Optimal. Leaf size=260 \[ -\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {752, 652, 628,
632, 212} \begin {gather*} -\frac {2 (b+2 c x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {-2 x \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+8 a c d e+3 b^2 d e}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {8 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac {(d+e x) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 628
Rule 632
Rule 652
Rule 752
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^4} \, dx &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {\int \frac {2 \left (5 c d^2-e (3 b d-a e)\right )+4 e (2 c d-b e) x}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {\left (2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {\left (4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {\left (8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-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 )^3}\\ &=-\frac {(d+e x) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {3 b^2 d e+8 a c d e-5 b \left (c d^2+a e^2\right )-2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-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 )^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 258, normalized size = 0.99 \begin {gather*} \frac {1}{3} \left (\frac {\left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {6 \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac {a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)}{c \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {24 c \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+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 )^{7/2}}\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. \(648\) vs.
\(2(252)=504\).
time = 0.94, size = 649, normalized size = 2.50
method | result | size |
default | \(\frac {\frac {4 c^{3} \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {10 c^{2} \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) b \,x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {2 \left (16 a c +11 b^{2}\right ) c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{3}}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b \left (16 a c +b^{2}\right ) \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {\left (4 a^{3} c^{2} e^{2}-22 a^{2} b^{2} c \,e^{2}+44 a^{2} b \,c^{2} d e -44 a^{2} c^{3} d^{2}-a \,b^{4} e^{2}+18 a \,b^{3} c d e -18 a \,c^{2} d^{2} b^{2}-b^{5} d e +b^{4} c \,d^{2}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {26 a^{3} b c \,e^{2}-64 a^{3} c^{2} d e +a^{2} b^{3} e^{2}-18 a^{2} b^{2} c d e +66 a^{2} b \,c^{2} d^{2}+a \,b^{4} d e -13 a \,b^{3} c \,d^{2}+b^{5} d^{2}}{192 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {8 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {4 a c -b^{2}}}\) | \(649\) |
risch | \(\frac {\frac {4 c^{3} \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {10 c^{2} \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) b \,x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {2 \left (16 a c +11 b^{2}\right ) c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{3}}{3 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b \left (16 a c +b^{2}\right ) \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {\left (4 a^{3} c^{2} e^{2}-22 a^{2} b^{2} c \,e^{2}+44 a^{2} b \,c^{2} d e -44 a^{2} c^{3} d^{2}-a \,b^{4} e^{2}+18 a \,b^{3} c d e -18 a \,c^{2} d^{2} b^{2}-b^{5} d e +b^{4} c \,d^{2}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {26 a^{3} b c \,e^{2}-64 a^{3} c^{2} d e +a^{2} b^{3} e^{2}-18 a^{2} b^{2} c d e +66 a^{2} b \,c^{2} d^{2}+a \,b^{4} d e -13 a \,b^{3} c \,d^{2}+b^{5} d^{2}}{192 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {4 c^{2} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) a \,e^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}+\frac {4 c \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) b^{2} e^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {20 c^{2} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) b d e}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}+\frac {20 c^{3} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 a \,b^{4} c^{2}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} b \,c^{3}+48 a^{2} b^{3} c^{2}-12 a \,b^{5} c +b^{7}\right ) d^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {4 c^{2} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) a \,e^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {4 c \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) b^{2} e^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}+\frac {20 c^{2} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) b d e}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {20 c^{3} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 a \,b^{4} c^{2}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) d^{2}}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}\) | \(1340\) |
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. 1366 vs.
\(2 (259) = 518\).
time = 2.06, size = 2753, normalized size = 10.59 \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. 1635 vs.
\(2 (257) = 514\).
