Optimal. Leaf size=103 \[ \frac {b^4 x}{e^4}-\frac {(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac {2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac {6 b^2 (b d-a e)^2}{e^5 (d+e x)}-\frac {4 b^3 (b d-a e) \log (d+e x)}{e^5} \]
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Rubi [A]
time = 0.06, 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, 45}
\begin {gather*} -\frac {4 b^3 (b d-a e) \log (d+e x)}{e^5}-\frac {6 b^2 (b d-a e)^2}{e^5 (d+e x)}+\frac {2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac {(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac {b^4 x}{e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^4} \, dx\\ &=\int \left (\frac {b^4}{e^4}+\frac {(-b d+a e)^4}{e^4 (d+e x)^4}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^3}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^2}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)}\right ) \, dx\\ &=\frac {b^4 x}{e^4}-\frac {(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac {2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac {6 b^2 (b d-a e)^2}{e^5 (d+e x)}-\frac {4 b^3 (b d-a e) \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 163, normalized size = 1.58 \begin {gather*} -\frac {a^4 e^4+2 a^3 b e^3 (d+3 e x)+6 a^2 b^2 e^2 \left (d^2+3 d e x+3 e^2 x^2\right )-2 a b^3 d e \left (11 d^2+27 d e x+18 e^2 x^2\right )+b^4 \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 )+12 b^3 (b d-a e) (d+e x)^3 \log (d+e x)}{3 e^5 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 178, normalized size = 1.73
method | result | size |
default | \(\frac {b^{4} x}{e^{4}}-\frac {e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {6 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{5} \left (e x +d \right )}-\frac {2 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{5} \left (e x +d \right )^{2}}+\frac {4 b^{3} \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{5}}\) | \(178\) |
norman | \(\frac {\frac {b^{4} x^{4}}{e}-\frac {e^{4} a^{4}+2 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-22 a \,b^{3} d^{3} e +22 b^{4} d^{4}}{3 e^{5}}-\frac {3 \left (2 a^{2} b^{2} e^{2}-4 a \,b^{3} d e +4 b^{4} d^{2}\right ) x^{2}}{e^{3}}-\frac {\left (2 a^{3} b \,e^{3}+6 a^{2} b^{2} d \,e^{2}-18 d^{2} e a \,b^{3}+18 b^{4} d^{3}\right ) x}{e^{4}}}{\left (e x +d \right )^{3}}+\frac {4 b^{3} \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{5}}\) | \(180\) |
risch | \(\frac {b^{4} x}{e^{4}}+\frac {\left (-6 a^{2} b^{2} e^{3}+12 d \,e^{2} a \,b^{3}-6 d^{2} e \,b^{4}\right ) x^{2}-2 b \left (e^{3} a^{3}+3 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +5 b^{3} d^{3}\right ) x -\frac {e^{4} a^{4}+2 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-22 a \,b^{3} d^{3} e +13 b^{4} d^{4}}{3 e}}{e^{4} \left (e x +d \right )^{3}}+\frac {4 b^{3} \ln \left (e x +d \right ) a}{e^{4}}-\frac {4 b^{4} \ln \left (e x +d \right ) d}{e^{5}}\) | \(182\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 190, normalized size = 1.84 \begin {gather*} b^{4} x e^{\left (-4\right )} - 4 \, {\left (b^{4} d - a b^{3} e\right )} e^{\left (-5\right )} \log \left (x e + d\right ) - \frac {13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \, {\left (x^{3} e^{8} + 3 \, d x^{2} e^{7} + 3 \, d^{2} x e^{6} + d^{3} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs.
\(2 (104) = 208\).
time = 3.83, size = 275, normalized size = 2.67 \begin {gather*} -\frac {13 \, b^{4} d^{4} - {\left (3 \, b^{4} x^{4} - 18 \, a^{2} b^{2} x^{2} - 6 \, a^{3} b x - a^{4}\right )} e^{4} - {\left (9 \, b^{4} d x^{3} + 36 \, a b^{3} d x^{2} - 18 \, a^{2} b^{2} d x - 2 \, a^{3} b d\right )} e^{3} + 3 \, {\left (3 \, b^{4} d^{2} x^{2} - 18 \, a b^{3} d^{2} x + 2 \, a^{2} b^{2} d^{2}\right )} e^{2} + {\left (27 \, b^{4} d^{3} x - 22 \, a b^{3} d^{3}\right )} e + 12 \, {\left (b^{4} d^{4} - a b^{3} x^{3} e^{4} + {\left (b^{4} d x^{3} - 3 \, a b^{3} d x^{2}\right )} e^{3} + 3 \, {\left (b^{4} d^{2} x^{2} - a b^{3} d^{2} x\right )} e^{2} + {\left (3 \, b^{4} d^{3} x - a b^{3} d^{3}\right )} e\right )} \log \left (x e + d\right )}{3 \, {\left (x^{3} e^{8} + 3 \, d x^{2} e^{7} + 3 \, d^{2} x e^{6} + d^{3} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs.
\(2 (94) = 188\).
time = 1.07, size = 209, normalized size = 2.03 \begin {gather*} \frac {b^{4} x}{e^{4}} + \frac {4 b^{3} \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{5}} + \frac {- a^{4} e^{4} - 2 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} + 22 a b^{3} d^{3} e - 13 b^{4} d^{4} + x^{2} \left (- 18 a^{2} b^{2} e^{4} + 36 a b^{3} d e^{3} - 18 b^{4} d^{2} e^{2}\right ) + x \left (- 6 a^{3} b e^{4} - 18 a^{2} b^{2} d e^{3} + 54 a b^{3} d^{2} e^{2} - 30 b^{4} 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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Giac [A]
time = 2.59, size = 170, normalized size = 1.65 \begin {gather*} b^{4} x e^{\left (-4\right )} - 4 \, {\left (b^{4} d - a b^{3} e\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (5 \, b^{4} d^{3} e - 9 \, 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 )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 204, normalized size = 1.98 \begin {gather*} \frac {b^4\,x}{e^4}-\frac {\ln \left (d+e\,x\right )\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )}{e^5}-\frac {\frac {a^4\,e^4+2\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-22\,a\,b^3\,d^3\,e+13\,b^4\,d^4}{3\,e}+x\,\left (2\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-18\,a\,b^3\,d^2\,e+10\,b^4\,d^3\right )+x^2\,\left (6\,a^2\,b^2\,e^3-12\,a\,b^3\,d\,e^2+6\,b^4\,d^2\,e\right )}{d^3\,e^4+3\,d^2\,e^5\,x+3\,d\,e^6\,x^2+e^7\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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