Optimal. Leaf size=260 \[ \frac {2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {e^3 (2 c d-b e) x^2}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{c^3} \]
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Rubi [A]
time = 0.38, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {752, 814, 648,
632, 212, 642} \begin {gather*} \frac {2 e^2 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac {e^3 x^2 (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 \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 )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 752
Rule 814
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \frac {(d+e x)^2 \left (2 \left (c d^2-2 b d e+3 a e^2\right )-2 e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=-\frac {(d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\int \left (-\frac {2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right )}{c^2}-\frac {2 e^3 (2 c d-b e) x}{c}+\frac {2 \left (c^3 d^4+a b^2 e^4-2 c^2 d^2 e (b d-3 a e)-a c e^3 (2 b d+3 a e)-\left (b^2-4 a c\right ) e^3 (2 c d-b e) x\right )}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{-b^2+4 a c}\\ &=\frac {2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {e^3 (2 c d-b e) x^2}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \int \frac {c^3 d^4+a b^2 e^4-2 c^2 d^2 e (b d-3 a e)-a c e^3 (2 b d+3 a e)-\left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{a+b x+c x^2} \, dx}{c^2 \left (b^2-4 a c\right )}\\ &=\frac {2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {e^3 (2 c d-b e) x^2}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (e^3 (2 c d-b e)\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{c^3}-\frac {\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 ) \int \frac {1}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=\frac {2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {e^3 (2 c d-b e) x^2}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{c^3}+\frac {\left (2 \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 )}{c^3 \left (b^2-4 a c\right )}\\ &=\frac {2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac {e^3 (2 c d-b e) x^2}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 \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 )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac {e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{c^3}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 298, normalized size = 1.15 \begin {gather*} \frac {c e^4 x+\frac {-b^4 e^4 x+b^3 e^3 (-a e+4 c d x)+2 b^2 c e^2 \left (-3 c d^2 x+2 a e (d+e x)\right )-b c \left (-3 a^2 e^4+c^2 d^3 (d-4 e x)+6 a c d e^2 (d+2 e x)\right )-2 c^2 \left (c^2 d^4 x+a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {2 \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}}+e^3 (2 c d-b e) \log (a+x (b+c x))}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.97, size = 436, normalized size = 1.68
method | result | size |
default | \(\frac {e^{4} x}{c^{2}}-\frac {\frac {-\frac {\left (2 e^{4} a^{2} c^{2}-4 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 d^{2} e^{2} c^{3} a +b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {3 a^{2} b c \,e^{4}-8 a^{2} c^{2} d \,e^{3}-a \,b^{3} e^{4}+4 a \,b^{2} c d \,e^{3}-6 a b \,c^{2} d^{2} e^{2}+8 a \,c^{3} d^{3} e -d^{4} b \,c^{3}}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a b c \,e^{4}-8 d \,e^{3} c^{2} a -b^{3} e^{4}+2 b^{2} d \,e^{3} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {4 \left (3 e^{4} a^{2} c -a \,b^{2} e^{4}+2 a b c d \,e^{3}-6 d^{2} e^{2} c^{2} a +2 d^{3} e b \,c^{2}-d^{4} c^{3}-\frac {\left (4 a b c \,e^{4}-8 d \,e^{3} c^{2} a -b^{3} e^{4}+2 b^{2} d \,e^{3} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{2}}\) | \(436\) |
risch | \(\text {Expression too large to display}\) | \(5644\) |
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. 878 vs.
\(2 (260) = 520\).
