Optimal. Leaf size=259 \[ -\frac {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (5 b c d-2 b^2 e-2 a c e+5 c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {2 \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {8 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \]
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Rubi [A]
time = 0.29, antiderivative size = 259, 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 = {750, 818, 736,
632, 212} \begin {gather*} -\frac {2 (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {8 \left (a e^2-b d e+c d^2\right ) \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 {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (-2 a c e-2 b^2 e+5 c x (2 c d-b e)+5 b c d\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 736
Rule 750
Rule 818
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^4} \, dx &=-\frac {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {\int \frac {(d+e x)^3 (-10 c d+4 b e-2 c e x)}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (5 b c d-2 b^2 e-2 a c e+5 c (2 c d-b e) x\right )}{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 {(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (5 b c d-2 b^2 e-2 a c e+5 c (2 c d-b e) x\right )}{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 ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {\left (4 \left (c d^2-b d e+a e^2\right ) \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 {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (5 b c d-2 b^2 e-2 a c e+5 c (2 c d-b e) x\right )}{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 ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {\left (8 \left (c d^2-b d e+a e^2\right ) \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 {(b+2 c x) (d+e x)^4}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^3 \left (5 b c d-2 b^2 e-2 a c e+5 c (2 c d-b e) x\right )}{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 ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {8 \left (c d^2-b d e+a e^2\right ) \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 [B] Leaf count is larger than twice the leaf count of optimal. \(572\) vs. \(2(259)=518\).
time = 0.65, size = 572, normalized size = 2.21 \begin {gather*} \frac {1}{3} \left (\frac {6 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right ) (b+2 c x)}{c \left (-b^2+4 a c\right )^3 (a+x (b+c 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 )}{c^3 \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {b^5 e^4-b^4 c e^3 (4 d+e x)+b c^2 \left (17 a^2 e^4+5 c^2 d^3 (d-4 e x)+6 a c d e^2 (d-2 e x)\right )+b^3 c e^2 \left (-7 a e^2+2 c d (3 d-e x)\right )+2 b^2 c^2 e \left (a e^2 (9 d+5 e x)+c d^2 (-5 d+6 e x)\right )+2 c^3 \left (5 c^2 d^4 x+6 a c d^2 e^2 x-a^2 e^3 (24 d+7 e x)\right )}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac {24 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+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 )^{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. \(1204\) vs.
\(2(251)=502\).
time = 0.83, size = 1205, normalized size = 4.65
method | result | size |
default | \(\frac {\frac {4 \left (e^{4} a^{2} c +a \,b^{2} e^{4}-6 a b c d \,e^{3}+6 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+6 b^{2} c \,d^{2} e^{2}-10 d^{3} e b \,c^{2}+5 d^{4} c^{3}\right ) c^{2} x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {10 \left (e^{4} a^{2} c +a \,b^{2} e^{4}-6 a b c d \,e^{3}+6 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+6 b^{2} c \,d^{2} e^{2}-10 d^{3} e b \,c^{2}+5 d^{4} c^{3}\right ) b c \,x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {\left (32 a^{3} c^{3} e^{4}-102 a^{2} b^{2} c^{2} e^{4}+192 a^{2} b \,c^{3} d \,e^{3}-192 a^{2} c^{4} d^{2} e^{2}-10 a \,b^{4} c \,e^{4}+164 a \,b^{3} c^{2} d \,e^{3}-324 a \,b^{2} c^{3} d^{2} e^{2}+320 a b \,c^{4} d^{3} e -160 a \,c^{5} d^{4}-b^{6} e^{4}+22 b^{5} c d \,e^{3}-132 b^{4} c^{2} d^{2} e^{2}+220 b^{3} c^{3} d^{3} e -110 b^{2} d^{4} c^{4}\right ) x^{3}}{3 c \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {\left (16 a^{3} b \,c^{2} e^{4}-64 a^{3} c^{3} d \,e^{3}+17 a^{2} b^{3} c \,e^{4}-48 a^{2} b^{2} c^{2} d \,e^{3}+96 a^{2} b \,c^{3} d^{2} e^{2}+a \,b^{5} e^{4}-34 a \,b^{4} c d \,e^{3}+102 a \,b^{3} c^{2} d^{2} e^{2}-160 a \,b^{2} c^{3} d^{3} e +80 c^{4} d^{4} a b +6 b^{5} c \,d^{2} e^{2}-10 b^{4} c^{2} d^{3} e +5 b^{3} c^{3} d^{4}\right ) x^{2}}{c \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {\left (4 a^{4} c^{2} e^{4}-22 a^{3} b^{2} c \,e^{4}+40 a^{3} b \,c^{2} d \,e^{3}+24 a^{3} c^{3} d^{2} e^{2}-a^{2} b^{4} e^{4}+40 a^{2} b^{3} c d \,e^{3}-132 a^{2} b^{2} c^{2} d^{2} e^{2}+88 a^{2} b \,c^{3} d^{3} e -44 a^{2} c^{4} d^{4}-6 a \,b^{4} c \,d^{2} e^{2}+36 a \,b^{3} c^{2} d^{3} e -18 a \,b^{2} c^{3} d^{4}-2 b^{5} c \,d^{3} e +b^{4} c^{2} d^{4}\right ) x}{c \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {26 a^{4} b c \,e^{4}-64 a^{4} c^{2} d \,e^{3}+a^{3} b^{3} e^{4}-44 a^{3} b^{2} c d \,e^{3}+156 a^{3} b \,c^{2} d^{2} e^{2}-128 a^{3} c^{3} d^{3} e +6 a^{2} b^{3} c \,d^{2} e^{2}-36 a^{2} b^{2} c^{2} d^{3} e +66 a^{2} b \,c^{3} d^{4}+2 a \,b^{4} c \,d^{3} e -13 a \,b^{3} c^{2} d^{4}+b^{5} c \,d^{4}}{3 c \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {8 \left (e^{4} a^{2} c +a \,b^{2} e^{4}-6 a b c d \,e^{3}+6 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+6 b^{2} c \,d^{2} e^{2}-10 d^{3} e b \,c^{2}+5 d^{4} c^{3}\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}}}\) | \(1205\) |
risch | \(\text {Expression too large to display}\) | \(2656\) |
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. 2292 vs.
