Optimal. Leaf size=397 \[ \frac {-\left (\left (b^2-4 a c\right ) (c d-b e)\right )+c \left (b^2-4 a c\right ) e x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \]
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Rubi [A]
time = 0.71, antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {836, 814, 648,
632, 212, 642} \begin {gather*} -\frac {\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )}{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 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac {e \left (-2 c x \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )-b c \left (c d^2-7 a e^2\right )-8 a c^2 d e-2 b^3 e^2+3 b^2 c d e\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac {e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac {e^4 (2 c d-b e) \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 814
Rule 836
Rubi steps
\begin {align*} \int \frac {b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx &=-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 {\left (b^2-4 a c\right ) e (c d-2 b e)-3 c \left (b^2-4 a c\right ) e^2 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 {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \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 (b^2-4 a c\right ) e \left (c^3 d^3+b^3 e^3-c^2 d e (b d-5 a e)-b c e^2 (b d+4 a e)\right )+2 c \left (b^2-4 a c\right ) e^2 \left (c^2 d^2+b^2 e^2-c e (b d+3 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 {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \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^5 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {2 \left (b^2-4 a c\right ) e \left (c^4 d^4-b^4 e^4-2 c^3 d^2 e (b d-3 a e)-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) 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 {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e \int \frac {c^4 d^4-b^4 e^4-2 c^3 d^2 e (b d-3 a e)-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\left (e^4 (2 c d-b e)\right ) \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 (e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\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 ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) 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 {e \left (3 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (c d^2-7 a e^2\right )-2 c \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}-\frac {e^4 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 (2 c d-b e) \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 = 0.75, size = 356, normalized size = 0.90 \begin {gather*} \frac {1}{2} \left (\frac {-c d+b e+c e x}{\left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^2}+\frac {e \left (2 b^3 e^2+b^2 c e (-3 d+2 e x)+2 c^2 \left (c d^2 x+a e (4 d-3 e x)\right )+b c \left (-7 a e^2+c d (d-2 e x)\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}-\frac {2 e \left (-2 c^4 d^4+b^4 e^4+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (2 b d+a e)-2 b^2 c e^3 (b d+3 a e)\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (-c d^2+e (b d-a e)\right )^3}+\frac {2 e^4 (-2 c d+b e) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}+\frac {e^4 (2 c d-b e) \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. \(874\) vs.
