Optimal. Leaf size=388 \[ -\frac {e^2 (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{c^2 \left (b^2-4 a c\right )^2}-\frac {(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {e^5 \log \left (a+b x+c x^2\right )}{2 c^3} \]
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Rubi [A]
time = 0.70, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {752, 832, 787,
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 )}{c^3 \left (b^2-4 a c\right )^{5/2}}-\frac {e^2 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 )}{c^2 \left (b^2-4 a c\right )^2}-\frac {(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {e^5 \log \left (a+b x+c x^2\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 752
Rule 787
Rule 832
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {(d+e x)^3 \left (6 c d^2-e (7 b d-8 a e)-e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac {(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {(d+e x) \left (-2 \left (6 c^3 d^4-a b^2 e^4-c^2 d^2 e (15 b d-14 a e)+c e^2 \left (10 b^2 d^2-21 a b d e+16 a^2 e^2\right )\right )+2 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\right )}{a+b x+c x^2} \, dx}{2 c \left (b^2-4 a c\right )^2}\\ &=-\frac {e^2 (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}{c^2 \left (b^2-4 a c\right )^2}-\frac {(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\int \frac {-2 a e^2 (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right )-2 c d \left (6 c^3 d^4-a b^2 e^4-c^2 d^2 e (15 b d-14 a e)+c e^2 \left (10 b^2 d^2-21 a b d e+16 a^2 e^2\right )\right )+\left (2 c d 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 )-2 b e^2 (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right )-2 c e \left (6 c^3 d^4-a b^2 e^4-c^2 d^2 e (15 b d-14 a e)+c e^2 \left (10 b^2 d^2-21 a b d e+16 a^2 e^2\right )\right )\right ) x}{a+b x+c x^2} \, dx}{2 c^2 \left (b^2-4 a c\right )^2}\\ &=-\frac {e^2 (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}{c^2 \left (b^2-4 a c\right )^2}-\frac {(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {e^5 \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^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 c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac {e^2 (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}{c^2 \left (b^2-4 a c\right )^2}-\frac {(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {e^5 \log \left (a+b x+c x^2\right )}{2 c^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 )}{c^3 \left (b^2-4 a c\right )^2}\\ &=-\frac {e^2 (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}{c^2 \left (b^2-4 a c\right )^2}-\frac {(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\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 )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {e^5 \log \left (a+b x+c x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 628, normalized size = 1.62 \begin {gather*} \frac {\frac {b^5 e^5 x+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left (-2 c d^2 x+a e (d+e x)\right )-2 b^2 c e^2 \left (2 a^2 e^3+5 c^2 d^3 x-5 a c d e (d+2 e x)\right )+2 c^2 \left (a^3 e^5-c^3 d^5 x-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)\right )+b c^2 \left (-c^2 d^4 (d-5 e x)+5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {-b^6 e^5+b^5 c e^4 (5 d+4 e x)+b^4 c e^3 \left (11 a e^2-10 c d (d+e x)\right )+10 b^3 c^2 e^2 \left (c d^3-a e^2 (4 d+3 e x)\right )+4 c^3 \left (8 a^3 e^5+3 c^3 d^5 x+10 a c^2 d^3 e^2 x-5 a^2 c d e^3 (8 d+5 e x)\right )+2 b c^3 \left (3 c^2 d^4 (d-5 e x)+10 a c d^2 e^2 (d-3 e x)+5 a^2 e^4 (11 d+5 e x)\right )+b^2 c^2 e \left (-39 a^2 e^4-5 c^2 d^3 (3 d-4 e x)+10 a c d e^2 (5 d+8 e x)\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {2 c (2 c d-b e) \left (6 c^4 d^4+b^4 e^4+2 b^2 c e^3 (b d-5 a e)-4 c^3 d^2 e (3 b d-5 a e)+2 c^2 e^2 \left (2 b^2 d^2-10 a b d e+15 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}}+c e^5 \log (a+x (b+c x))}{2 c^4} \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. \(973\) vs.
