Optimal. Leaf size=429 \[ -\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^5 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \]
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Rubi [A]
time = 0.65, antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {754, 836, 814,
648, 632, 212, 642} \begin {gather*} -\frac {\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac {-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac {2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac {e^5 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 754
Rule 814
Rule 836
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)+3 c e (2 c d-b e) x}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {2 \left (6 c^4 d^4+b^4 e^4-c^3 d^2 e (9 b d-14 a e)+b^2 c e^3 (b d-8 a e)+c^2 e^2 \left (b^2 d^2-7 a b d e+16 a^2 e^2\right )\right )+2 c e (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{(d+e x) \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {2 \left (b^2-4 a c\right )^2 e^6}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {2 \left (6 c^5 d^5-b^5 e^5+9 a b^3 c e^5-23 a^2 b c^2 e^5-5 c^4 d^3 e (3 b d-4 a e)+10 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )-c \left (b^2-4 a c\right )^2 e^5 x\right )}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\int \frac {6 c^5 d^5-b^5 e^5+9 a b^3 c e^5-23 a^2 b c^2 e^5-5 c^4 d^3 e (3 b d-4 a e)+10 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )-c \left (b^2-4 a c\right )^2 e^5 x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^5 \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^5 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2-2 b^2 e^2-c e (3 b d-8 a e)\right )-2 c (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {e^5 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^5 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 429, normalized size = 1.00 \begin {gather*} \frac {1}{2} \left (\frac {-b^2 e+2 c (a e+c d x)+b c (d-e x)}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))^2}+\frac {2 b^4 e^3+b^3 c e^2 (d+2 e x)+4 c^2 \left (4 a^2 e^3+3 c^2 d^3 x+7 a c d e^2 x\right )+2 b c^2 \left (3 c d^2 (d-3 e x)+7 a e^2 (d-e x)\right )+b^2 c e \left (-15 a e^2+c d (-9 d+2 e x)\right )}{\left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}+\frac {2 \left (-12 c^5 d^5+b^5 e^5-10 a b^3 c e^5+30 a^2 b c^2 e^5+10 c^4 d^3 e (3 b d-4 a e)-20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2} \left (-c d^2+e (b d-a e)\right )^3}+\frac {2 e^5 \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}-\frac {e^5 \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs.
\(2(419)=838\).
time = 1.01, size = 1102, normalized size = 2.57
method | result | size |
default | \(-\frac {\frac {\frac {c^{2} \left (7 a^{2} b \,e^{5} c -14 d \,e^{4} a^{2} c^{2}-a \,b^{3} e^{5}-8 a \,b^{2} c d \,e^{4}+30 a b \,c^{2} d^{2} e^{3}-20 a \,c^{3} d^{3} e^{2}+b^{4} d \,e^{4}-10 b^{2} c^{2} d^{3} e^{2}+15 b \,c^{3} d^{4} e -6 c^{4} d^{5}\right ) x^{3}}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}-\frac {c \left (16 a^{3} c^{2} e^{5}-29 a^{2} b^{2} c \,e^{5}+26 a^{2} b \,c^{2} d \,e^{4}+16 a^{2} c^{3} d^{2} e^{3}+4 a \,b^{4} e^{5}+32 a \,b^{3} c d \,e^{4}-98 a \,b^{2} c^{2} d^{2} e^{3}+60 a b \,c^{3} d^{3} e^{2}-4 b^{5} d \,e^{4}+b^{4} c \,d^{2} e^{3}+30 b^{3} c^{2} d^{3} e^{2}-45 b^{2} c^{3} d^{4} e +18 b \,c^{4} d^{5}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}+\frac {\left (a^{3} b \,c^{2} e^{5}-18 a^{3} c^{3} d \,e^{4}+6 a^{2} b^{3} c \,e^{5}-10 a^{2} b^{2} c^{2} d \,e^{4}+34 a^{2} b \,c^{3} d^{2} e^{3}-28 a^{2} c^{4} d^{3} e^{2}-a \,b^{5} e^{5}-6 a \,b^{4} c d \,e^{4}+18 a \,b^{3} c^{2} d^{2} e^{3}-26 a \,b^{2} c^{3} d^{3} e^{2}+25 a b \,c^{4} d^{4} e -10 a \,c^{5} d^{5}+b^{6} d \,e^{4}-b^{5} c \,d^{2} e^{3}-3 b^{4} c^{2} d^{3} e^{2}+5 b^{3} c^{3} d^{4} e -2 c^{4} b^{2} d^{5}\right ) x}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}-\frac {24 a^{4} c^{2} e^{5}-21 a^{3} b^{2} c \,e^{5}-14 a^{3} b \,c^{2} d \,e^{4}+32 a^{3} c^{3} d^{2} e^{3}+3 a^{2} b^{4} e^{5}+27 a^{2} b^{3} c d \,e^{4}-50 a^{2} b^{2} c^{2} d^{2} e^{3}+12 a^{2} b \,c^{3} d^{3} e^{2}+8 a^{2} c^{4} d^{4} e -4 a \,b^{5} d \,e^{4}-a \,b^{4} c \,d^{2} e^{3}+24 a \,b^{3} c^{2} d^{3} e^{2}-29 a \,b^{2} c^{3} d^{4} e +10 a b \,c^{4} d^{5}+b^{6} d^{2} e^{3}-3 b^{5} c \,d^{3} e^{2}+3 b^{4} c^{2} d^{4} e -b^{3} c^{3} d^{5}}{2 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {\left (16 a^{2} c^{3} e^{5}-8 a \,b^{2} c^{2} e^{5}+b^{4} c \,e^{5}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (23 a^{2} b \,c^{2} e^{5}-30 a^{2} c^{3} d \,e^{4}-9 a \,b^{3} c \,e^{5}+30 a b \,c^{3} d^{2} e^{3}-20 a \,c^{4} d^{3} e^{2}+b^{5} e^{5}-10 b^{2} c^{3} d^{3} e^{2}+15 b \,c^{4} d^{4} e -6 c^{5} d^{5}-\frac {\left (16 a^{2} c^{3} e^{5}-8 a \,b^{2} c^{2} e^{5}+b^{4} c \,e^{5}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}+\frac {e^{5} \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}\) | \(1102\) |
risch | \(\text {Expression too large to display}\) | \(5780\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 1386 vs.