time = 2.23, size = 1776, normalized size = 6.83 \begin {gather*} \left [-\frac {2 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{4} x + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{4} - {\left (2 \, c^{5} d^{4} x^{2} + 2 \, b c^{4} d^{4} x + 2 \, a c^{4} d^{4} - {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x\right )} e^{4} + 2 \, {\left ({\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} d x^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2}\right )} d x + {\left (a b^{3} c - 6 \, a^{2} b c^{2}\right )} d\right )} e^{3} + 12 \, {\left (a c^{4} d^{2} x^{2} + a b c^{3} d^{2} x + a^{2} c^{3} d^{2}\right )} e^{2} - 4 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x + a b c^{3} d^{3}\right )} e\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (a b^{5} - 7 \, a^{2} b^{3} c + 12 \, a^{3} b c^{2} - {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{3} - {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} x\right )} e^{4} - 4 \, {\left ({\left (b^{5} c - 7 \, a b^{3} c^{2} + 12 \, a^{2} b c^{3}\right )} d x + {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d\right )} e^{3} + 6 \, {\left ({\left (b^{4} c^{2} - 6 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{2} x + {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d^{2}\right )} e^{2} - 4 \, {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{3} x + 2 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} d^{3}\right )} e + {\left ({\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x\right )} e^{4} - 2 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{2} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x}, -\frac {2 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{4} x + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{4} - 2 \, {\left (2 \, c^{5} d^{4} x^{2} + 2 \, b c^{4} d^{4} x + 2 \, a c^{4} d^{4} - {\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{2} + {\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x\right )} e^{4} + 2 \, {\left ({\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} d x^{2} + {\left (b^{4} c - 6 \, a b^{2} c^{2}\right )} d x + {\left (a b^{3} c - 6 \, a^{2} b c^{2}\right )} d\right )} e^{3} + 12 \, {\left (a c^{4} d^{2} x^{2} + a b c^{3} d^{2} x + a^{2} c^{3} d^{2}\right )} e^{2} - 4 \, {\left (b c^{4} d^{3} x^{2} + b^{2} c^{3} d^{3} x + a b c^{3} d^{3}\right )} e\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (a b^{5} - 7 \, a^{2} b^{3} c + 12 \, a^{3} b c^{2} - {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{3} - {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 9 \, a b^{4} c + 26 \, a^{2} b^{2} c^{2} - 24 \, a^{3} c^{3}\right )} x\right )} e^{4} - 4 \, {\left ({\left (b^{5} c - 7 \, a b^{3} c^{2} + 12 \, a^{2} b c^{3}\right )} d x + {\left (a b^{4} c - 6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d\right )} e^{3} + 6 \, {\left ({\left (b^{4} c^{2} - 6 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{2} x + {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d^{2}\right )} e^{2} - 4 \, {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{3} x + 2 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4}\right )} d^{3}\right )} e + {\left ({\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{2} + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} x\right )} e^{4} - 2 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{a b^{4} c^{3} - 8 \, a^{2} b^{2} c^{4} + 16 \, a^{3} c^{5} + {\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} x^{2} + {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} x}\right ] \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. 1924 vs.
\(2 (252) = 504\).
time = 12.53, size = 1924, normalized size = 7.40 \begin {gather*} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a^{2} c^{2} e^{4} - 6 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 2 b^{3} c d e^{3} + 4 b c^{3} d^{3} e - 2 c^{4} d^{4}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x + \frac {- 10 a^{2} b c e^{4} - 16 a^{2} c^{4} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a^{2} c^{2} e^{4} - 6 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 2 b^{3} c d e^{3} + 4 b c^{3} d^{3} e - 2 c^{4} d^{4}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 32 a^{2} c^{2} d e^{3} + 2 a b^{3} e^{4} + 8 a b^{2} c^{3} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a^{2} c^{2} e^{4} - 6 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 2 b^{3} c d e^{3} + 4 b c^{3} d^{3} e - 2 c^{4} d^{4}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - 4 a b^{2} c d e^{3} - 12 a b