\(2 (261) = 522\).
time = 2.72, size = 4605, normalized size = 17.78 \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. 1129 vs.
\(2 (261) = 522\).
time = 0.79, size = 1129, normalized size = 4.36 \begin {gather*} -\frac {8 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} + 6 \, a c^{2} d^{2} e^{2} - b^{3} d e^{3} - 6 \, a b c d e^{3} + a b^{2} e^{4} + a^{2} c e^{4}\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^{6} d^{4} x^{5} - 120 \, b c^{5} d^{3} x^{5} e + 150 \, b c^{5} d^{4} x^{4} + 72 \, b^{2} c^{4} d^{2} x^{5} e^{2} + 72 \, a c^{5} d^{2} x^{5} e^{2} - 300 \, b^{2} c^{4} d^{3} x^{4} e + 110 \, b^{2} c^{4} d^{4} x^{3} + 160 \, a c^{5} d^{4} x^{3} - 12 \, b^{3} c^{3} d x^{5} e^{3} - 72 \, a b c^{4} d x^{5} e^{3} + 180 \, b^{3} c^{3} d^{2} x^{4} e^{2} + 180 \, a b c^{4} d^{2} x^{4} e^{2} - 220 \, b^{3} c^{3} d^{3} x^{3} e - 320 \, a b c^{4} d^{3} x^{3} e + 15 \, b^{3} c^{3} d^{4} x^{2} + 240 \, a b c^{4} d^{4} x^{2} + 12 \, a b^{2} c^{3} x^{5} e^{4} + 12 \, a^{2} c^{4} x^{5} e^{4} - 30 \, b^{4} c^{2} d x^{4} e^{3} - 180 \, a b^{2} c^{3} d x^{4} e^{3} + 132 \, b^{4} c^{2} d^{2} x^{3} e^{2} + 324 \, a b^{2} c^{3} d^{2} x^{3} e^{2} + 192 \, a^{2} c^{4} d^{2} x^{3} e^{2} - 30 \, b^{4} c^{2} d^{3} x^{2} e - 480 \, a b^{2} c^{3} d^{3} x^{2} e - 3 \, b^{4} c^{2} d^{4} x + 54 \, a b^{2} c^{3} d^{4} x + 132 \, a^{2} c^{4} d^{4} x + 30 \, a b^{3} c^{2} x^{4} e^{4} + 30 \, a^{2} b c^{3} x^{4} e^{4} - 22 \, b^{5} c d x^{3} e^{3} - 164 \, a b^{3} c^{2} d x^{3} e^{3} - 192 \, a^{2} b c^{3} d x^{3} e^{3} + 18 \, b^{5} c d^{2} x^{2} e^{2} + 306 \, a b^{3} c^{2} d^{2} x^{2} e^{2} + 288 \, a^{2} b c^{3} d^{2} x^{2} e^{2} + 6 \, b^{5} c d^{3} x e - 108 \, a b^{3} c^{2} d^{3} x e - 264 \, a^{2} b c^{3} d^{3} x e + b^{5} c d^{4} - 13 \, a b^{3} c^{2} d^{4} + 66 \, a^{2} b c^{3} d^{4} + b^{6} x^{3} e^{4} + 10 \, a b^{4} c x^{3} e^{4} + 102 \, a^{2} b^{2} c^{2} x^{3} e^{4} - 32 \, a^{3} c^{3} x^{3} e^{4} - 102 \, a b^{4} c d x^{2} e^{3} - 144 \, a^{2} b^{2} c^{2} d x^{2} e^{3} - 192 \, a^{3} c^{3} d x^{2} e^{3} + 18 \, a b^{4} c d^{2} x e^{2} + 396 \, a^{2} b^{2} c^{2} d^{2} x e^{2} - 72 \, a^{3} c^{3} d^{2} x e^{2} + 2 \, a b^{4} c d^{3} e - 36 \, a^{2} b^{2} c^{2} d^{3} e - 128 \, a^{3} c^{3} d^{3} e + 3 \, a b^{5} x^{2} e^{4} + 51 \, a^{2} b^{3} c x^{2} e^{4} + 48 \, a^{3} b c^{2} x^{2} e^{4} - 120 \, a^{2} b^{3} c d x e^{3} - 120 \, a^{3} b c^{2} d x e^{3} + 6 \, a^{2} b^{3} c d^{2} e^{2} + 156 \, a^{3} b c^{2} d^{2} e^{2} + 3 \, a^{2} b^{4} x e^{4} + 66 \, a^{3} b^{2} c x e^{4} - 12 \, a^{4} c^{2} x e^{4} - 44 \, a^{3} b^{2} c d