Optimal. Leaf size=259 \[ \frac {(2 c d-b e) n}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {\sqrt {b^2-4 a c} (2 c d-b e) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log (d+e x)}{2 e \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log \left (a+b x+c x^2\right )}{4 e \left (c d^2-b d e+a e^2\right )^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2} \]
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Rubi [A]
time = 0.25, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2605, 814, 648,
632, 212, 642} \begin {gather*} \frac {n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{4 e \left (a e^2-b d e+c d^2\right )^2}-\frac {n \log (d+e x) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{2 e \left (a e^2-b d e+c d^2\right )^2}+\frac {n \sqrt {b^2-4 a c} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 \left (a e^2-b d e+c d^2\right )^2}+\frac {n (2 c d-b e)}{2 e (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 814
Rule 2605
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^3} \, dx &=-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}+\frac {n \int \frac {b+2 c x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx}{2 e}\\ &=-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}+\frac {n \int \left (\frac {e (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {e \left (-2 c^2 d^2-b^2 e^2+2 c e (b d+a e)\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {-2 b^2 c d e+4 a c^2 d e+b^3 e^2+b c \left (c d^2-3 a e^2\right )+c \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}\right ) \, dx}{2 e}\\ &=\frac {(2 c d-b e) n}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log (d+e x)}{2 e \left (c d^2-b d e+a e^2\right )^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}+\frac {n \int \frac {-2 b^2 c d e+4 a c^2 d e+b^3 e^2+b c \left (c d^2-3 a e^2\right )+c \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x}{a+b x+c x^2} \, dx}{2 e \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {(2 c d-b e) n}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log (d+e x)}{2 e \left (c d^2-b d e+a e^2\right )^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}-\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{4 \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{4 e \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {(2 c d-b e) n}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log (d+e x)}{2 e \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log \left (a+b x+c x^2\right )}{4 e \left (c d^2-b d e+a e^2\right )^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}+\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) n\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {(2 c d-b e) n}{2 e \left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {\sqrt {b^2-4 a c} (2 c d-b e) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log (d+e x)}{2 e \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \log \left (a+b x+c x^2\right )}{4 e \left (c d^2-b d e+a e^2\right )^2}-\frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 215, normalized size = 0.83 \begin {gather*} \frac {\frac {n (d+e x) \left (2 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )-2 \sqrt {b^2-4 a c} e (-2 c d+b e) (d+e x) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) (d+e x) \log (d+e x)+\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) (d+e x) \log (a+x (b+c x))\right )}{\left (c d^2+e (-b d+a e)\right )^2}-2 \log \left (d (a+x (b+c x))^n\right )}{4 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.18, size = 14679, normalized size = 56.68
method | result | size |
risch | \(\text {Expression too large to display}\) | \(14679\) |
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. 658 vs.