c^{2} d^{2} e^{2} - b^{4} c^{2} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} - \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a^{2} c^{2} e^{4} - 6 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 2 b^{3} c d e^{3} + 4 b c^{3} d^{3} e - 2 c^{4} d^{4}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 4 b^{2} c^{2} d^{3} e - 2 b c^{3} d^{4}}{12 a^{2} c^{2} e^{4} - 12 a b^{2} c e^{4} + 24 a b c^{2} d e^{3} - 24 a c^{3} d^{2} e^{2} + 2 b^{4} e^{4} - 4 b^{3} c d e^{3} + 8 b c^{3} d^{3} e - 4 c^{4} d^{4}} \right )} + \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a^{2} c^{2} e^{4} - 6 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 2 b^{3} c d e^{3} + 4 b c^{3} d^{3} e - 2 c^{4} d^{4}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) \log {\left (x + \frac {- 10 a^{2} b c e^{4} - 16 a^{2} c^{4} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a^{2} c^{2} e^{4} - 6 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 2 b^{3} c d e^{3} + 4 b c^{3} d^{3} e - 2 c^{4} d^{4}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 32 a^{2} c^{2} d e^{3} + 2 a b^{3} e^{4} + 8 a b^{2} c^{3} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a^{2} c^{2} e^{4} - 6 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 2 b^{3} c d e^{3} + 4 b c^{3} d^{3} e - 2 c^{4} d^{4}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) - 4 a b^{2} c d e^{3} - 12 a b c^{2} d^{2} e^{2} - b^{4} c^{2} \left (- \frac {e^{3} \left (b e - 2 c d\right )}{c^{3}} + \frac {\sqrt {- \left (4 a c - b^{2}\right )^{3}} \cdot \left (6 a^{2} c^{2} e^{4} - 6 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 2 b^{3} c d e^{3} + 4 b c^{3} d^{3} e - 2 c^{4} d^{4}\right )}{c^{3} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}\right ) + 4 b^{2} c^{2} d^{3} e - 2 b c^{3} d^{4}}{12 a^{2} c^{2} e^{4} - 12 a b^{2} c e^{4} + 24 a b c^{2} d e^{3} - 24 a c^{3} d^{2} e^{2} + 2 b^{4} e^{4} - 4 b^{3} c d e^{3} + 8 b c^{3} d^{3} e - 4 c^{4} d^{4}} \right )} + \frac {- 3 a^{2} b c e^{4} + 8 a^{2} c^{2} d e^{3} + a b^{3} e^{4} - 4 a b^{2} c d e^{3} + 6 a b c^{2} d^{2} e^{2} - 8 a c^{3} d^{3} e + b c^{3} d^{4} + x \left (2 a^{2} c^{2} e^{4} - 4 a b^{2} c e^{4} + 12 a b c^{2} d e^{3} - 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 4 b^{3} c d e^{3} + 6 b^{2} c^{2} d^{2} e^{2} - 4 b c^{3} d^{3} e + 2 c^{4} d^{4}\right )}{4 a^{2} c^{4} - a b^{2} c^{3} + x^{2} \cdot \left (4 a c^{5} - b^{2} c^{4}\right ) + x \left (4 a b c^{4} - b^{3} c^{3}\right )} + \frac {e^{4} x}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.99, size = 355, normalized size = 1.37 \begin {gather*} -\frac {2 \, {\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 12 \, a c^{3} d^{2} e^{2} + 2 \, b^{3} c d e^{3} - 12 \, a b c^{2} d e^{3} - b^{4} e^{4} + 6 \, a b^{2} c e^{4} - 6 \, a^{2} c^{2} e^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {x e^{4}}{c^{2}} + \frac {{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{c^{3}} - \frac {\frac {{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 12 \, a c^{3} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + 12 \, a b c^{2} d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} x}{c} + \frac {b c^{3} d^{4} - 8 \, a c^{3} d^{3} e + 6 \, a b c^{2} d^{2} e^{2} - 4 \, a b^{2} c d e^{3} + 8 \, a^{2} c^{2} d e^{3} + a b^{3} e^{4} - 3 \, a^{2} b c e^{4}}{c}}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.87, size = 525, normalized size = 2.02 \begin {gather*} \frac {\frac {-3\,a^2\,b\,c\,e^4+8\,a^2\,c^2\,d\,e^3+a\,b^3\,e^4-4\,a\,b^2\,c\,d\,e^3+6\,a\,b\,c^2\,d^2\,e^2-8\,a\,c^3\,d^3\,e+b\,c^3\,d^4}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (2\,a^2\,c^2\,e^4-4\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3-12\,a\,c^3\,d^2\,e^2+b^4\,e^4-4\,b^3\,c\,d\,e^3+6\,b^2\,c^2\,d^2\,e^2-4\,b\,c^3\,d^3\,e+2\,c^4\,d^4\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^3\,x^2+b\,c^2\,x+a\,c^2}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-128\,a^3\,b\,c^3\,e^4+256\,d\,a^3\,c^4\,e^3+96\,a^2\,b^3\,c^2\,e^4-192\,d\,a^2\,b^2\,c^3\,e^3-24\,a\,b^5\,c\,e^4+48\,d\,a\,b^4\,c^2\,e^3+2\,b^7\,e^4-4\,d\,b^6\,c\,e^3\right )}{2\,\left (64\,a^3\,c^6-48\,a^2\,b^2\,c^5+12\,a\,b^4\,c^4-b^6\,c^3\right )}+\frac {e^4\,x}{c^2}-\frac {2\,\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^2-4\,a\,b\,c^3}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (6\,a^2\,c^2\,e^4-6\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3-12\,a\,c^3\,d^2\,e^2+b^4\,e^4-2\,b^3\,c\,d\,e^3+4\,b\,c^3\,d^3\,e-2\,c^4\,d^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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