\(2(387)=774\).
time = 1.52, size = 875, normalized size = 2.20
method | result | size |
default | \(\frac {\frac {\frac {c^{2} e \left (3 e^{4} a^{2} c -a \,b^{2} e^{4}-2 a b c d \,e^{3}+2 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}-2 b^{2} c \,d^{2} e^{2}+2 d^{3} e b \,c^{2}-d^{4} c^{3}\right ) x^{3}}{4 a c -b^{2}}+\frac {c e \left (13 c \,e^{4} a^{2} b -8 a^{2} c^{2} d \,e^{3}-4 a \,b^{3} e^{4}-8 a \,b^{2} c d \,e^{3}+18 a b \,c^{2} d^{2} e^{2}-8 a \,c^{3} d^{3} e +4 b^{4} d \,e^{3}-9 b^{3} c \,d^{2} e^{2}+8 b^{2} c^{2} d^{3} e -3 d^{4} b \,c^{3}\right ) x^{2}}{8 a c -2 b^{2}}+\frac {e \left (5 a^{3} c^{2} e^{4}+2 a^{2} b^{2} c \,e^{4}-10 a^{2} b \,c^{2} d \,e^{3}+6 a^{2} c^{3} d^{2} e^{2}-a \,b^{4} e^{4}+6 a \,b^{2} c^{2} d^{2} e^{2}-6 a b \,c^{3} d^{3} e +a \,c^{4} d^{4}+b^{5} d \,e^{3}-3 b^{4} c \,d^{2} e^{2}+3 b^{3} c^{2} d^{3} e -b^{2} c^{3} d^{4}\right ) x}{4 a c -b^{2}}+\frac {11 a^{3} b c \,e^{5}-12 a^{3} c^{2} d \,e^{4}-3 a^{2} b^{3} e^{5}-11 a^{2} b^{2} c d \,e^{4}+30 a^{2} b \,c^{2} d^{2} e^{3}-16 a^{2} c^{3} d^{3} e^{2}+4 a \,b^{4} d \,e^{4}-5 a \,b^{3} c \,d^{2} e^{3}-6 a \,b^{2} c^{2} d^{3} e^{2}+11 a b \,c^{3} d^{4} e -4 a \,c^{4} d^{5}-b^{5} d^{2} e^{3}+3 b^{4} c \,d^{3} e^{2}-3 b^{3} c^{2} d^{4} e +b^{2} c^{3} d^{5}}{8 a c -2 b^{2}}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {e \left (\frac {\left (-4 a b \,c^{2} e^{4}+8 d \,e^{3} c^{3} a +b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (3 e^{4} a^{2} c^{2}-5 a \,b^{2} c \,e^{4}+10 a b \,c^{2} d \,e^{3}-6 d^{2} e^{2} c^{3} a +b^{4} e^{4}-2 b^{3} c d \,e^{3}+2 b \,c^{3} d^{3} e -c^{4} d^{4}-\frac {\left (-4 a b \,c^{2} e^{4}+8 d \,e^{3} c^{3} a +b^{3} c \,e^{4}-2 b^{2} c^{2} d \,e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )}{4 a c -b^{2}}}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}+\frac {\left (b e -2 c d \right ) e^{4} \ln \left (e x +d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right )^{3}}\) | \(875\) |
risch | \(\text {Expression too large to display}\) | \(56735\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 3212 vs.
\(2 (394) = 788\).
time = 152.71, size = 6444, normalized size = 16.23 \begin {gather*} \text {Too large to display} \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. 1117 vs.
\(2 (394) = 788\).
time = 1.16, size = 1117, normalized size = 2.81 \begin {gather*} \frac {{\left (2 \, c d e^{4} - b e^{5}\right )} \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 {{\left (2 \, c d e^{5} - b e^{6}\right )} \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 (2 \, c^{4} d^{4} e - 4 \, b c^{3} d^{3} e^{2} + 12 \, a c^{3} d^{2} e^{3} + 2 \, b^{3} c d e^{4} - 12 \, a b c^{2} d e^{4} - b^{4} e^{5} + 6 \, a b^{2} c e^{5} - 6 \, a^{2} c^{2} e^{5}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} d^{6} - 4 \, a c^{4} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 12 \, a b