\(2 (426) = 852\).
time = 1.04, size = 1386, normalized size = 3.23 \begin {gather*} -\frac {e^{5} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )}} + \frac {e^{6} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e - 3 \, b c^{2} d^{5} e^{2} + 3 \, b^{2} c d^{4} e^{3} + 3 \, a c^{2} d^{4} e^{3} - b^{3} d^{3} e^{4} - 6 \, a b c d^{3} e^{4} + 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} c d^{2} e^{5} - 3 \, a^{2} b d e^{6} + a^{3} e^{7}} + \frac {{\left (12 \, c^{5} d^{5} - 30 \, b c^{4} d^{4} e + 20 \, b^{2} c^{3} d^{3} e^{2} + 40 \, a c^{4} d^{3} e^{2} - 60 \, a b c^{3} d^{2} e^{3} + 60 \, a^{2} c^{3} d e^{4} - b^{5} e^{5} + 10 \, a b^{3} c e^{5} - 30 \, a^{2} b c^{2} e^{5}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{3} d^{6} - 8 \, a b^{2} c^{4} d^{6} + 16 \, a^{2} c^{5} d^{6} - 3 \, b^{5} c^{2} d^{5} e + 24 \, a b^{3} c^{3} d^{5} e - 48 \, a^{2} b c^{4} d^{5} e + 3 \, b^{6} c d^{4} e^{2} - 21 \, a b^{4} c^{2} d^{4} e^{2} + 24 \, a^{2} b^{2} c^{3} d^{4} e^{2} + 48 \, a^{3} c^{4} d^{4} e^{2} - b^{7} d^{3} e^{3} + 2 \, a b^{5} c d^{3} e^{3} + 32 \, a^{2} b^{3} c^{2} d^{3} e^{3} - 96 \, a^{3} b c^{3} d^{3} e^{3} + 3 \, a b^{6} d^{2} e^{4} - 21 \, a^{2} b^{4} c d^{2} e^{4} + 24 \, a^{3} b^{2} c^{2} d^{2} e^{4} + 48 \, a^{4} c^{3} d^{2} e^{4} - 3 \, a^{2} b^{5} d e^{5} + 24 \, a^{3} b^{3} c d e^{5} - 48 \, a^{4} b c^{2} d e^{5} + a^{3} b^{4} e^{6} - 8 \, a^{4} b^{2} c e^{6} + 16 \, a^{5} c^{2} e^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b^{3} c^{3} d^{5} - 10 \, a b c^{4} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 29 \, a b^{2} c^{3} d^{4} e - 8 \, a^{2} c^{4} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - 24 \, a b^{3} c^{2} d^{3} e^{2} - 12 \, a^{2} b c^{3} d^{3} e^{2} - b^{6} d^{2} e^{3} + a b^{4} c d^{2} e^{3} + 50 \, a^{2} b^{2} c^{2} d^{2} e^{3} - 32 \, a^{3} c^{3} d^{2} e^{3} + 4 \, a b^{5} d e^{4} - 27 \, a^{2} b^{3} c d e^{4} + 14 \, a^{3} b c^{2} d e^{4} - 3 \, a^{2} b^{4} e^{5} + 21 \, a^{3} b^{2} c e^{5} - 24 \, a^{4} c^{2} e^{5} - 2 \, {\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} + 20 \, a c^{5} d^{3} e^{2} - 30 \, a b c^{4} d^{2} e^{3} - b^{4} c^{2} d e^{4} + 8 \, a b^{2} c^{3} d e^{4} + 14 \, a^{2} c^{4} d e^{4} + a b^{3} c^{2} e^{5} - 7 \, a^{2} b c^{3} e^{5}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} + 60 \, a b c^{4} d^{3} e^{2} + b^{4} c^{2} d^{2} e^{3} - 98 \, a b^{2} c^{3} d^{2} e^{3} + 16 \, a^{2} c^{4} d^{2} e^{3} - 4 \, b^{5} c d e^{4} + 32 \, a b^{3} c^{2} d e^{4} + 26 \, a^{2} b c^{3} d e^{4} + 4 \, a b^{4} c e^{5} - 29 \, a^{2} b^{2} c^{2} e^{5} + 16 \, a^{3} c^{3} e^{5}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d^{5} + 10 \, a c^{5} d^{5} - 5 \, b^{3} c^{3} d^{4} e - 25 \, a b c^{4} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + 26 \, a b^{2} c^{3} d^{3} e^{2} + 28 \, a^{2} c^{4} d^{3} e^{2} + b^{5} c d^{2} e^{3} - 18 \, a b^{3} c^{2} d^{2} e^{3} - 34 \, a^{2} b c^{3} d^{2} e^{3} - b^{6} d e^{4} + 6 \, a b^{4} c d e^{4} + 10 \, a^{2} b^{2} c^{2} d e^{4} + 18 \, a^{3} c^{3} d e^{4} + a b^{5} e^{5} - 6 \, a^{2} b^{3} c e^{5} - a^{3} b c^{2} e^{5}\right )} x}{2 \, {\left (c d^{2} - b d e + a e^{2}\right )}^{3} {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.59, size = 2500, normalized size = 5.83 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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