\(2 (248) = 496\).
time = 1.34, size = 1337, normalized size = 5.16 \begin {gather*} \left [\frac {4 \, c^{2} d^{4} n - 2 \, a b n x e^{4} - {\left (2 \, c d^{3} n e - b n x^{2} e^{4} + 2 \, {\left (c d n x^{2} - b d n x\right )} e^{3} + {\left (4 \, c d^{2} n x - b d^{2} n\right )} e^{2}\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 ) - 2 \, {\left (a b d n - {\left (b^{2} + 2 \, a c\right )} d n x\right )} e^{3} - 2 \, {\left (3 \, b c d^{2} n x - {\left (b^{2} + 2 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x - 3 \, b c d^{3} n\right )} e + {\left ({\left ({\left (b^{2} - 2 \, a c\right )} n x^{2} - 2 \, a^{2} n\right )} e^{4} - 2 \, {\left (b c d n x^{2} - 2 \, a b d n - {\left (b^{2} - 2 \, a c\right )} d n x\right )} e^{3} + {\left (2 \, c^{2} d^{2} n x^{2} - 4 \, b c d^{2} n x - {\left (b^{2} + 6 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x + b c d^{3} n\right )} e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (2 \, c^{2} d^{4} n + {\left (b^{2} - 2 \, a c\right )} n x^{2} e^{4} - 2 \, {\left (b c d n x^{2} - {\left (b^{2} - 2 \, a c\right )} d n x\right )} e^{3} + {\left (2 \, c^{2} d^{2} n x^{2} - 4 \, b c d^{2} n x + {\left (b^{2} - 2 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x - b c d^{3} n\right )} e\right )} \log \left (x e + d\right ) - 2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (d\right )}{4 \, {\left (c^{2} d^{6} e + a^{2} x^{2} e^{7} - 2 \, {\left (a b d x^{2} - a^{2} d x\right )} e^{6} - {\left (4 \, a b d^{2} x - {\left (b^{2} + 2 \, a c\right )} d^{2} x^{2} - a^{2} d^{2}\right )} e^{5} - 2 \, {\left (b c d^{3} x^{2} + a b d^{3} - {\left (b^{2} + 2 \, a c\right )} d^{3} x\right )} e^{4} + {\left (c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} e^{3} + 2 \, {\left (c^{2} d^{5} x - b c d^{5}\right )} e^{2}\right )}}, \frac {4 \, c^{2} d^{4} n - 2 \, a b n x e^{4} + 2 \, {\left (2 \, c d^{3} n e - b n x^{2} e^{4} + 2 \, {\left (c d n x^{2} - b d n x\right )} e^{3} + {\left (4 \, c d^{2} n x - b d^{2} n\right )} e^{2}\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 ) - 2 \, {\left (a b d n - {\left (b^{2} + 2 \, a c\right )} d n x\right )} e^{3} - 2 \, {\left (3 \, b c d^{2} n x - {\left (b^{2} + 2 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x - 3 \, b c d^{3} n\right )} e + {\left ({\left ({\left (b^{2} - 2 \, a c\right )} n x^{2} - 2 \, a^{2} n\right )} e^{4} - 2 \, {\left (b c d n x^{2} - 2 \, a b d n - {\left (b^{2} - 2 \, a c\right )} d n x\right )} e^{3} + {\left (2 \, c^{2} d^{2} n x^{2} - 4 \, b c d^{2} n x - {\left (b^{2} + 6 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x + b c d^{3} n\right )} e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (2 \, c^{2} d^{4} n + {\left (b^{2} - 2 \, a c\right )} n x^{2} e^{4} - 2 \, {\left (b c d n x^{2} - {\left (b^{2} - 2 \, a c\right )} d n x\right )} e^{3} + {\left (2 \, c^{2} d^{2} n x^{2} - 4 \, b c d^{2} n x + {\left (b^{2} - 2 \, a c\right )} d^{2} n\right )} e^{2} + 2 \, {\left (2 \, c^{2} d^{3} n x - b c d^{3} n\right )} e\right )} \log \left (x e + d\right ) - 2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (d\right )}{4 \, {\left (c^{2} d^{6} e + a^{2} x^{2} e^{7} - 2 \, {\left (a b d x^{2} - a^{2} d x\right )} e^{6} - {\left (4 \, a b d^{2} x - {\left (b^{2} + 2 \, a c\right )} d^{2} x^{2} - a^{2} d^{2}\right )} e^{5} - 2 \, {\left (b c d^{3} x^{2} + a b d^{3} - {\left (b^{2} + 2 \, a c\right )} d^{3} x\right )} e^{4} + {\left (c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} e^{3} + 2 \, {\left (c^{2} d^{5} x - b c d^{5}\right )} e^{2}\right )}}\right ] \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. 887 vs.
\(2 (248) = 496\).
