Optimal. Leaf size=103 \[ -\frac {b^3 x (3 b d-4 a e)}{e^4}+\frac {6 b^2 (b d-a e)^2 \log (d+e x)}{e^5}+\frac {4 b (b d-a e)^3}{e^5 (d+e x)}-\frac {(b d-a e)^4}{2 e^5 (d+e x)^2}+\frac {b^4 x^2}{2 e^3} \]
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Rubi [A] time = 0.09, antiderivative size = 103, 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 {b^3 x (3 b d-4 a e)}{e^4}+\frac {6 b^2 (b d-a e)^2 \log (d+e x)}{e^5}+\frac {4 b (b d-a e)^3}{e^5 (d+e x)}-\frac {(b d-a e)^4}{2 e^5 (d+e x)^2}+\frac {b^4 x^2}{2 e^3} \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)^3} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^3} \, dx\\ &=\int \left (-\frac {b^3 (3 b d-4 a e)}{e^4}+\frac {b^4 x}{e^3}+\frac {(-b d+a e)^4}{e^4 (d+e x)^3}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^2}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac {b^3 (3 b d-4 a e) x}{e^4}+\frac {b^4 x^2}{2 e^3}-\frac {(b d-a e)^4}{2 e^5 (d+e x)^2}+\frac {4 b (b d-a e)^3}{e^5 (d+e x)}+\frac {6 b^2 (b d-a e)^2 \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 167, normalized size = 1.62 \begin {gather*} \frac {-a^4 e^4-4 a^3 b e^3 (d+2 e x)+6 a^2 b^2 d e^2 (3 d+4 e x)+4 a b^3 e \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+12 b^2 (d+e x)^2 (b d-a e)^2 \log (d+e x)+b^4 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )}{2 e^5 (d+e x)^2} \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)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 291, normalized size = 2.83 \begin {gather*} \frac {b^{4} e^{4} x^{4} + 7 \, b^{4} d^{4} - 20 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - a^{4} e^{4} - 4 \, {\left (b^{4} d e^{3} - 2 \, a b^{3} e^{4}\right )} x^{3} - {\left (11 \, b^{4} d^{2} e^{2} - 16 \, a b^{3} d e^{3}\right )} x^{2} + 2 \, {\left (b^{4} d^{3} e - 8 \, a b^{3} d^{2} e^{2} + 12 \, a^{2} b^{2} d e^{3} - 4 \, a^{3} b e^{4}\right )} x + 12 \, {\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{4} d^{3} e - 2 \, a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 175, normalized size = 1.70 \begin {gather*} 6 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (b^{4} x^{2} e^{3} - 6 \, b^{4} d x e^{2} + 8 \, a b^{3} x e^{3}\right )} e^{\left (-6\right )} + \frac {{\left (7 \, b^{4} d^{4} - 20 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - a^{4} e^{4} + 8 \, {\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 )} e^{\left (-5\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 245, normalized size = 2.38 \begin {gather*} -\frac {a^{4}}{2 \left (e x +d \right )^{2} e}+\frac {2 a^{3} b d}{\left (e x +d \right )^{2} e^{2}}-\frac {3 a^{2} b^{2} d^{2}}{\left (e x +d \right )^{2} e^{3}}+\frac {2 a \,b^{3} d^{3}}{\left (e x +d \right )^{2} e^{4}}-\frac {b^{4} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {b^{4} x^{2}}{2 e^{3}}-\frac {4 a^{3} b}{\left (e x +d \right ) e^{2}}+\frac {12 a^{2} b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {6 a^{2} b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {12 a \,b^{3} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {12 a \,b^{3} d \ln \left (e x +d \right )}{e^{4}}+\frac {4 a \,b^{3} x}{e^{3}}+\frac {4 b^{4} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {6 b^{4} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {3 b^{4} d x}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 191, normalized size = 1.85 \begin {gather*} \frac {7 \, b^{4} d^{4} - 20 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - a^{4} e^{4} + 8 \, {\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}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac {b^{4} e x^{2} - 2 \, {\left (3 \, b^{4} d - 4 \, a b^{3} e\right )} x}{2 \, e^{4}} + \frac {6 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\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 = 0.10, size = 196, normalized size = 1.90 \begin {gather*} x\,\left (\frac {4\,a\,b^3}{e^3}-\frac {3\,b^4\,d}{e^4}\right )-\frac {\frac {a^4\,e^4+4\,a^3\,b\,d\,e^3-18\,a^2\,b^2\,d^2\,e^2+20\,a\,b^3\,d^3\,e-7\,b^4\,d^4}{2\,e}-x\,\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 )}{d^2\,e^4+2\,d\,e^5\,x+e^6\,x^2}+\frac {b^4\,x^2}{2\,e^3}+\frac {\ln \left (d+e\,x\right )\,\left (6\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e+6\,b^4\,d^2\right )}{e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 185, normalized size = 1.80 \begin {gather*} \frac {b^{4} x^{2}}{2 e^{3}} + \frac {6 b^{2} \left (a e - b d\right )^{2} \log {\left (d + e x \right )}}{e^{5}} + x \left (\frac {4 a b^{3}}{e^{3}} - \frac {3 b^{4} d}{e^{4}}\right ) + \frac {- a^{4} e^{4} - 4 a^{3} b d e^{3} + 18 a^{2} b^{2} d^{2} e^{2} - 20 a b^{3} d^{3} e + 7 b^{4} d^{4} + x \left (- 8 a^{3} b e^{4} + 24 a^{2} b^{2} d e^{3} - 24 a b^{3} d^{2} e^{2} + 8 b^{4} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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