Optimal. Leaf size=150 \[ \frac {2 a+b (d+e x)^2}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {3 b \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {3 b c \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} e} \]
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Rubi [A]
time = 0.13, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1156, 1128,
652, 628, 632, 212} \begin {gather*} \frac {2 a+b (d+e x)^2}{4 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {3 b \left (b+2 c (d+e x)^2\right )}{4 e \left (b^2-4 a c\right )^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {3 b c \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 628
Rule 632
Rule 652
Rule 1128
Rule 1156
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^3}{\left (a+b x^2+c x^4\right )^3} \, dx,x,d+e x\right )}{e}\\ &=\frac {\text {Subst}\left (\int \frac {x}{\left (a+b x+c x^2\right )^3} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {2 a+b (d+e x)^2}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx,x,(d+e x)^2\right )}{4 \left (b^2-4 a c\right ) e}\\ &=\frac {2 a+b (d+e x)^2}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {3 b \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {(3 b c) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 \left (b^2-4 a c\right )^2 e}\\ &=\frac {2 a+b (d+e x)^2}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {3 b \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {(3 b c) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right )^2 e}\\ &=\frac {2 a+b (d+e x)^2}{4 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )^2}-\frac {3 b \left (b+2 c (d+e x)^2\right )}{4 \left (b^2-4 a c\right )^2 e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {3 b c \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} e}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 146, normalized size = 0.97 \begin {gather*} \frac {-\frac {3 b \left (b+2 c (d+e x)^2\right )}{a+b (d+e x)^2+c (d+e x)^4}+\frac {\left (b^2-4 a c\right ) \left (2 a+b (d+e x)^2\right )}{\left (a+(d+e x)^2 \left (b+c (d+e x)^2\right )\right )^2}-\frac {12 b c \tan ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{4 \left (b^2-4 a c\right )^2 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.21, size = 544, normalized size = 3.63
method | result | size |
default | \(\frac {-\frac {3 c^{2} e^{5} b \,x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {9 e^{4} b \,c^{2} d \,x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {9 b c \,e^{3} \left (10 c \,d^{2}+b \right ) x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 d \,e^{2} c b \left (10 c \,d^{2}+3 b \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {b e \left (45 c^{2} d^{4}+27 b c \,d^{2}+5 a c +b^{2}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d b \left (9 c^{2} d^{4}+9 b c \,d^{2}+5 a c +b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {6 b \,c^{2} d^{6}+9 b^{2} c \,d^{4}+10 a b c \,d^{2}+2 b^{3} d^{2}+8 a^{2} c +a \,b^{2}}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a \right )^{2}}+\frac {3 b c \left (\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (-\textit {\_R} e -d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) e}\) | \(544\) |
risch | \(\frac {-\frac {3 c^{2} e^{5} b \,x^{6}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {9 e^{4} b \,c^{2} d \,x^{5}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {9 b c \,e^{3} \left (10 c \,d^{2}+b \right ) x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {3 d \,e^{2} c b \left (10 c \,d^{2}+3 b \right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {b e \left (45 c^{2} d^{4}+27 b c \,d^{2}+5 a c +b^{2}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {d b \left (9 c^{2} d^{4}+9 b c \,d^{2}+5 a c +b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {6 b \,c^{2} d^{6}+9 b^{2} c \,d^{4}+10 a b c \,d^{2}+2 b^{3} d^{2}+8 a^{2} c +a \,b^{2}}{4 e \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+d^{4} c +2 d e b x +d^{2} b +a \right )^{2}}-\frac {3 c b \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e^{2}-16 e^{2} c^{2} a^{2} b +8 a c \,e^{2} b^{3}-e^{2} b^{5}\right ) x^{2}+\left (-2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d e -32 a^{2} b \,c^{2} d e +16 a \,b^{3} c d e -2 b^{5} d e \right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}-16 a^{2} b \,c^{2} d^{2}+8 a \,b^{3} c \,d^{2}-b^{5} d^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right )}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} e}+\frac {3 c b \ln \left (\left (-\left (-4 a c +b^{2}\right )^{\frac {5}{2}} e^{2}+16 e^{2} c^{2} a^{2} b -8 a c \,e^{2} b^{3}+e^{2} b^{5}\right ) x^{2}+\left (-2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} d e +32 a^{2} b \,c^{2} d e -16 a \,b^{3} c d e +2 b^{5} d e \right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}} d^{2}+16 a^{2} b \,c^{2} d^{2}-8 a \,b^{3} c \,d^{2}+b^{5} d^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right )}{2 \left (-4 a c +b^{2}\right )^{\frac {5}{2}} e}\) | \(750\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 1787 vs.