time = 3.82, size = 1635, normalized size = 6.29 \begin {gather*} - 4 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (x + \frac {- 1024 a^{4} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 1024 a^{3} b^{2} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) - 384 a^{2} b^{4} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 64 a b^{6} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 4 a b c^{2} e^{2} - 4 b^{8} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 4 b^{3} c e^{2} - 20 b^{2} c^{2} d e + 20 b c^{3} d^{2}}{8 a c^{3} e^{2} + 8 b^{2} c^{2} e^{2} - 40 b c^{3} d e + 40 c^{4} d^{2}} \right )} + 4 c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (x + \frac {1024 a^{4} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) - 1024 a^{3} b^{2} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 384 a^{2} b^{4} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) - 64 a b^{6} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 4 a b c^{2} e^{2} + 4 b^{8} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) + 4 b^{3} c e^{2} - 20 b^{2} c^{2} d e + 20 b c^{3} d^{2}}{8 a c^{3} e^{2} + 8 b^{2} c^{2} e^{2} - 40 b c^{3} d e + 40 c^{4} d^{2}} \right )} + \frac {26 a^{3} b c e^{2} - 64 a^{3} c^{2} d e + a^{2} b^{3} e^{2} - 18 a^{2} b^{2} c d e + 66 a^{2} b c^{2} d^{2} + a b^{4} d e - 13 a b^{3} c d^{2} + b^{5} d^{2} + x^{5} \cdot \left (12 a c^{4} e^{2} + 12 b^{2} c^{3} e^{2} - 60 b c^{4} d e + 60 c^{5} d^{2}\right ) + x^{4} \cdot \left (30 a b c^{3} e^{2} + 30 b^{3} c^{2} e^{2} - 150 b^{2} c^{3} d e + 150 b c^{4} d^{2}\right ) + x^{3} \cdot \left (32 a^{2} c^{3} e^{2} + 54 a b^{2} c^{2} e^{2} - 160 a b c^{3} d e + 160 a c^{4} d^{2} + 22 b^{4} c e^{2} - 110 b^{3} c^{2} d e + 110 b^{2} c^{3} d^{2}\right ) + x^{2} \cdot \left (48 a^{2} b c^{2} e^{2} + 51 a b^{3} c e^{2} - 240 a b^{2} c^{2} d e + 240 a b c^{3} d^{2} + 3 b^{5} e^{2} - 15 b^{4} c d e + 15 b^{3} c^{2} d^{2}\right ) + x \left (- 12 a^{3} c^{2} e^{2} + 66 a^{2} b^{2} c e^{2} - 132 a^{2} b c^{2} d e + 132 a^{2} c^{3} d^{2} + 3 a b^{4} e^{2} - 54 a b^{3} c d e + 54 a b^{2} c^{2} d^{2} + 3 b^{5} d e - 3 b^{4} c d^{2}\right )}{192 a^{6} c^{3} - 144 a^{5} b^{2} c^{2} + 36 a^{4} b^{4} c - 3 a^{3} b^{6} + x^{6} \cdot \left (192 a^{3} c^{6} - 144 a^{2} b^{2} c^{5} + 36 a b^{4} c^{4} - 3 b^{6} c^{3}\right ) + x^{5} \cdot \left (576 a^{3} b c^{5} - 432 a^{2} b^{3} c^{4} + 108 a b^{5} c^{3} - 9 b^{7} c^{2}\right ) + x^{4} \cdot \left (576 a^{4} c^{5} + 144 a^{3} b^{2} c^{4} - 324 a^{2} b^{4} c^{3} + 99 a b^{6} c^{2} - 9 b^{8} c\right ) + x^{3} \cdot \left (1152 a^{4} b c^{4} - 672 a^{3} b^{3} c^{3} + 72 a^{2} b^{5} c^{2} + 18 a b^{7} c - 3 b^{9}\right ) + x^{2} \cdot \left (576 a^{5} c^{4} + 144 a^{4} b^{2} c^{3} - 324 a^{3} b^{4} c^{2} + 99 a^{2} b^{6} c - 9 a b^{8}\right ) + x \left (576 a^{5} b c^{3} - 432 a^{4} b^{3} c^{2} + 108 a^{3} b^{5} c - 9 a^{2} b^{7}\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. 603 vs.
\(2 (259) = 518\).
time = 0.84, size = 603, normalized size = 2.32 \begin {gather*} -\frac {8 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {60 \, c^{5} d^{2} x^{5} - 60 \, b c^{4} d x^{5} e + 150 \, b c^{4} d^{2} x^{4} + 12 \, b^{2} c^{3} x^{5} e^{2} + 12 \, a c^{4} x^{5} e^{2} - 150 \, b^{2} c^{3} d x^{4} e + 110 \, b^{2} c^{3} d^{2} x^{3} + 160 \, a c^{4} d^{2} x^{3} + 30 \, b^{3} c^{2} x^{4} e^{2} + 30 \, a b c^{3} x^{4} e^{2} - 110 \, b^{3} c^{2} d x^{3} e - 160 \, a b c^{3} d x^{3} e + 15 \, b^{3} c^{2} d^{2} x^{2} + 240 \, a b c^{3} d^{2} x^{2} + 22 \, b^{4} c x^{3} e^{2} + 54 \, a b^{2} c^{2} x^{3} e^{2} + 32 \, a^{2} c^{3} x^{3} e^{2} - 15 \, b^{4} c d x^{2} e - 240 \, a b^{2} c^{2} d x^{2} e - 3 \, b^{4} c d^{2} x + 54 \, a b^{2} c^{2} d^{2} x + 132 \, a^{2} c^{3} d^{2} x + 3 \, b^{5} x^{2} e^{2} + 51 \, a b^{3} c x^{2} e^{2} + 48 \, a^{2} b c^{2} x^{2} e^{2} + 3 \, b^{5} d x e - 54 \, a b^{3} c d x e - 132 \, a^{2} b c^{2} d x e + b^{5} d^{2} - 13 \, a b^{3} c d^{2} + 66 \, a^{2} b c^{2} d^{2} + 3 \, a b^{4} x e^{2} + 66 \, a^{2} b^{2} c x e^{2} - 12 \, a^{3} c^{2} x e^{2} + a b^{4} d e - 18 \, a^{2} b^{2} c d e - 64 \, a^{3} c^{2} d e + a^{2} b^{3} e^{2} + 26 \, a^{3} b c e^{2}}{3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.37, size = 872, normalized size = 3.35 \begin {gather*} -\frac {\frac {26\,a^3\,b\,c\,e^2-64\,a^3\,c^2\,d\,e+a^2\,b^3\,e^2-18\,a^2\,b^2\,c\,d\,e+66\,a^2\,b\,c^2\,d^2+a\,b^4\,d\,e-13\,a\,b^3\,c\,d^2+b^5\,d^2}{3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x\,\left (-4\,a^3\,c^2\,e^2+22\,a^2\,b^2\,c\,e^2-44\,a^2\,b\,c^2\,d\,e+44\,a^2\,c^3\,d^2+a\,b^4\,e^2-18\,a\,b^3\,c\,d\,e+18\,a\,b^2\,c^2\,d^2+b^5\,d\,e-b^4\,c\,d^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {2\,x^3\,\left (11\,b^2\,c+16\,a\,c^2\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^2\,\left (b^3+16\,a\,c\,b\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {4\,c^3\,x^5\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {10\,b\,c^2\,x^4\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}-\frac {8\,c\,\mathrm {atan}\left (\frac {\left (\frac {8\,c^2\,x\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {4\,c\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{4\,b^2\,c\,e^2-20\,b\,c^2\,d\,e+20\,c^3\,d^2+4\,a\,c^2\,e^2}\right )\,\left (b^2\,e^2-5\,b\,c\,d\,e+5\,c^2\,d^2+a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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