\(2(378)=756\).
time = 1.03, size = 974, normalized size = 2.51
method | result | size |
default | \(\frac {\frac {\left (25 a^{2} b \,c^{2} e^{5}-50 a^{2} c^{3} d \,e^{4}-15 a \,b^{3} c \,e^{5}+40 a \,b^{2} c^{2} d \,e^{4}-30 a b \,c^{3} d^{2} e^{3}+20 a \,c^{4} d^{3} e^{2}+2 b^{5} e^{5}-5 b^{4} c d \,e^{4}+10 b^{2} c^{3} d^{3} e^{2}-15 b \,c^{4} d^{4} e +6 c^{5} d^{5}\right ) x^{3}}{c^{2} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}+\frac {\left (32 a^{3} c^{3} e^{5}+11 a^{2} b^{2} c^{2} e^{5}+10 a^{2} b \,c^{3} d \,e^{4}-160 a^{2} c^{4} d^{2} e^{3}-19 a \,b^{4} c \,e^{5}+40 a \,b^{3} c^{2} d \,e^{4}-10 a \,b^{2} c^{3} d^{2} e^{3}+60 a b \,c^{4} d^{3} e^{2}+3 b^{6} e^{5}-5 b^{5} c d \,e^{4}-10 b^{4} c^{2} d^{2} e^{3}+30 b^{3} c^{3} d^{3} e^{2}-45 b^{2} c^{4} d^{4} e +18 b \,c^{5} d^{5}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) c^{3}}+\frac {\left (31 a^{3} b \,c^{2} e^{5}-30 a^{3} c^{3} d \,e^{4}-22 a^{2} b^{3} c \,e^{5}+50 a^{2} b^{2} c^{2} d \,e^{4}-50 a^{2} b \,c^{3} d^{2} e^{3}-20 a^{2} c^{4} d^{3} e^{2}+3 a \,b^{5} e^{5}-5 a \,b^{4} c d \,e^{4}-10 a \,b^{3} c^{2} d^{2} e^{3}+50 a \,b^{2} c^{3} d^{3} e^{2}-25 a b \,c^{4} d^{4} e +10 a \,c^{5} d^{5}-5 b^{3} c^{3} d^{4} e +2 c^{4} b^{2} d^{5}\right ) x}{\left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right ) c^{3}}+\frac {24 a^{4} c^{2} e^{5}-21 a^{3} b^{2} c \,e^{5}+50 a^{3} b \,c^{2} d \,e^{4}-80 a^{3} c^{3} d^{2} e^{3}+3 a^{2} b^{4} e^{5}-5 a^{2} b^{3} c d \,e^{4}-10 a^{2} b^{2} c^{2} d^{2} e^{3}+60 a^{2} b \,c^{3} d^{3} e^{2}-40 a^{2} c^{4} d^{4} e -5 a \,b^{2} c^{3} d^{4} e +10 a b \,c^{4} d^{5}-b^{3} c^{3} d^{5}}{2 c^{3} \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^{2} e^{5}-8 a \,b^{2} e^{5} c +b^{4} e^{5}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-7 a^{2} b \,e^{5} c +30 d \,e^{4} a^{2} c^{2}+a \,b^{3} e^{5}-30 a b \,c^{2} d^{2} e^{3}+20 a \,c^{3} d^{3} e^{2}+10 b^{2} c^{2} d^{3} e^{2}-15 b \,c^{3} d^{4} e +6 c^{4} d^{5}-\frac {\left (16 a^{2} c^{2} e^{5}-8 a \,b^{2} e^{5} c +b^{4} 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}}}}{c^{2} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}\) | \(974\) |
risch | \(\text {Expression too large to display}\) | \(5551\) |
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. 1884 vs.
