Optimal. Leaf size=119 \[ -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^4 (d+e x)^2}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac {3 c (2 c d-b e)}{e^4 (d+e x)}+\frac {2 c^2 \log (d+e x)}{e^4} \]
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Rubi [A] time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^4 (d+e x)^2}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac {3 c (2 c d-b e)}{e^4 (d+e x)}+\frac {2 c^2 \log (d+e x)}{e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^4} \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^4}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^3 (d+e x)^3}-\frac {3 c (2 c d-b e)}{e^3 (d+e x)^2}+\frac {2 c^2}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^4 (d+e x)^3}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{2 e^4 (d+e x)^2}+\frac {3 c (2 c d-b e)}{e^4 (d+e x)}+\frac {2 c^2 \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 111, normalized size = 0.93 \begin {gather*} \frac {-2 c e \left (a e (d+3 e x)+3 b \left (d^2+3 d e x+3 e^2 x^2\right )\right )-b e^2 (2 a e+b (d+3 e x))+2 c^2 d \left (11 d^2+27 d e x+18 e^2 x^2\right )+12 c^2 (d+e x)^3 \log (d+e x)}{6 e^4 (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 {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 179, normalized size = 1.50 \begin {gather*} \frac {22 \, c^{2} d^{3} - 6 \, b c d^{2} e - 2 \, a b e^{3} - {\left (b^{2} + 2 \, a c\right )} d e^{2} + 18 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} x^{2} + 3 \, {\left (18 \, c^{2} d^{2} e - 6 \, b c d e^{2} - {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x + 12 \, {\left (c^{2} e^{3} x^{3} + 3 \, c^{2} d e^{2} x^{2} + 3 \, c^{2} d^{2} e x + c^{2} d^{3}\right )} \log \left (e x + d\right )}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 123, normalized size = 1.03 \begin {gather*} 2 \, c^{2} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (18 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} + 3 \, {\left (18 \, c^{2} d^{2} - 6 \, b c d e - b^{2} e^{2} - 2 \, a c e^{2}\right )} x + {\left (22 \, c^{2} d^{3} - 6 \, b c d^{2} e - b^{2} d e^{2} - 2 \, a c d e^{2} - 2 \, a b e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 188, normalized size = 1.58 \begin {gather*} -\frac {a b}{3 \left (e x +d \right )^{3} e}+\frac {2 a c d}{3 \left (e x +d \right )^{3} e^{2}}+\frac {b^{2} d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {b c \,d^{2}}{\left (e x +d \right )^{3} e^{3}}+\frac {2 c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {a c}{\left (e x +d \right )^{2} e^{2}}-\frac {b^{2}}{2 \left (e x +d \right )^{2} e^{2}}+\frac {3 b c d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 c^{2} d^{2}}{\left (e x +d \right )^{2} e^{4}}-\frac {3 b c}{\left (e x +d \right ) e^{3}}+\frac {6 c^{2} d}{\left (e x +d \right ) e^{4}}+\frac {2 c^{2} \ln \left (e x +d \right )}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 146, normalized size = 1.23 \begin {gather*} \frac {22 \, c^{2} d^{3} - 6 \, b c d^{2} e - 2 \, a b e^{3} - {\left (b^{2} + 2 \, a c\right )} d e^{2} + 18 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} x^{2} + 3 \, {\left (18 \, c^{2} d^{2} e - 6 \, b c d e^{2} - {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac {2 \, c^{2} \log \left (e x + d\right )}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.83, size = 144, normalized size = 1.21 \begin {gather*} \frac {2\,c^2\,\ln \left (d+e\,x\right )}{e^4}-\frac {\frac {b^2\,d\,e^2+6\,b\,c\,d^2\,e+2\,a\,b\,e^3-22\,c^2\,d^3+2\,a\,c\,d\,e^2}{6\,e^4}+\frac {x\,\left (b^2\,e^2+6\,b\,c\,d\,e-18\,c^2\,d^2+2\,a\,c\,e^2\right )}{2\,e^3}+\frac {3\,c\,x^2\,\left (b\,e-2\,c\,d\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.82, size = 158, normalized size = 1.33 \begin {gather*} \frac {2 c^{2} \log {\left (d + e x \right )}}{e^{4}} + \frac {- 2 a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} - 6 b c d^{2} e + 22 c^{2} d^{3} + x^{2} \left (- 18 b c e^{3} + 36 c^{2} d e^{2}\right ) + x \left (- 6 a c e^{3} - 3 b^{2} e^{3} - 18 b c d e^{2} + 54 c^{2} d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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