Optimal. Leaf size=159 \[ \frac {e^4}{(d+e x) (b d-a e)^5}+\frac {5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac {5 b e^4 \log (d+e x)}{(b d-a e)^6}+\frac {4 b e^3}{(a+b x) (b d-a e)^5}-\frac {3 b e^2}{2 (a+b x)^2 (b d-a e)^4}+\frac {2 b e}{3 (a+b x)^3 (b d-a e)^3}-\frac {b}{4 (a+b x)^4 (b d-a e)^2} \]
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Rubi [A] time = 0.14, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 44} \begin {gather*} \frac {e^4}{(d+e x) (b d-a e)^5}+\frac {4 b e^3}{(a+b x) (b d-a e)^5}-\frac {3 b e^2}{2 (a+b x)^2 (b d-a e)^4}+\frac {5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac {5 b e^4 \log (d+e x)}{(b d-a e)^6}+\frac {2 b e}{3 (a+b x)^3 (b d-a e)^3}-\frac {b}{4 (a+b x)^4 (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin {align*} \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {1}{(a+b x)^5 (d+e x)^2} \, dx\\ &=\int \left (\frac {b^2}{(b d-a e)^2 (a+b x)^5}-\frac {2 b^2 e}{(b d-a e)^3 (a+b x)^4}+\frac {3 b^2 e^2}{(b d-a e)^4 (a+b x)^3}-\frac {4 b^2 e^3}{(b d-a e)^5 (a+b x)^2}+\frac {5 b^2 e^4}{(b d-a e)^6 (a+b x)}-\frac {e^5}{(b d-a e)^5 (d+e x)^2}-\frac {5 b e^5}{(b d-a e)^6 (d+e x)}\right ) \, dx\\ &=-\frac {b}{4 (b d-a e)^2 (a+b x)^4}+\frac {2 b e}{3 (b d-a e)^3 (a+b x)^3}-\frac {3 b e^2}{2 (b d-a e)^4 (a+b x)^2}+\frac {4 b e^3}{(b d-a e)^5 (a+b x)}+\frac {e^4}{(b d-a e)^5 (d+e x)}+\frac {5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac {5 b e^4 \log (d+e x)}{(b d-a e)^6}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 144, normalized size = 0.91 \begin {gather*} \frac {\frac {12 e^4 (b d-a e)}{d+e x}+\frac {48 b e^3 (b d-a e)}{a+b x}-\frac {18 b e^2 (b d-a e)^2}{(a+b x)^2}+\frac {8 b e (b d-a e)^3}{(a+b x)^3}-\frac {3 b (b d-a e)^4}{(a+b x)^4}+60 b e^4 \log (a+b x)-60 b e^4 \log (d+e x)}{12 (b d-a e)^6} \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 {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.43, size = 1083, normalized size = 6.81 \begin {gather*} -\frac {3 \, b^{5} d^{5} - 20 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 65 \, a^{4} b d e^{4} + 12 \, a^{5} e^{5} - 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \, {\left (b^{5} d^{2} e^{3} + 6 \, a b^{4} d e^{4} - 7 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (b^{5} d^{3} e^{2} - 12 \, a b^{4} d^{2} e^{3} - 15 \, a^{2} b^{3} d e^{4} + 26 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 8 \, a b^{4} d^{3} e^{2} + 36 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} e^{5} x^{5} + a^{4} b d e^{4} + {\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{3} d e^{4} + 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} e^{5} x^{5} + a^{4} b d e^{4} + {\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{3} d e^{4} + 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (a^{4} b^{6} d^{7} - 6 \, a^{5} b^{5} d^{6} e + 15 \, a^{6} b^{4} d^{5} e^{2} - 20 \, a^{7} b^{3} d^{4} e^{3} + 15 \, a^{8} b^{2} d^{3} e^{4} - 6 \, a^{9} b d^{2} e^{5} + a^{10} d e^{6} + {\left (b^{10} d^{6} e - 6 \, a b^{9} d^{5} e^{2} + 15 \, a^{2} b^{8} d^{4} e^{3} - 20 \, a^{3} b^{7} d^{3} e^{4} + 15 \, a^{4} b^{6} d^{2} e^{5} - 6 \, a^{5} b^{5} d e^{6} + a^{6} b^{4} e^{7}\right )} x^{5} + {\left (b^{10} d^{7} - 2 \, a b^{9} d^{6} e - 9 \, a^{2} b^{8} d^{5} e^{2} + 40 \, a^{3} b^{7} d^{4} e^{3} - 65 \, a^{4} b^{6} d^{3} e^{4} + 54 \, a^{5} b^{5} d^{2} e^{5} - 23 \, a^{6} b^{4} d e^{6} + 4 \, a^{7} b^{3} e^{7}\right )} x^{4} + 2 \, {\left (2 \, a b^{9} d^{7} - 9 \, a^{2} b^{8} d^{6} e + 12 \, a^{3} b^{7} d^{5} e^{2} + 5 \, a^{4} b^{6} d^{4} e^{3} - 30 \, a^{5} b^{5} d^{3} e^{4} + 33 \, a^{6} b^{4} d^{2} e^{5} - 16 \, a^{7} b^{3} d e^{6} + 3 \, a^{8} b^{2} e^{7}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{8} d^{7} - 16 \, a^{3} b^{7} d^{6} e + 33 \, a^{4} b^{6} d^{5} e^{2} - 30 \, a^{5} b^{5} d^{4} e^{3} + 5 \, a^{6} b^{4} d^{3} e^{4} + 12 \, a^{7} b^{3} d^{2} e^{5} - 9 \, a^{8} b^{2} d e^{6} + 2 \, a^{9} b e^{7}\right )} x^{2} + {\left (4 \, a^{3} b^{7} d^{7} - 23 \, a^{4} b^{6} d^{6} e + 54 \, a^{5} b^{5} d^{5} e^{2} - 65 \, a^{6} b^{4} d^{4} e^{3} + 40 \, a^{7} b^{3} d^{3} e^{4} - 9 \, a^{8} b^{2} d^{2} e^{5} - 2 \, a^{9} b d e^{6} + a^{10} e^{7}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 358, normalized size = 2.25 \begin {gather*} \frac {5 \, b e^{5} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} + \frac {e^{9}}{{\left (b^{5} d^{5} e^{5} - 5 \, a b^{4} d^{4} e^{6} + 10 \, a^{2} b^{3} d^{3} e^{7} - 10 \, a^{3} b^{2} d^{2} e^{8} + 5 \, a^{4} b d e^{9} - a^{5} e^{10}\right )} {\left (x e + d\right )}} + \frac {77 \, b^{5} e^{4} - \frac {260 \, {\left (b^{5} d e^{5} - a b^{4} e^{6}\right )} e^{\left (-1\right )}}{x e + d} + \frac {300 \, {\left (b^{5} d^{2} e^{6} - 2 \, a b^{4} d e^{7} + a^{2} b^{3} e^{8}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {120 \, {\left (b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{12 \, {\left (b d - a e\right )}^{6} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 155, normalized size = 0.97 \begin {gather*} \frac {5 b \,e^{4} \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}-\frac {5 b \,e^{4} \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}-\frac {4 b \,e^{3}}{\left (a e -b d \right )^{5} \left (b x +a \right )}-\frac {e^{4}}{\left (a e -b d \right )^{5} \left (e x +d \right )}-\frac {3 b \,e^{2}}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}-\frac {2 b e}{3 \left (a e -b d \right )^{3} \left (b x +a \right )^{3}}-\frac {b}{4 \left (a e -b d \right )^{2} \left (b x +a \right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.86, size = 858, normalized size = 5.40 \begin {gather*} \frac {5 \, b e^{4} \log \left (b x + a\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} - \frac {5 \, b e^{4} \log \left (e x + d\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac {60 \, b^{4} e^{4} x^{4} - 3 \, b^{4} d^{4} + 17 \, a b^{3} d^{3} e - 43 \, a^{2} b^{2} d^{2} e^{2} + 77 \, a^{3} b d e^{3} + 12 \, a^{4} e^{4} + 30 \, {\left (b^{4} d e^{3} + 7 \, a b^{3} e^{4}\right )} x^{3} - 10 \, {\left (b^{4} d^{2} e^{2} - 11 \, a b^{3} d e^{3} - 26 \, a^{2} b^{2} e^{4}\right )} x^{2} + 5 \, {\left (b^{4} d^{3} e - 7 \, a b^{3} d^{2} e^{2} + 29 \, a^{2} b^{2} d e^{3} + 25 \, a^{3} b e^{4}\right )} x}{12 \, {\left (a^{4} b^{5} d^{6} - 5 \, a^{5} b^{4} d^{5} e + 10 \, a^{6} b^{3} d^{4} e^{2} - 10 \, a^{7} b^{2} d^{3} e^{3} + 5 \, a^{8} b d^{2} e^{4} - a^{9} d e^{5} + {\left (b^{9} d^{5} e - 5 \, a b^{8} d^{4} e^{2} + 10 \, a^{2} b^{7} d^{3} e^{3} - 10 \, a^{3} b^{6} d^{2} e^{4} + 5 \, a^{4} b^{5} d e^{5} - a^{5} b^{4} e^{6}\right )} x^{5} + {\left (b^{9} d^{6} - a b^{8} d^{5} e - 10 \, a^{2} b^{7} d^{4} e^{2} + 30 \, a^{3} b^{6} d^{3} e^{3} - 35 \, a^{4} b^{5} d^{2} e^{4} + 19 \, a^{5} b^{4} d e^{5} - 4 \, a^{6} b^{3} e^{6}\right )} x^{4} + 2 \, {\left (2 \, a b^{8} d^{6} - 7 \, a^{2} b^{7} d^{5} e + 5 \, a^{3} b^{6} d^{4} e^{2} + 10 \, a^{4} b^{5} d^{3} e^{3} - 20 \, a^{5} b^{4} d^{2} e^{4} + 13 \, a^{6} b^{3} d e^{5} - 3 \, a^{7} b^{2} e^{6}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{7} d^{6} - 13 \, a^{3} b^{6} d^{5} e + 20 \, a^{4} b^{5} d^{4} e^{2} - 10 \, a^{5} b^{4} d^{3} e^{3} - 5 \, a^{6} b^{3} d^{2} e^{4} + 7 \, a^{7} b^{2} d e^{5} - 2 \, a^{8} b e^{6}\right )} x^{2} + {\left (4 \, a^{3} b^{6} d^{6} - 19 \, a^{4} b^{5} d^{5} e + 35 \, a^{5} b^{4} d^{4} e^{2} - 30 \, a^{6} b^{3} d^{3} e^{3} + 10 \, a^{7} b^{2} d^{2} e^{4} + a^{8} b d e^{5} - a^{9} e^{6}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 763, normalized size = 4.80 \begin {gather*} \frac {10\,b\,e^4\,\mathrm {atanh}\left (\frac {a^6\,e^6-4\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4-5\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e-b^6\,d^6}{{\left (a\,e-b\,d\right )}^6}+\frac {2\,b\,e\,x\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^6}\right )}{{\left (a\,e-b\,d\right )}^6}-\frac {\frac {12\,a^4\,e^4+77\,a^3\,b\,d\,e^3-43\,a^2\,b^2\,d^2\,e^2+17\,a\,b^3\,d^3\,e-3\,b^4\,d^4}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,e\,x\,\left (25\,a^3\,b\,e^3+29\,a^2\,b^2\,d\,e^2-7\,a\,b^3\,d^2\,e+b^4\,d^3\right )}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,b^4\,e^4\,x^4}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}+\frac {5\,e^3\,x^3\,\left (d\,b^4+7\,a\,e\,b^3\right )}{2\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,e^2\,x^2\,\left (26\,a^2\,b^2\,e^2+11\,a\,b^3\,d\,e-b^4\,d^2\right )}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}}{x^4\,\left (d\,b^4+4\,a\,e\,b^3\right )+a^4\,d+x\,\left (e\,a^4+4\,b\,d\,a^3\right )+x^2\,\left (4\,e\,a^3\,b+6\,d\,a^2\,b^2\right )+x^3\,\left (6\,e\,a^2\,b^2+4\,d\,a\,b^3\right )+b^4\,e\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.55, size = 1178, normalized size = 7.41 \begin {gather*} - \frac {5 b e^{4} \log {\left (x + \frac {- \frac {5 a^{7} b e^{11}}{\left (a e - b d\right )^{6}} + \frac {35 a^{6} b^{2} d e^{10}}{\left (a e - b d\right )^{6}} - \frac {105 a^{5} b^{3} d^{2} e^{9}}{\left (a e - b d\right )^{6}} + \frac {175 a^{4} b^{4} d^{3} e^{8}}{\left (a e - b d\right )^{6}} - \frac {175 a^{3} b^{5} d^{4} e^{7}}{\left (a e - b d\right )^{6}} + \frac {105 a^{2} b^{6} d^{5} e^{6}}{\left (a e - b d\right )^{6}} - \frac {35 a b^{7} d^{6} e^{5}}{\left (a e - b d\right )^{6}} + 5 a b e^{5} + \frac {5 b^{8} d^{7} e^{4}}{\left (a e - b d\right )^{6}} + 5 b^{2} d e^{4}}{10 b^{2} e^{5}} \right )}}{\left (a e - b d\right )^{6}} + \frac {5 b e^{4} \log {\left (x + \frac {\frac {5 a^{7} b e^{11}}{\left (a e - b d\right )^{6}} - \frac {35 a^{6} b^{2} d e^{10}}{\left (a e - b d\right )^{6}} + \frac {105 a^{5} b^{3} d^{2} e^{9}}{\left (a e - b d\right )^{6}} - \frac {175 a^{4} b^{4} d^{3} e^{8}}{\left (a e - b d\right )^{6}} + \frac {175 a^{3} b^{5} d^{4} e^{7}}{\left (a e - b d\right )^{6}} - \frac {105 a^{2} b^{6} d^{5} e^{6}}{\left (a e - b d\right )^{6}} + \frac {35 a b^{7} d^{6} e^{5}}{\left (a e - b d\right )^{6}} + 5 a b e^{5} - \frac {5 b^{8} d^{7} e^{4}}{\left (a e - b d\right )^{6}} + 5 b^{2} d e^{4}}{10 b^{2} e^{5}} \right )}}{\left (a e - b d\right )^{6}} + \frac {- 12 a^{4} e^{4} - 77 a^{3} b d e^{3} + 43 a^{2} b^{2} d^{2} e^{2} - 17 a b^{3} d^{3} e + 3 b^{4} d^{4} - 60 b^{4} e^{4} x^{4} + x^{3} \left (- 210 a b^{3} e^{4} - 30 b^{4} d e^{3}\right ) + x^{2} \left (- 260 a^{2} b^{2} e^{4} - 110 a b^{3} d e^{3} + 10 b^{4} d^{2} e^{2}\right ) + x \left (- 125 a^{3} b e^{4} - 145 a^{2} b^{2} d e^{3} + 35 a b^{3} d^{2} e^{2} - 5 b^{4} d^{3} e\right )}{12 a^{9} d e^{5} - 60 a^{8} b d^{2} e^{4} + 120 a^{7} b^{2} d^{3} e^{3} - 120 a^{6} b^{3} d^{4} e^{2} + 60 a^{5} b^{4} d^{5} e - 12 a^{4} b^{5} d^{6} + x^{5} \left (12 a^{5} b^{4} e^{6} - 60 a^{4} b^{5} d e^{5} + 120 a^{3} b^{6} d^{2} e^{4} - 120 a^{2} b^{7} d^{3} e^{3} + 60 a b^{8} d^{4} e^{2} - 12 b^{9} d^{5} e\right ) + x^{4} \left (48 a^{6} b^{3} e^{6} - 228 a^{5} b^{4} d e^{5} + 420 a^{4} b^{5} d^{2} e^{4} - 360 a^{3} b^{6} d^{3} e^{3} + 120 a^{2} b^{7} d^{4} e^{2} + 12 a b^{8} d^{5} e - 12 b^{9} d^{6}\right ) + x^{3} \left (72 a^{7} b^{2} e^{6} - 312 a^{6} b^{3} d e^{5} + 480 a^{5} b^{4} d^{2} e^{4} - 240 a^{4} b^{5} d^{3} e^{3} - 120 a^{3} b^{6} d^{4} e^{2} + 168 a^{2} b^{7} d^{5} e - 48 a b^{8} d^{6}\right ) + x^{2} \left (48 a^{8} b e^{6} - 168 a^{7} b^{2} d e^{5} + 120 a^{6} b^{3} d^{2} e^{4} + 240 a^{5} b^{4} d^{3} e^{3} - 480 a^{4} b^{5} d^{4} e^{2} + 312 a^{3} b^{6} d^{5} e - 72 a^{2} b^{7} d^{6}\right ) + x \left (12 a^{9} e^{6} - 12 a^{8} b d e^{5} - 120 a^{7} b^{2} d^{2} e^{4} + 360 a^{6} b^{3} d^{3} e^{3} - 420 a^{5} b^{4} d^{4} e^{2} + 228 a^{4} b^{5} d^{5} e - 48 a^{3} b^{6} d^{6}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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