Optimal. Leaf size=255 \[ -\frac {\left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right ) x}{e^6}+\frac {3 c \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right ) x^2}{2 e^5}-\frac {c^2 (c d-b e) x^3}{e^4}+\frac {c^3 x^4}{4 e^3}-\frac {\left (c d^2-b d e+a e^2\right )^3}{2 e^7 (d+e x)^2}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \log (d+e x)}{e^7} \]
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Rubi [A]
time = 0.23, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712}
\begin {gather*} -\frac {x \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )}{e^6}+\frac {3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac {3 c x^2 \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )}{2 e^5}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^3}{2 e^7 (d+e x)^2}-\frac {c^2 x^3 (c d-b e)}{e^4}+\frac {c^3 x^4}{4 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx &=\int \left (\frac {-10 c^3 d^3+b^3 e^3-3 b c e^2 (3 b d-2 a e)+9 c^2 d e (2 b d-a e)}{e^6}+\frac {3 c \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right ) x}{e^5}-\frac {3 c^2 (c d-b e) x^2}{e^4}+\frac {c^3 x^3}{e^3}+\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^3}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^2}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {\left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right ) x}{e^6}+\frac {3 c \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right ) x^2}{2 e^5}-\frac {c^2 (c d-b e) x^3}{e^4}+\frac {c^3 x^4}{4 e^3}-\frac {\left (c d^2-b d e+a e^2\right )^3}{2 e^7 (d+e x)^2}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 265, normalized size = 1.04 \begin {gather*} \frac {4 e \left (-10 c^3 d^3+b^3 e^3+9 c^2 d e (2 b d-a e)+3 b c e^2 (-3 b d+2 a e)\right ) x+6 c e^2 \left (2 c^2 d^2+b^2 e^2+c e (-3 b d+a e)\right ) x^2+4 c^2 e^3 (-c d+b e) x^3+c^3 e^4 x^4-\frac {2 \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^2}+\frac {12 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^2}{d+e x}+12 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{4 e^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 461, normalized size = 1.81
method | result | size |
norman | \(\frac {\frac {\left (6 a b c \,e^{3}-6 d \,e^{2} c^{2} a +b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right ) x^{3}}{e^{4}}-\frac {e^{6} a^{3}+3 a^{2} b d \,e^{5}-9 e^{4} d^{2} a^{2} c -9 a \,b^{2} d^{2} e^{4}+54 a b c \,d^{3} e^{3}-54 d^{4} e^{2} c^{2} a +9 b^{3} d^{3} e^{3}-54 b^{2} c \,d^{4} e^{2}+90 b \,c^{2} d^{5} e -45 d^{6} c^{3}}{2 e^{7}}+\frac {c^{3} x^{6}}{4 e}-\frac {\left (3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+36 a b c \,d^{2} e^{3}-36 d^{3} e^{2} c^{2} a +6 b^{3} d^{2} e^{3}-36 b^{2} c \,d^{3} e^{2}+60 b \,c^{2} d^{4} e -30 d^{5} c^{3}\right ) x}{e^{6}}+\frac {c \left (6 a c \,e^{2}+6 b^{2} e^{2}-10 b c d e +5 c^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {c^{2} \left (2 b e -c d \right ) x^{5}}{2 e^{2}}}{\left (e x +d \right )^{2}}+\frac {3 \left (e^{4} a^{2} c +a \,b^{2} e^{4}-6 a b c d \,e^{3}+6 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+6 b^{2} c \,d^{2} e^{2}-10 d^{3} e b \,c^{2}+5 d^{4} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(441\) |
default | \(\frac {\frac {1}{4} c^{3} x^{4} e^{3}+b \,c^{2} e^{3} x^{3}-c^{3} d \,e^{2} x^{3}+\frac {3}{2} a \,c^{2} e^{3} x^{2}+\frac {3}{2} b^{2} c \,e^{3} x^{2}-\frac {9}{2} b \,c^{2} d \,e^{2} x^{2}+3 c^{3} d^{2} e \,x^{2}+6 a b c \,e^{3} x -9 d \,e^{2} c^{2} a x +b^{3} e^{3} x -9 b^{2} d \,e^{2} c x +18 b \,c^{2} d^{2} e x -10 c^{3} d^{3} x}{e^{6}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{e^{7} \left (e x +d \right )}+\frac {\left (3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 e^{4} d^{2} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{2 e^{7} \left (e x +d \right )^{2}}\) | \(461\) |
