e^{2} - 2 c d^{2} - 3 c d e x}{6 a^{3} c^{2} d^{2} e^{3} + 18 a^{2} c^{3} d^{3} e^{2} x + 18 a c^{4} d^{4} e x^{2} + 6 c^{5} d^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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