Optimal. Leaf size=54 \[ \frac {\left (c d^2-a e^2\right ) (a e+c d x)^4}{4 c^2 d^2}+\frac {e (a e+c d x)^5}{5 c^2 d^2} \]
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Rubi [A] time = 0.03, 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 {\left (c d^2-a e^2\right ) (a e+c d x)^4}{4 c^2 d^2}+\frac {e (a e+c d x)^5}{5 c^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^2} \, dx &=\int (a e+c d x)^3 (d+e x) \, dx\\ &=\int \left (\frac {\left (c d^2-a e^2\right ) (a e+c d x)^3}{c d}+\frac {e (a e+c d x)^4}{c d}\right ) \, dx\\ &=\frac {\left (c d^2-a e^2\right ) (a e+c d x)^4}{4 c^2 d^2}+\frac {e (a e+c d x)^5}{5 c^2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 79, normalized size = 1.46 \begin {gather*} \frac {1}{20} x \left (10 a^3 e^3 (2 d+e x)+10 a^2 c d e^2 x (3 d+2 e x)+5 a c^2 d^2 e x^2 (4 d+3 e x)+c^3 d^3 x^3 (5 d+4 e x)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.16, size = 149, normalized size = 2.76 \begin {gather*} \frac {(d+e x)^2 \left (10 a^3 e^6-30 a^2 c d^2 e^4+20 a^2 c d e^4 (d+e x)+30 a c^2 d^4 e^2-40 a c^2 d^3 e^2 (d+e x)+15 a c^2 d^2 e^2 (d+e x)^2-10 c^3 d^6+20 c^3 d^5 (d+e x)-15 c^3 d^4 (d+e x)^2+4 c^3 d^3 (d+e x)^3\right )}{20 e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 95, normalized size = 1.76 \begin {gather*} \frac {1}{5} \, c^{3} d^{3} e x^{5} + a^{3} d e^{3} x + \frac {1}{4} \, {\left (c^{3} d^{4} + 3 \, a c^{2} d^{2} e^{2}\right )} x^{4} + {\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 146, normalized size = 2.70 \begin {gather*} \frac {1}{20} \, {\left (4 \, c^{3} d^{3} - \frac {15 \, {\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} e^{\left (-1\right )}}{x e + d} + \frac {20 \, {\left (c^{3} d^{5} e^{2} - 2 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {10 \, {\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 )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )} {\left (x e + d\right )}^{5} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 136, normalized size = 2.52 \begin {gather*} \frac {c^{3} d^{3} e \,x^{5}}{5}+a^{3} d \,e^{3} x +\frac {\left (2 a \,c^{2} d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right ) c^{2} d^{2}\right ) x^{4}}{4}+\frac {\left (a^{2} c d \,e^{3}+a \,c^{2} d^{3} e +2 \left (a \,e^{2}+c \,d^{2}\right ) a c d e \right ) x^{3}}{3}+\frac {\left (2 a^{2} c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right ) a^{2} e^{2}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 95, normalized size = 1.76 \begin {gather*} \frac {1}{5} \, c^{3} d^{3} e x^{5} + a^{3} d e^{3} x + \frac {1}{4} \, {\left (c^{3} d^{4} + 3 \, a c^{2} d^{2} e^{2}\right )} x^{4} + {\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 91, normalized size = 1.69 \begin {gather*} x^2\,\left (\frac {a^3\,e^4}{2}+\frac {3\,c\,a^2\,d^2\,e^2}{2}\right )+x^4\,\left (\frac {c^3\,d^4}{4}+\frac {3\,a\,c^2\,d^2\,e^2}{4}\right )+\frac {c^3\,d^3\,e\,x^5}{5}+a^3\,d\,e^3\,x+a\,c\,d\,e\,x^3\,\left (c\,d^2+a\,e^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 100, normalized size = 1.85 \begin {gather*} a^{3} d e^{3} x + \frac {c^{3} d^{3} e x^{5}}{5} + x^{4} \left (\frac {3 a c^{2} d^{2} e^{2}}{4} + \frac {c^{3} d^{4}}{4}\right ) + x^{3} \left (a^{2} c d e^{3} + a c^{2} d^{3} e\right ) + x^{2} \left (\frac {a^{3} e^{4}}{2} + \frac {3 a^{2} c d^{2} e^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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