c^{3} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - 9 \, a b^{2} c^{2} d^{4} e^{2} - 12 \, a^{2} c^{3} d^{4} e^{2} - b^{5} d^{3} e^{3} - 2 \, a b^{3} c d^{3} e^{3} + 24 \, a^{2} b c^{2} d^{3} e^{3} + 3 \, a b^{4} d^{2} e^{4} - 9 \, a^{2} b^{2} c d^{2} e^{4} - 12 \, a^{3} c^{2} d^{2} e^{4} - 3 \, a^{2} b^{3} d e^{5} + 12 \, a^{3} b c d e^{5} + a^{3} b^{2} e^{6} - 4 \, a^{4} c e^{6}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b^{2} c^{3} d^{5} - 4 \, a c^{4} d^{5} - 3 \, b^{3} c^{2} d^{4} e + 11 \, a b c^{3} d^{4} e + 3 \, b^{4} c d^{3} e^{2} - 6 \, a b^{2} c^{2} d^{3} e^{2} - 16 \, a^{2} c^{3} d^{3} e^{2} - b^{5} d^{2} e^{3} - 5 \, a b^{3} c d^{2} e^{3} + 30 \, a^{2} b c^{2} d^{2} e^{3} + 4 \, a b^{4} d e^{4} - 11 \, a^{2} b^{2} c d e^{4} - 12 \, a^{3} c^{2} d e^{4} - 3 \, a^{2} b^{3} e^{5} + 11 \, a^{3} b c e^{5} - 2 \, {\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2} + 2 \, b^{2} c^{3} d^{2} e^{3} - 2 \, a c^{4} d^{2} e^{3} - b^{3} c^{2} d e^{4} + 2 \, a b c^{3} d e^{4} + a b^{2} c^{2} e^{5} - 3 \, a^{2} c^{3} e^{5}\right )} x^{3} - {\left (3 \, b c^{4} d^{4} e - 8 \, b^{2} c^{3} d^{3} e^{2} + 8 \, a c^{4} d^{3} e^{2} + 9 \, b^{3} c^{2} d^{2} e^{3} - 18 \, a b c^{3} d^{2} e^{3} - 4 \, b^{4} c d e^{4} + 8 \, a b^{2} c^{2} d e^{4} + 8 \, a^{2} c^{3} d e^{4} + 4 \, a b^{3} c e^{5} - 13 \, a^{2} b c^{2} e^{5}\right )} x^{2} - 2 \, {\left (b^{2} c^{3} d^{4} e - a c^{4} d^{4} e - 3 \, b^{3} c^{2} d^{3} e^{2} + 6 \, a b c^{3} d^{3} e^{2} + 3 \, b^{4} c d^{2} e^{3} - 6 \, a b^{2} c^{2} d^{2} e^{3} - 6 \, a^{2} c^{3} d^{2} e^{3} - b^{5} d e^{4} + 10 \, a^{2} b c^{2} d e^{4} + a b^{4} e^{5} - 2 \, a^{2} b^{2} c e^{5} - 5 \, a^{3} 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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.46, size = 2461, normalized size = 6.20 \begin {gather*} \frac {\ln \left (\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3+4\,a\,b\,c+8\,a\,c^2\,x-2\,b^2\,c\,x\right )\,\left (b^7\,e^5+b^4\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-64\,a^3\,b\,c^3\,e^5+128\,a^3\,c^4\,d\,e^4-2\,c^4\,d^4\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+48\,a^2\,b^3\,c^2\,e^5+6\,a^2\,c^2\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-12\,a\,b^5\,c\,e^5-2\,b^6\,c\,d\,e^4-6\,a\,b^2\,c\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a\,b^4\,c^2\,d\,e^4-2\,b^3\,c\,d\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-96\,a^2\,b^2\,c^3\,d\,e^4-12\,a\,c^3\,d^2\,e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+4\,b\,c^3\,d^3\,e^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a\,b\,c^2\,d\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )}{2\,\left (64\,a^6\,c^3\,e^6-48\,a^5\,b^2\,c^2\,e^6-192\,a^5\,b\,c^3\,d\,e^5+192\,a^5\,c^4\,d^2\,e^4+12\,a^4\,b^4\,c\,e^6+144\,a^4\,b^3\,c^2\,d\,e^5+48\,a^4\,b^2\,c^3\,d^2\,e^4-384\,a^4\,b\,c^4\,d^3\,e^3+192\,a^4\,c^5\,d^4\,e^2-a^3\,b^6\,e^6-36\,a^3\,b^5\,c\,d\,e^5-108\,a^3\,b^4\,c^2\,d^2\,e^4+224\,a^3\,b^3\,c^3\,d^3\,e^3+48\,a^3\,b^2\,c^4\,d^4\,e^2-192\,a^3\,b\,c^5\,d^5\,e+64\,a^3\,c^6\,d^6+3\,a^2\,b^7\,d\,e^5+33\,a^2\,b^6\,c\,d^2\,e^4-24\,a^2\,b^5\,c^2\,d^3\,e^3-108\,a^2\,b^4\,c^3\,d^4\,e^2+144\,a^2\,b^3\,c^4\,d^5\,e-48\,a^2\,b^2\,c^5\,d^6-3\,a\,b^8\,d^2\,e^4-6\,a\,b^7\,c\,d^3\,e^3+33\,a\,b^6\,c^2\,d^4\,e^2-36\,a\,b^5\,c^3\,d^5\,e+12\,a\,b^4\,c^4\,d^6+b^9\,d^3\,e^3-3\,b^8\,c\,d^4\,e^2+3\,b^7\,c^2\,d^5\,e-b^6\,c^3\,d^6\right )}-\frac {\frac {-11\,a^2\,b\,c\,e^3+12\,a^2\,c^2\,d\,e^2+3\,a\,b^3\,e^3-7\,a\,b\,c^2\,d^2\,e+4\,a\,c^3\,d^3-b^4\,d\,e^2+2\,b^3\,c\,d^2\,e-b^2\,c^2\,d^3}{2\,\left (4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4\right )}-\frac {x\,\left (5\,a^2\,c^2\,e^3+2\,a\,b^2\,c\,e^3-5\,a\,b\,c^2\,d\,e^2+a\,c^3\,d^2\,e-b^4\,e^3+2\,b^3\,c\,d\,e^2-b^2\,c^2\,d^2\,e\right )}{4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4}+\frac {x^2\,\left (4\,b^3\,c\,e^3-5\,b^2\,c^2\,d\,e^2+3\,b\,c^3\,d^2\,e-13\,a\,b\,c^2\,e^3+8\,a\,c^3\,d\,e^2\right )}{2\,\left (4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4\right )}+\frac {e\,x^3\,\left (b^2\,c^2\,e^2-b\,c^3\,d\,e+c^4\,d^2-3\,a\,c^3\,e^2\right )}{4\,a^3\,c\,e^4-a^2\,b^2\,e^4-8\,a^2\,b\,c\,d\,e^3+8\,a^2\,c^2\,d^2\,e^2+2\,a\,b^3\,d\,e^3+2\,a\,b^2\,c\,d^2\,e^2-8\,a\,b\,c^2\,d^3\,e+4\,a\,c^3\,d^4-b^4\,d^2\,e^2+2\,b^3\,c\,d^3\,e-b^2\,c^2\,d^4}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (b\,e^5-2\,c\,d\,e^4\right )}{a^3\,e^6-3\,a^2\,b\,d\,e^5+3\,a^2\,c\,d^2\,e^4+3\,a\,b^2\,d^2\,e^4-6\,a\,b\,c\,d^3\,e^3+3\,a\,c^2\,d^4\,e^2-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}+\frac {\ln \left (b^3+\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c-8\,a\,c^2\,x+2\,b^2\,c\,x\right )\,\left (\frac {b^7\,e^5}{2}-\frac {b^4\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}}{2}-32\,a^3\,b\,c^3\,e^5+64\,a^3\,c^4\,d\,e^4+c^4\,d^4\,e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+24\,a^2\,b^3\,c^2\,e^5-3\,a^2\,c^2\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-6\,a\,b^5\,c\,e^5-b^6\,c\,d\,e^4+3\,a\,b^2\,c\,e^5\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+12\,a\,b^4\,c^2\,d\,e^4+b^3\,c\,d\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-48\,a^2\,b^2\,c^3\,d\,e^4+6\,a\,c^3\,d^2\,e^3\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-2\,b\,c^3\,d^3\,e^2\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-6\,a\,b\,c^2\,d\,e^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}\right )}{64\,a^6\,c^3\,e^6-48\,a^5\,b^2\,c^2\,e^6-192\,a^5\,b\,c^3\,d\,e^5+192\,a^5\,c^4\,d^2\,e^4+12\,a^4\,b^4\,c\,e^6+144\,a^4\,b^3\,c^2\,d\,e^5+48\,a^4\,b^2\,c^3\,d^2\,e^4-384\,a^4\,b\,c^4\,d^3\,e^3+192\,a^4\,c^5\,d^4\,e^2-a^3\,b^6\,e^6-36\,a^3\,b^5\,c\,d\,e^5-108\,a^3\,b^4\,c^2\,d^2\,e^4+224\,a^3\,b^3\,c^3\,d^3\,e^3+48\,a^3\,b^2\,c^4\,d^4\,e^2-192\,a^3\,b\,c^5\,d^5\,e+64\,a^3\,c^6\,d^6+3\,a^2\,b^7\,d\,e^5+33\,a^2\,b^6\,c\,d^2\,e^4-24\,a^2\,b^5\,c^2\,d^3\,e^3-108\,a^2\,b^4\,c^3\,d^4\,e^2+144\,a^2\,b^3\,c^4\,d^5\,e-48\,a^2\,b^2\,c^5\,d^6-3\,a\,b^8\,d^2\,e^4-6\,a\,b^7\,c\,d^3\,e^3+33\,a\,b^6\,c^2\,d^4\,e^2-36\,a\,b^5\,c^3\,d^5\,e+12\,a\,b^4\,c^4\,d^6+b^9\,d^3\,e^3-3\,b^8\,c\,d^4\,e^2+3\,b^7\,c^2\,d^5\,e-b^6\,c^3\,d^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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