\(2 (146) = 292\).
time = 0.41, size = 3701, normalized size = 24.67 \begin {gather*} \text {Too large to display} \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. 1671 vs.
\(2 (134) = 268\).
time = 7.81, size = 1671, normalized size = 11.14 \begin {gather*} \frac {3 b c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 192 a^{3} b c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{2} b^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a b^{5} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{7} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{2} c + 6 b c^{2} d^{2}}{6 b c^{2} e^{2}} \right )}}{2 e} - \frac {3 b c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {192 a^{3} b c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{2} b^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a b^{5} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 3 b^{7} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{2} c + 6 b c^{2} d^{2}}{6 b c^{2} e^{2}} \right )}}{2 e} + \frac {- 8 a^{2} c - a b^{2} - 10 a b c d^{2} - 2 b^{3} d^{2} - 9 b^{2} c d^{4} - 6 b c^{2} d^{6} - 36 b c^{2} d e^{5} x^{5} - 6 b c^{2} e^{6} x^{6} + x^{4} \left (- 9 b^{2} c e^{4} - 90 b c^{2} d^{2} e^{4}\right ) + x^{3} \left (- 36 b^{2} c d e^{3} - 120 b c^{2} d^{3} e^{3}\right ) + x^{2} \left (- 10 a b c e^{2} - 2 b^{3} e^{2} - 54 b^{2} c d^{2} e^{2} - 90 b c^{2} d^{4} e^{2}\right ) + x \left (- 20 a b c d e - 4 b^{3} d e - 36 b^{2} c d^{3} e - 36 b c^{2} d^{5} e\right )}{64 a^{4} c^{2} e - 32 a^{3} b^{2} c e + 128 a^{3} b c^{2} d^{2} e + 128 a^{3} c^{3} d^{4} e + 4 a^{2} b^{4} e - 64 a^{2} b^{3} c d^{2} e + 128 a^{2} b c^{3} d^{6} e + 64 a^{2} c^{4} d^{8} e + 8 a b^{5} d^{2} e - 24 a b^{4} c d^{4} e - 64 a b^{3} c^{2} d^{6} e - 32 a b^{2} c^{3} d^{8} e + 4 b^{6} d^{4} e + 8 b^{5} c d^{6} e + 4 b^{4} c^{2} d^{8} e + x^{8} \cdot \left (64 a^{2} c^{4} e^{9} - 32 a b^{2} c^{3} e^{9} + 4 b^{4} c^{2} e^{9}\right ) + x^{7} \cdot \left (512 a^{2} c^{4} d e^{8} - 256 a b^{2} c^{3} d e^{8} + 32 b^{4} c^{2} d e^{8}\right ) + x^{6} \cdot \left (128 a^{2} b c^{3} e^{7} + 1792 a^{2} c^{4} d^{2} e^{7} - 64 a b^{3} c^{2} e^{7} - 896 a b^{2} c^{3} d^{2} e^{7} + 8 b^{5} c e^{7} + 112 b^{4} c^{2} d^{2} e^{7}\right ) + x^{5} \cdot \left (768 a^{2} b c^{3} d e^{6} + 3584 a^{2} c^{4} d^{3} e^{6} - 384 a b^{3} c^{2} d e^{6} - 1792 a b^{2} c^{3} d^{3} e^{6} + 48 b^{5} c d e^{6} + 224 b^{4} c^{2} d^{3} e^{6}\right ) + x^{4} \cdot \left (128 a^{3} c^{3} e^{5} + 1920 a^{2} b c^{3} d^{2} e^{5} + 4480 a^{2} c^{4} d^{4} e^{5} - 24 a b^{4} c e^{5} - 960 a b^{3} c^{2} d^{2} e^{5} - 2240 a b^{2} c^{3} d^{4} e^{5} + 4 b^{6} e^{5} + 120 b^{5} c d^{2} e^{5} + 280 b^{4} c^{2} d^{4} e^{5}\right ) + x^{3} \cdot \left (512 a^{3} c^{3} d e^{4} + 2560 a^{2} b c^{3} d^{3} e^{4} + 3584 a^{2} c^{4} d^{5} e^{4} - 96 a b^{4} c d e^{4} - 1280 a b^{3} c^{2} d^{3} e^{4} - 1792 a b^{2} c^{3} d^{5} e^{4} + 16 b^{6} d e^{4} + 160 b^{5} c d^{3} e^{4} + 224 b^{4} c^{2} d^{5} e^{4}\right ) + x^{2} \cdot \left (128 a^{3} b c^{2} e^{3} + 768 a^{3} c^{3} d^{2} e^{3} - 64 a^{2} b^{3} c e^{3} + 1920 a^{2} b c^{3} d^{4} e^{3} + 1792 a^{2} c^{4} d^{6} e^{3} + 8 a b^{5} e^{3} - 144 a b^{4} c d^{2} e^{3} - 960 a b^{3} c^{2} d^{4} e^{3} - 896 a b^{2} c^{3} d^{6} e^{3} + 24 b^{6} d^{2} e^{3} + 120 b^{5} c d^{4} e^{3} + 112 b^{4} c^{2} d^{6} e^{3}\right ) + x \left (256 a^{3} b c^{2} d e^{2} + 512 a^{3} c^{3} d^{3} e^{2} - 128 a^{2} b^{3} c d e^{2} + 768 a^{2} b c^{3} d^{5} e^{2} + 512 a^{2} c^{4} d^{7} e^{2} + 16 a b^{5} d e^{2} - 96 a b^{4} c d^{3} e^{2} - 384 a b^{3} c^{2} d^{5} e^{2} - 256 a b^{2} c^{3} d^{7} e^{2} + 16 b^{6} d^{3} e^{2} + 48 b^{5} c d^{5} e^{2} + 32 b^{4} c^{2} d^{7} e^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs.
\(2 (146) = 292\).
time = 4.02, size = 365, normalized size = 2.43 \begin {gather*} -\frac {3 \, b c \arctan \left (\frac {2 \, c d^{2} + 2 \, {\left (x^{2} e + 2 \, d x\right )} c e + b}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-1\right )}}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {6 \, b c^{2} d^{6} + 18 \, {\left (x^{2} e + 2 \, d x\right )} b c^{2} d^{4} e + 18 \, {\left (x^{2} e + 2 \, d x\right )}^{2} b c^{2} d^{2} e^{2} + 9 \, b^{2} c d^{4} + 6 \, {\left (x^{2} e + 2 \, d x\right )}^{3} b c^{2} e^{3} + 18 \, {\left (x^{2} e + 2 \, d x\right )} b^{2} c d^{2} e + 9 \, {\left (x^{2} e + 2 \, d x\right )}^{2} b^{2} c e^{2} + 2 \, b^{3} d^{2} + 10 \, a b c d^{2} + 2 \, {\left (x^{2} e + 2 \, d x\right )} b^{3} e + 10 \, {\left (x^{2} e + 2 \, d x\right )} a b c e + a b^{2} + 8 \, a^{2} c}{4 \, {\left (c d^{4} + 2 \, {\left (x^{2} e + 2 \, d x\right )} c d^{2} e + {\left (x^{2} e + 2 \, d x\right )}^{2} c e^{2} + b d^{2} + {\left (x^{2} e + 2 \, d x\right )} b e + a\right )}^{2} {\left (b^{4} e - 8 \, a b^{2} c e + 16 \, a^{2} c^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.85, size = 1182, normalized size = 7.88 \begin {gather*} -\frac {\frac {9\,x^4\,\left (b^2\,c\,e^3+10\,b\,c^2\,d^2\,e^3\right )}{4\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {8\,a^2\,c+a\,b^2+10\,a\,b\,c\,d^2+2\,b^3\,d^2+9\,b^2\,c\,d^4+6\,b\,c^2\,d^6}{4\,e\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (e\,b^3+27\,e\,b^2\,c\,d^2+45\,e\,b\,c^2\,d^4+5\,a\,e\,b\,c\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,d\,x^3\,\left (3\,b^2\,c\,e^2+10\,b\,c^2\,d^2\,e^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {d\,x\,\left (b^3+9\,b^2\,c\,d^2+9\,b\,c^2\,d^4+5\,a\,b\,c\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {3\,b\,c^2\,e^5\,x^6}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {9\,b\,c^2\,d\,e^4\,x^5}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}}{x^2\,\left (6\,b^2\,d^2\,e^2+30\,b\,c\,d^4\,e^2+2\,a\,b\,e^2+28\,c^2\,d^6\,e^2+12\,a\,c\,d^2\,e^2\right )+x^6\,\left (28\,c^2\,d^2\,e^6+2\,b\,c\,e^6\right )+x\,\left (4\,e\,b^2\,d^3+12\,e\,b\,c\,d^5+4\,a\,e\,b\,d+8\,e\,c^2\,d^7+8\,a\,e\,c\,d^3\right )+x^3\,\left (4\,b^2\,d\,e^3+40\,b\,c\,d^3\,e^3+56\,c^2\,d^5\,e^3+8\,a\,c\,d\,e^3\right )+x^5\,\left (56\,c^2\,d^3\,e^5+12\,b\,c\,d\,e^5\right )+x^4\,\left (b^2\,e^4+30\,b\,c\,d^2\,e^4+70\,c^2\,d^4\,e^4+2\,a\,c\,e^4\right )+a^2+b^2\,d^4+c^2\,d^8+c^2\,e^8\,x^8+2\,a\,b\,d^2+2\,a\,c\,d^4+2\,b\,c\,d^6+8\,c^2\,d\,e^7\,x^7}-\frac {3\,b\,c\,\mathrm {atan}\left (\frac {\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )\,\left (x^2\,\left (\frac {9\,b^2\,c^4\,e^8}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {9\,b^3\,c^2\,\left (32\,a^2\,b\,c^4\,e^{10}-16\,a\,b^3\,c^3\,e^{10}+2\,b^5\,c^2\,e^{10}\right )}{2\,a\,e^2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+x\,\left (\frac {18\,b^2\,c^4\,d\,e^7}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {9\,b^3\,c^2\,\left (32\,d\,a^2\,b\,c^4\,e^9-16\,d\,a\,b^3\,c^3\,e^9+2\,d\,b^5\,c^2\,e^9\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+\frac {9\,b^3\,c^2\,\left (64\,a^3\,c^4\,e^8-32\,a^2\,b^2\,c^3\,e^8+32\,a^2\,b\,c^4\,d^2\,e^8+4\,a\,b^4\,c^2\,e^8-16\,a\,b^3\,c^3\,d^2\,e^8+2\,b^5\,c^2\,d^2\,e^8\right )}{2\,a\,e^2\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {9\,b^2\,c^4\,d^2\,e^6}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )}{18\,b^2\,c^4\,e^6}\right )}{e\,{\left (4\,a\,c-b^2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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