risch | \(\frac {c^{3} x^{4}}{4 e^{3}}+\frac {b \,c^{2} x^{3}}{e^{3}}-\frac {c^{3} d \,x^{3}}{e^{4}}+\frac {3 a \,c^{2} x^{2}}{2 e^{3}}+\frac {3 b^{2} c \,x^{2}}{2 e^{3}}-\frac {9 b \,c^{2} d \,x^{2}}{2 e^{4}}+\frac {3 c^{3} d^{2} x^{2}}{e^{5}}+\frac {6 a b c x}{e^{3}}-\frac {9 d \,c^{2} a x}{e^{4}}+\frac {b^{3} x}{e^{3}}-\frac {9 b^{2} d c x}{e^{4}}+\frac {18 b \,c^{2} d^{2} x}{e^{5}}-\frac {10 c^{3} d^{3} x}{e^{6}}+\frac {\left (-3 a^{2} b \,e^{5}+6 d \,e^{4} a^{2} c +6 a \,b^{2} d \,e^{4}-18 a b c \,d^{2} e^{3}+12 d^{3} e^{2} c^{2} a -3 b^{3} d^{2} e^{3}+12 b^{2} c \,d^{3} e^{2}-15 b \,c^{2} d^{4} e +6 d^{5} c^{3}\right ) x -\frac {e^{6} a^{3}+3 a^{2} b d \,e^{5}-9 e^{4} d^{2} a^{2} c -9 a \,b^{2} d^{2} e^{4}+30 a b c \,d^{3} e^{3}-21 d^{4} e^{2} c^{2} a +5 b^{3} d^{3} e^{3}-21 b^{2} c \,d^{4} e^{2}+27 b \,c^{2} d^{5} e -11 d^{6} c^{3}}{2 e}}{e^{6} \left (e x +d \right )^{2}}+\frac {3 \ln \left (e x +d \right ) a^{2} c}{e^{3}}+\frac {3 \ln \left (e x +d \right ) a \,b^{2}}{e^{3}}-\frac {18 \ln \left (e x +d \right ) a b c d}{e^{4}}+\frac {18 \ln \left (e x +d \right ) d^{2} c^{2} a}{e^{5}}-\frac {3 \ln \left (e x +d \right ) b^{3} d}{e^{4}}+\frac {18 \ln \left (e x +d \right ) b^{2} c \,d^{2}}{e^{5}}-\frac {30 \ln \left (e x +d \right ) d^{3} b \,c^{2}}{e^{6}}+\frac {15 \ln \left (e x +d \right ) d^{4} c^{3}}{e^{7}}\) | \(501\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 420, normalized size = 1.65 \begin {gather*} 3 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + a b^{2} e^{4} + a^{2} c e^{4} + 6 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{2} - {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{4} \, {\left (c^{3} x^{4} e^{3} - 4 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{3} + 6 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3} + a c^{2} e^{3}\right )} x^{2} - 4 \, {\left (10 \, c^{3} d^{3} - 18 \, b c^{2} d^{2} e - b^{3} e^{3} - 6 \, a b c e^{3} + 9 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d\right )} x\right )} e^{\left (-6\right )} + \frac {11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} - 3 \, a^{2} b d e^{5} - 5 \, {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} - a^{3} e^{6} + 9 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2} + 6 \, {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{3} - a^{2} b e^{6} - {\left (b^{3} e^{4} + 6 \, a b c e^{4}\right )} d^{2} + 2 \, {\left (a b^{2} e^{5} + a^{2} c e^{5}\right )} d\right )} x}{2 \, {\left (x^{2} e^{9} + 2 \, d x e^{8} + d^{2} e^{7}\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. 604 vs.
\(2 (250) = 500\).
time = 3.49, size = 604, normalized size = 2.37 \begin {gather*} \frac {22 \, c^{3} d^{6} + {\left (c^{3} x^{6} + 4 \, b c^{2} x^{5} + 6 \, {\left (b^{2} c + a c^{2}\right )} x^{4} - 12 \, a^{2} b x + 4 \, {\left (b^{3} + 6 \, a b c\right )} x^{3} - 2 \, a^{3}\right )} e^{6} - 2 \, {\left (c^{3} d x^{5} + 5 \, b c^{2} d x^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + 3 \, a^{2} b d - 4 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} - 12 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} + {\left (5 \, c^{3} d^{2} x^{4} + 40 \, b c^{2} d^{2} x^{3} - 66 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} - 8 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x + 18 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{4} - 2 \, {\left (10 \, c^{3} d^{3} x^{3} - 63 \, b c^{2} d^{3} x^{2} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x + 5 \, {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} - 2 \, {\left (34 \, c^{3} d^{4} x^{2} - 6 \, b c^{2} d^{4} x - 21 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} - 2 \, {\left (8 \, c^{3} d^{5} x + 27 \, b c^{2} d^{5}\right )} e + 12 \, {\left (5 \, c^{3} d^{6} + {\left (a b^{2} + a^{2} c\right )} x^{2} e^{6} - {\left ({\left (b^{3} + 6 \, a b c\right )} d x^{2} - 2 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} + {\left (6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} - 2 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x + {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{4} - {\left (10 \, b c^{2} d^{3} x^{2} - 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x + {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} + {\left (5 \, c^{3} d^{4} x^{2} - 20 \, b c^{2} d^{4} x + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} + 10 \, {\left (c^{3} d^{5} x - b c^{2} d^{5}\right )} e\right )} \log \left (x e + d\right )}{4 \, {\left (x^{2} e^{9} + 2 \, d x e^{8} + d^{2} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.23, size = 466, normalized size = 1.83 \begin {gather*} \frac {c^{3} x^{4}}{4 e^{3}} + x^{3} \left (\frac {b c^{2}}{e^{3}} - \frac {c^{3} d}{e^{4}}\right ) + x^{2} \cdot \left (\frac {3 a c^{2}}{2 e^{3}} + \frac {3 b^{2} c}{2 e^{3}} - \frac {9 b c^{2} d}{2 e^{4}} + \frac {3 c^{3} d^{2}}{e^{5}}\right ) + x \left (\frac {6 a b c}{e^{3}} - \frac {9 a c^{2} d}{e^{4}} + \frac {b^{3}}{e^{3}} - \frac {9 b^{2} c d}{e^{4}} + \frac {18 b c^{2} d^{2}}{e^{5}} - \frac {10 c^{3} d^{3}}{e^{6}}\right ) + \frac {- a^{3} e^{6} - 3 a^{2} b d e^{5} + 9 a^{2} c d^{2} e^{4} + 9 a b^{2} d^{2} e^{4} - 30 a b c d^{3} e^{3} + 21 a c^{2} d^{4} e^{2} - 5 b^{3} d^{3} e^{3} + 21 b^{2} c d^{4} e^{2} - 27 b c^{2} d^{5} e + 11 c^{3} d^{6} + x \left (- 6 a^{2} b e^{6} + 12 a^{2} c d e^{5} + 12 a b^{2} d e^{5} - 36 a b c d^{2} e^{4} + 24 a c^{2} d^{3} e^{3} - 6 b^{3} d^{2} e^{4} + 24 b^{2} c d^{3} e^{3} - 30 b c^{2} d^{4} e^{2} + 12 c^{3} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac {3 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.32, size = 432, normalized size = 1.69 \begin {gather*} 3 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} + 6 \, a c^{2} d^{2} e^{2} - b^{3} d e^{3} - 6 \, a b c d e^{3} + a b^{2} e^{4} + a^{2} c e^{4}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{4} \, {\left (c^{3} x^{4} e^{9} - 4 \, c^{3} d x^{3} e^{8} + 12 \, c^{3} d^{2} x^{2} e^{7} - 40 \, c^{3} d^{3} x e^{6} + 4 \, b c^{2} x^{3} e^{9} - 18 \, b c^{2} d x^{2} e^{8} + 72 \, b c^{2} d^{2} x e^{7} + 6 \, b^{2} c x^{2} e^{9} + 6 \, a c^{2} x^{2} e^{9} - 36 \, b^{2} c d x e^{8} - 36 \, a c^{2} d x e^{8} + 4 \, b^{3} x e^{9} + 24 \, a b c x e^{9}\right )} e^{\left (-12\right )} + \frac {{\left (11 \, c^{3} d^{6} - 27 \, b c^{2} d^{5} e + 21 \, b^{2} c d^{4} e^{2} + 21 \, a c^{2} d^{4} e^{2} - 5 \, b^{3} d^{3} e^{3} - 30 \, a b c d^{3} e^{3} + 9 \, a b^{2} d^{2} e^{4} + 9 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} - a^{3} e^{6} + 6 \, {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + 4 \, a c^{2} d^{3} e^{3} - b^{3} d^{2} e^{4} - 6 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} - a^{2} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.76, size = 521, normalized size = 2.04 \begin {gather*} \frac {x\,\left (-3\,a^2\,b\,e^5+6\,a^2\,c\,d\,e^4+6\,a\,b^2\,d\,e^4-18\,a\,b\,c\,d^2\,e^3+12\,a\,c^2\,d^3\,e^2-3\,b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2-15\,b\,c^2\,d^4\,e+6\,c^3\,d^5\right )-\frac {a^3\,e^6+3\,a^2\,b\,d\,e^5-9\,a^2\,c\,d^2\,e^4-9\,a\,b^2\,d^2\,e^4+30\,a\,b\,c\,d^3\,e^3-21\,a\,c^2\,d^4\,e^2+5\,b^3\,d^3\,e^3-21\,b^2\,c\,d^4\,e^2+27\,b\,c^2\,d^5\,e-11\,c^3\,d^6}{2\,e}}{d^2\,e^6+2\,d\,e^7\,x+e^8\,x^2}+x^3\,\left (\frac {b\,c^2}{e^3}-\frac {c^3\,d}{e^4}\right )+x\,\left (\frac {b^3+6\,a\,c\,b}{e^3}-\frac {c^3\,d^3}{e^6}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{e}+\frac {3\,c^3\,d^2}{e^5}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^3}\right )}{e}-\frac {3\,d^2\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{e^2}\right )-x^2\,\left (\frac {3\,d\,\left (\frac {3\,b\,c^2}{e^3}-\frac {3\,c^3\,d}{e^4}\right )}{2\,e}+\frac {3\,c^3\,d^2}{2\,e^5}-\frac {3\,c\,\left (b^2+a\,c\right )}{2\,e^3}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,a^2\,c\,e^4+3\,a\,b^2\,e^4-18\,a\,b\,c\,d\,e^3+18\,a\,c^2\,d^2\,e^2-3\,b^3\,d\,e^3+18\,b^2\,c\,d^2\,e^2-30\,b\,c^2\,d^3\,e+15\,c^3\,d^4\right )}{e^7}+\frac {c^3\,x^4}{4\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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