e^{3} - 64 \, a^{4} c^{2} d e^{3} + a^{3} b^{3} e^{4} + 26 \, a^{4} b c e^{4}}{3 \, {\left (b^{6} c - 12 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\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 = 1.81, size = 1463, normalized size = 5.65 \begin {gather*} -\frac {\frac {26\,a^4\,b\,c\,e^4-64\,a^4\,c^2\,d\,e^3+a^3\,b^3\,e^4-44\,a^3\,b^2\,c\,d\,e^3+156\,a^3\,b\,c^2\,d^2\,e^2-128\,a^3\,c^3\,d^3\,e+6\,a^2\,b^3\,c\,d^2\,e^2-36\,a^2\,b^2\,c^2\,d^3\,e+66\,a^2\,b\,c^3\,d^4+2\,a\,b^4\,c\,d^3\,e-13\,a\,b^3\,c^2\,d^4+b^5\,c\,d^4}{3\,c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^3\,\left (-32\,a^3\,c^3\,e^4+102\,a^2\,b^2\,c^2\,e^4-192\,a^2\,b\,c^3\,d\,e^3+192\,a^2\,c^4\,d^2\,e^2+10\,a\,b^4\,c\,e^4-164\,a\,b^3\,c^2\,d\,e^3+324\,a\,b^2\,c^3\,d^2\,e^2-320\,a\,b\,c^4\,d^3\,e+160\,a\,c^5\,d^4+b^6\,e^4-22\,b^5\,c\,d\,e^3+132\,b^4\,c^2\,d^2\,e^2-220\,b^3\,c^3\,d^3\,e+110\,b^2\,c^4\,d^4\right )}{3\,c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {4\,c^2\,x^5\,\left (a^2\,c\,e^4+a\,b^2\,e^4-6\,a\,b\,c\,d\,e^3+6\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+6\,b^2\,c\,d^2\,e^2-10\,b\,c^2\,d^3\,e+5\,c^3\,d^4\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {x^2\,\left (16\,a^3\,b\,c^2\,e^4-64\,a^3\,c^3\,d\,e^3+17\,a^2\,b^3\,c\,e^4-48\,a^2\,b^2\,c^2\,d\,e^3+96\,a^2\,b\,c^3\,d^2\,e^2+a\,b^5\,e^4-34\,a\,b^4\,c\,d\,e^3+102\,a\,b^3\,c^2\,d^2\,e^2-160\,a\,b^2\,c^3\,d^3\,e+80\,a\,b\,c^4\,d^4+6\,b^5\,c\,d^2\,e^2-10\,b^4\,c^2\,d^3\,e+5\,b^3\,c^3\,d^4\right )}{c\,\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^4\,c^2\,e^4+22\,a^3\,b^2\,c\,e^4-40\,a^3\,b\,c^2\,d\,e^3-24\,a^3\,c^3\,d^2\,e^2+a^2\,b^4\,e^4-40\,a^2\,b^3\,c\,d\,e^3+132\,a^2\,b^2\,c^2\,d^2\,e^2-88\,a^2\,b\,c^3\,d^3\,e+44\,a^2\,c^4\,d^4+6\,a\,b^4\,c\,d^2\,e^2-36\,a\,b^3\,c^2\,d^3\,e+18\,a\,b^2\,c^3\,d^4+2\,b^5\,c\,d^3\,e-b^4\,c^2\,d^4\right )}{c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {10\,b\,c\,x^4\,\left (a^2\,c\,e^4+a\,b^2\,e^4-6\,a\,b\,c\,d\,e^3+6\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+6\,b^2\,c\,d^2\,e^2-10\,b\,c^2\,d^3\,e+5\,c^3\,d^4\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\,\mathrm {atan}\left (\frac {\left (\frac {4\,\left (c\,d^2-b\,d\,e+a\,e^2\right )\,\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 )}+\frac {8\,c\,x\,\left (c\,d^2-b\,d\,e+a\,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}}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4}\right )\,\left (c\,d^2-b\,d\,e+a\,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.
[In]
[Out]
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