\(2 (381) = 762\).
time = 3.67, size = 3788, normalized size = 9.76 \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. 805 vs.
\(2 (381) = 762\).
time = 0.62, size = 805, normalized size = 2.07 \begin {gather*} \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} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {e^{5} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac {b^{3} c^{3} d^{5} - 10 \, a b c^{4} d^{5} + 5 \, a b^{2} c^{3} d^{4} e + 40 \, a^{2} c^{4} d^{4} e - 60 \, a^{2} b c^{3} d^{3} e^{2} + 10 \, a^{2} b^{2} c^{2} d^{2} e^{3} + 80 \, a^{3} c^{3} d^{2} e^{3} + 5 \, a^{2} b^{3} c d e^{4} - 50 \, 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} - 5 \, b^{4} c^{2} d e^{4} + 40 \, a b^{2} c^{3} d e^{4} - 50 \, a^{2} c^{4} d e^{4} + 2 \, b^{5} c e^{5} - 15 \, a b^{3} c^{2} e^{5} + 25 \, 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} - 10 \, b^{4} c^{2} d^{2} e^{3} - 10 \, a b^{2} c^{3} d^{2} e^{3} - 160 \, a^{2} c^{4} d^{2} e^{3} - 5 \, b^{5} c d e^{4} + 40 \, a b^{3} c^{2} d e^{4} + 10 \, a^{2} b c^{3} d e^{4} + 3 \, b^{6} e^{5} - 19 \, a b^{4} c e^{5} + 11 \, a^{2} b^{2} c^{2} e^{5} + 32 \, 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 + 50 \, a b^{2} c^{3} d^{3} e^{2} - 20 \, a^{2} c^{4} d^{3} e^{2} - 10 \, a b^{3} c^{2} d^{2} e^{3} - 50 \, a^{2} b c^{3} d^{2} e^{3} - 5 \, a b^{4} c d e^{4} + 50 \, a^{2} b^{2} c^{2} d e^{4} - 30 \, a^{3} c^{3} d e^{4} + 3 \, a b^{5} e^{5} - 22 \, a^{2} b^{3} c e^{5} + 31 \, a^{3} b c^{2} e^{5}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.03, size = 1486, normalized size = 3.83 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\left (\frac {x\,\left (b\,e-2\,c\,d\right )\,\left (30\,a^2\,c^2\,e^4-10\,a\,b^2\,c\,e^4-20\,a\,b\,c^2\,d\,e^3+20\,a\,c^3\,d^2\,e^2+b^4\,e^4+2\,b^3\,c\,d\,e^3+4\,b^2\,c^2\,d^2\,e^2-12\,b\,c^3\,d^3\,e+6\,c^4\,d^4\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^5}+\frac {\left (b\,e-2\,c\,d\right )\,\left (16\,a^2\,b\,c^4-8\,a\,b^3\,c^3+b^5\,c^2\right )\,\left (30\,a^2\,c^2\,e^4-10\,a\,b^2\,c\,e^4-20\,a\,b\,c^2\,d\,e^3+20\,a\,c^3\,d^2\,e^2+b^4\,e^4+2\,b^3\,c\,d\,e^3+4\,b^2\,c^2\,d^2\,e^2-12\,b\,c^3\,d^3\,e+6\,c^4\,d^4\right )}{2\,c^5\,{\left (4\,a\,c-b^2\right )}^5\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (32\,a^2\,c^5\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,b^4\,c^3\,{\left (4\,a\,c-b^2\right )}^{5/2}-16\,a\,b^2\,c^4\,{\left (4\,a\,c-b^2\right )}^{5/2}\right )}{-30\,a^2\,b\,c^2\,e^5+60\,a^2\,c^3\,d\,e^4+10\,a\,b^3\,c\,e^5-60\,a\,b\,c^3\,d^2\,e^3+40\,a\,c^4\,d^3\,e^2-b^5\,e^5+20\,b^2\,c^3\,d^3\,e^2-30\,b\,c^4\,d^4\,e+12\,c^5\,d^5}\right )\,\left (b\,e-2\,c\,d\right )\,\left (30\,a^2\,c^2\,e^4-10\,a\,b^2\,c\,e^4-20\,a\,b\,c^2\,d\,e^3+20\,a\,c^3\,d^2\,e^2+b^4\,e^4+2\,b^3\,c\,d\,e^3+4\,b^2\,c^2\,d^2\,e^2-12\,b\,c^3\,d^3\,e+6\,c^4\,d^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-1024\,a^5\,c^5\,e^5+1280\,a^4\,b^2\,c^4\,e^5-640\,a^3\,b^4\,c^3\,e^5+160\,a^2\,b^6\,c^2\,e^5-20\,a\,b^8\,c\,e^5+b^{10}\,e^5\right )}{2\,\left (1024\,a^5\,c^8-1280\,a^4\,b^2\,c^7+640\,a^3\,b^4\,c^6-160\,a^2\,b^6\,c^5+20\,a\,b^8\,c^4-b^{10}\,c^3\right )}-\frac {\frac {-24\,a^4\,c^2\,e^5+21\,a^3\,b^2\,c\,e^5-50\,a^3\,b\,c^2\,d\,e^4+80\,a^3\,c^3\,d^2\,e^3-3\,a^2\,b^4\,e^5+5\,a^2\,b^3\,c\,d\,e^4+10\,a^2\,b^2\,c^2\,d^2\,e^3-60\,a^2\,b\,c^3\,d^3\,e^2+40\,a^2\,c^4\,d^4\,e+5\,a\,b^2\,c^3\,d^4\,e-10\,a\,b\,c^4\,d^5+b^3\,c^3\,d^5}{2\,c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x\,\left (-31\,a^3\,b\,c^2\,e^5+30\,a^3\,c^3\,d\,e^4+22\,a^2\,b^3\,c\,e^5-50\,a^2\,b^2\,c^2\,d\,e^4+50\,a^2\,b\,c^3\,d^2\,e^3+20\,a^2\,c^4\,d^3\,e^2-3\,a\,b^5\,e^5+5\,a\,b^4\,c\,d\,e^4+10\,a\,b^3\,c^2\,d^2\,e^3-50\,a\,b^2\,c^3\,d^3\,e^2+25\,a\,b\,c^4\,d^4\,e-10\,a\,c^5\,d^5+5\,b^3\,c^3\,d^4\,e-2\,b^2\,c^4\,d^5\right )}{c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {x^2\,\left (32\,a^3\,c^3\,e^5+11\,a^2\,b^2\,c^2\,e^5+10\,a^2\,b\,c^3\,d\,e^4-160\,a^2\,c^4\,d^2\,e^3-19\,a\,b^4\,c\,e^5+40\,a\,b^3\,c^2\,d\,e^4-10\,a\,b^2\,c^3\,d^2\,e^3+60\,a\,b\,c^4\,d^3\,e^2+3\,b^6\,e^5-5\,b^5\,c\,d\,e^4-10\,b^4\,c^2\,d^2\,e^3+30\,b^3\,c^3\,d^3\,e^2-45\,b^2\,c^4\,d^4\,e+18\,b\,c^5\,d^5\right )}{2\,c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {x^3\,\left (25\,a^2\,b\,c^2\,e^5-50\,a^2\,c^3\,d\,e^4-15\,a\,b^3\,c\,e^5+40\,a\,b^2\,c^2\,d\,e^4-30\,a\,b\,c^3\,d^2\,e^3+20\,a\,c^4\,d^3\,e^2+2\,b^5\,e^5-5\,b^4\,c\,d\,e^4+10\,b^2\,c^3\,d^3\,e^2-15\,b\,c^4\,d^4\,e+6\,c^5\,d^5\right )}{c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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