Optimal. Leaf size=139 \[ -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^5 (d+e x)}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac {2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac {c^2 x}{e^4} \]
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Rubi [A] time = 0.12, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {698} \begin {gather*} -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^5 (d+e x)}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac {2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac {c^2 x}{e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx &=\int \left (\frac {c^2}{e^4}+\frac {\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^4}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^3}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^2}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)}\right ) \, dx\\ &=\frac {c^2 x}{e^4}-\frac {\left (c d^2-b d e+a e^2\right )^2}{3 e^5 (d+e x)^3}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^5 (d+e x)^2}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^5 (d+e x)}-\frac {2 c (2 c d-b e) \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 176, normalized size = 1.27 \begin {gather*} \frac {-e^2 \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )+c e \left (b d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 a e \left (d^2+3 d e x+3 e^2 x^2\right )\right )-6 c (d+e x)^3 (2 c d-b e) \log (d+e x)+c^2 \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )}{3 e^5 (d+e 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 {\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.38, size = 282, normalized size = 2.03 \begin {gather*} \frac {3 \, c^{2} e^{4} x^{4} + 9 \, c^{2} d e^{3} x^{3} - 13 \, c^{2} d^{4} + 11 \, b c d^{3} e - a b d e^{3} - a^{2} e^{4} - {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 3 \, {\left (3 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 3 \, {\left (9 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x - 6 \, {\left (2 \, c^{2} d^{4} - b c d^{3} e + {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 3 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} x^{2} + 3 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 170, normalized size = 1.22 \begin {gather*} c^{2} x e^{\left (-4\right )} - 2 \, {\left (2 \, c^{2} d - b c e\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (13 \, c^{2} d^{4} - 11 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + a b d e^{3} + 3 \, {\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4} + 2 \, a c e^{4}\right )} x^{2} + a^{2} e^{4} + 3 \, {\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3} + 2 \, a c d e^{3} + a b e^{4}\right )} x\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 279, normalized size = 2.01 \begin {gather*} -\frac {a^{2}}{3 \left (e x +d \right )^{3} e}+\frac {2 a b d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 a c \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}-\frac {b^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {2 b c \,d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {c^{2} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {a b}{\left (e x +d \right )^{2} e^{2}}+\frac {2 a c d}{\left (e x +d \right )^{2} e^{3}}+\frac {b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 b c \,d^{2}}{\left (e x +d \right )^{2} e^{4}}+\frac {2 c^{2} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {2 a c}{\left (e x +d \right ) e^{3}}-\frac {b^{2}}{\left (e x +d \right ) e^{3}}+\frac {6 b c d}{\left (e x +d \right ) e^{4}}+\frac {2 b c \ln \left (e x +d \right )}{e^{4}}-\frac {6 c^{2} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {4 c^{2} d \ln \left (e x +d \right )}{e^{5}}+\frac {c^{2} x}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 194, normalized size = 1.40 \begin {gather*} -\frac {13 \, c^{2} d^{4} - 11 \, b c d^{3} e + a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 3 \, {\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 3 \, {\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{3 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac {c^{2} x}{e^{4}} - \frac {2 \, {\left (2 \, c^{2} d - b c e\right )} \log \left (e x + d\right )}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.74, size = 203, normalized size = 1.46 \begin {gather*} \frac {c^2\,x}{e^4}-\frac {\frac {a^2\,e^4+a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2-11\,b\,c\,d^3\,e+13\,c^2\,d^4}{3\,e}+x\,\left (b^2\,d\,e^2-9\,b\,c\,d^2\,e+a\,b\,e^3+10\,c^2\,d^3+2\,a\,c\,d\,e^2\right )+x^2\,\left (b^2\,e^3-6\,b\,c\,d\,e^2+6\,c^2\,d^2\,e+2\,a\,c\,e^3\right )}{d^3\,e^4+3\,d^2\,e^5\,x+3\,d\,e^6\,x^2+e^7\,x^3}-\frac {\ln \left (d+e\,x\right )\,\left (4\,c^2\,d-2\,b\,c\,e\right )}{e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.85, size = 218, normalized size = 1.57 \begin {gather*} \frac {c^{2} x}{e^{4}} + \frac {2 c \left (b e - 2 c d\right ) \log {\left (d + e x \right )}}{e^{5}} + \frac {- a^{2} e^{4} - a b d e^{3} - 2 a c d^{2} e^{2} - b^{2} d^{2} e^{2} + 11 b c d^{3} e - 13 c^{2} d^{4} + x^{2} \left (- 6 a c e^{4} - 3 b^{2} e^{4} + 18 b c d e^{3} - 18 c^{2} d^{2} e^{2}\right ) + x \left (- 3 a b e^{4} - 6 a c d e^{3} - 3 b^{2} d e^{3} + 27 b c d^{2} e^{2} - 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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