Optimal. Leaf size=104 \[ -\frac {2 b^3 (d+e x)^2 (b d-a e)}{e^5}+\frac {6 b^2 x (b d-a e)^2}{e^4}-\frac {(b d-a e)^4}{e^5 (d+e x)}-\frac {4 b (b d-a e)^3 \log (d+e x)}{e^5}+\frac {b^4 (d+e x)^3}{3 e^5} \]
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Rubi [A] time = 0.10, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} -\frac {2 b^3 (d+e x)^2 (b d-a e)}{e^5}+\frac {6 b^2 x (b d-a e)^2}{e^4}-\frac {(b d-a e)^4}{e^5 (d+e x)}-\frac {4 b (b d-a e)^3 \log (d+e x)}{e^5}+\frac {b^4 (d+e x)^3}{3 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^2} \, dx\\ &=\int \left (\frac {6 b^2 (b d-a e)^2}{e^4}+\frac {(-b d+a e)^4}{e^4 (d+e x)^2}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)}-\frac {4 b^3 (b d-a e) (d+e x)}{e^4}+\frac {b^4 (d+e x)^2}{e^4}\right ) \, dx\\ &=\frac {6 b^2 (b d-a e)^2 x}{e^4}-\frac {(b d-a e)^4}{e^5 (d+e x)}-\frac {2 b^3 (b d-a e) (d+e x)^2}{e^5}+\frac {b^4 (d+e x)^3}{3 e^5}-\frac {4 b (b d-a e)^3 \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 165, normalized size = 1.59 \begin {gather*} \frac {-3 a^4 e^4+12 a^3 b d e^3+18 a^2 b^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 a b^3 e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )-12 b (d+e x) (b d-a e)^3 \log (d+e x)+b^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )}{3 e^5 (d+e x)} \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^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 267, normalized size = 2.57 \begin {gather*} \frac {b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 12 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 12 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} - 2 \, {\left (b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 3 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 3 \, {\left (3 \, b^{4} d^{3} e - 8 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3}\right )} x - 12 \, {\left (b^{4} d^{4} - 3 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} - a^{3} b d e^{3} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{6} x + d e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 239, normalized size = 2.30 \begin {gather*} \frac {1}{3} \, {\left (b^{4} - \frac {6 \, {\left (b^{4} d e - a b^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )} {\left (x e + d\right )}^{3} e^{\left (-5\right )} + 4 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} e^{\left (-5\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {b^{4} d^{4} e^{3}}{x e + d} - \frac {4 \, a b^{3} d^{3} e^{4}}{x e + d} + \frac {6 \, a^{2} b^{2} d^{2} e^{5}}{x e + d} - \frac {4 \, a^{3} b d e^{6}}{x e + d} + \frac {a^{4} e^{7}}{x e + d}\right )} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 230, normalized size = 2.21 \begin {gather*} \frac {b^{4} x^{3}}{3 e^{2}}+\frac {2 a \,b^{3} x^{2}}{e^{2}}-\frac {b^{4} d \,x^{2}}{e^{3}}-\frac {a^{4}}{\left (e x +d \right ) e}+\frac {4 a^{3} b d}{\left (e x +d \right ) e^{2}}+\frac {4 a^{3} b \ln \left (e x +d \right )}{e^{2}}-\frac {6 a^{2} b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {12 a^{2} b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {6 a^{2} b^{2} x}{e^{2}}+\frac {4 a \,b^{3} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {12 a \,b^{3} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {8 a \,b^{3} d x}{e^{3}}-\frac {b^{4} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {4 b^{4} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {3 b^{4} d^{2} x}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 183, normalized size = 1.76 \begin {gather*} -\frac {b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}}{e^{6} x + d e^{5}} + \frac {b^{4} e^{2} x^{3} - 3 \, {\left (b^{4} d e - 2 \, a b^{3} e^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{4} d^{2} - 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x}{3 \, e^{4}} - \frac {4 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\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 = 0.55, size = 203, normalized size = 1.95 \begin {gather*} x^2\,\left (\frac {2\,a\,b^3}{e^2}-\frac {b^4\,d}{e^3}\right )-x\,\left (\frac {2\,d\,\left (\frac {4\,a\,b^3}{e^2}-\frac {2\,b^4\,d}{e^3}\right )}{e}-\frac {6\,a^2\,b^2}{e^2}+\frac {b^4\,d^2}{e^4}\right )+\frac {b^4\,x^3}{3\,e^2}-\frac {\ln \left (d+e\,x\right )\,\left (-4\,a^3\,b\,e^3+12\,a^2\,b^2\,d\,e^2-12\,a\,b^3\,d^2\,e+4\,b^4\,d^3\right )}{e^5}-\frac {a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}{e\,\left (x\,e^5+d\,e^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 155, normalized size = 1.49 \begin {gather*} \frac {b^{4} x^{3}}{3 e^{2}} + \frac {4 b \left (a e - b d\right )^{3} \log {\left (d + e x \right )}}{e^{5}} + x^{2} \left (\frac {2 a b^{3}}{e^{2}} - \frac {b^{4} d}{e^{3}}\right ) + x \left (\frac {6 a^{2} b^{2}}{e^{2}} - \frac {8 a b^{3} d}{e^{3}} + \frac {3 b^{4} d^{2}}{e^{4}}\right ) + \frac {- a^{4} e^{4} + 4 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} + 4 a b^{3} d^{3} e - b^{4} d^{4}}{d e^{5} + e^{6} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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