time = 3.35, size = 887, normalized size = 3.42 \begin {gather*} \frac {{\left (2 \, c^{2} d^{2} n - 2 \, b c d n e + b^{2} n e^{2} - 2 \, a c n e^{2}\right )} \log \left (c x^{2} + b x + a\right )}{4 \, {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3} + 2 \, a c d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )}} - \frac {{\left (2 \, b^{2} c d n - 8 \, a c^{2} d n - b^{3} n e + 4 \, a b c n e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c^{2} d^{2} n x^{2} e^{2} \log \left (x e + d\right ) + 4 \, c^{2} d^{3} n x e \log \left (x e + d\right ) - 2 \, c^{2} d^{3} n x e + c^{2} d^{4} n \log \left (c x^{2} + b x + a\right ) - 2 \, b c d^{3} n e \log \left (c x^{2} + b x + a\right ) + 2 \, c^{2} d^{4} n \log \left (x e + d\right ) - 2 \, b c d n x^{2} e^{3} \log \left (x e + d\right ) - 4 \, b c d^{2} n x e^{2} \log \left (x e + d\right ) - 2 \, b c d^{3} n e \log \left (x e + d\right ) - 2 \, c^{2} d^{4} n + 3 \, b c d^{2} n x e^{2} + 3 \, b c d^{3} n e + b^{2} d^{2} n e^{2} \log \left (c x^{2} + b x + a\right ) + 2 \, a c d^{2} n e^{2} \log \left (c x^{2} + b x + a\right ) + b^{2} n x^{2} e^{4} \log \left (x e + d\right ) - 2 \, a c n x^{2} e^{4} \log \left (x e + d\right ) + 2 \, b^{2} d n x e^{3} \log \left (x e + d\right ) - 4 \, a c d n x e^{3} \log \left (x e + d\right ) + b^{2} d^{2} n e^{2} \log \left (x e + d\right ) - 2 \, a c d^{2} n e^{2} \log \left (x e + d\right ) + c^{2} d^{4} \log \left (d\right ) - 2 \, b c d^{3} e \log \left (d\right ) - b^{2} d n x e^{3} - 2 \, a c d n x e^{3} - b^{2} d^{2} n e^{2} - 2 \, a c d^{2} n e^{2} - 2 \, a b d n e^{3} \log \left (c x^{2} + b x + a\right ) + b^{2} d^{2} e^{2} \log \left (d\right ) + 2 \, a c d^{2} e^{2} \log \left (d\right ) + a b n x e^{4} + a b d n e^{3} + a^{2} n e^{4} \log \left (c x^{2} + b x + a\right ) - 2 \, a b d e^{3} \log \left (d\right ) + a^{2} e^{4} \log \left (d\right )}{2 \, {\left (c^{2} d^{4} x^{2} e^{3} + 2 \, c^{2} d^{5} x e^{2} + c^{2} d^{6} e - 2 \, b c d^{3} x^{2} e^{4} - 4 \, b c d^{4} x e^{3} - 2 \, b c d^{5} e^{2} + b^{2} d^{2} x^{2} e^{5} + 2 \, a c d^{2} x^{2} e^{5} + 2 \, b^{2} d^{3} x e^{4} + 4 \, a c d^{3} x e^{4} + b^{2} d^{4} e^{3} + 2 \, a c d^{4} e^{3} - 2 \, a b d x^{2} e^{6} - 4 \, a b d^{2} x e^{5} - 2 \, a b d^{3} e^{4} + a^{2} x^{2} e^{7} + 2 \, a^{2} d x e^{6} + a^{2} d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.74, size = 1715, normalized size = 6.62 \begin {gather*} \frac {\ln \left (3\,b^2\,c^3\,d^4-12\,a\,c^4\,d^4-2\,b^5\,e^4\,x-12\,a^3\,c^2\,e^4-2\,a\,b^4\,e^4+2\,b^4\,e^4\,x\,\sqrt {b^2-4\,a\,c}+6\,c^4\,d^4\,x\,\sqrt {b^2-4\,a\,c}+11\,a^2\,b^2\,c\,e^4-2\,b^3\,c^2\,d^3\,e+b^4\,c\,d^2\,e^2+40\,a^2\,c^3\,d^2\,e^2+2\,a\,b^3\,e^4\,\sqrt {b^2-4\,a\,c}+3\,b\,c^3\,d^4\,\sqrt {b^2-4\,a\,c}+8\,a\,b\,c^3\,d^3\,e+6\,a\,b^3\,c\,d\,e^3+12\,a\,b^3\,c\,e^4\,x-32\,a\,c^4\,d^3\,e\,x+8\,b^4\,c\,d\,e^3\,x-5\,a^2\,b\,c\,e^4\,\sqrt {b^2-4\,a\,c}-16\,a\,c^3\,d^3\,e\,\sqrt {b^2-4\,a\,c}-24\,a^2\,b\,c^2\,d\,e^3-16\,a^2\,b\,c^2\,e^4\,x+32\,a^2\,c^3\,d\,e^3\,x+8\,b^2\,c^3\,d^3\,e\,x+16\,a^2\,c^2\,d\,e^3\,\sqrt {b^2-4\,a\,c}-2\,b^2\,c^2\,d^3\,e\,\sqrt {b^2-4\,a\,c}+b^3\,c\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}+6\,a^2\,c^2\,e^4\,x\,\sqrt {b^2-4\,a\,c}-14\,a\,b^2\,c^2\,d^2\,e^2-12\,b^3\,c^2\,d^2\,e^2\,x+14\,a\,b\,c^2\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}-20\,a\,c^3\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}+14\,b^2\,c^2\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}-10\,a\,b^2\,c\,d\,e^3\,\sqrt {b^2-4\,a\,c}-8\,a\,b^2\,c\,e^4\,x\,\sqrt {b^2-4\,a\,c}-12\,b\,c^3\,d^3\,e\,x\,\sqrt {b^2-4\,a\,c}-8\,b^3\,c\,d\,e^3\,x\,\sqrt {b^2-4\,a\,c}+48\,a\,b\,c^3\,d^2\,e^2\,x-40\,a\,b^2\,c^2\,d\,e^3\,x+20\,a\,b\,c^2\,d\,e^3\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (e\,\left (\frac {c\,d\,n\,\sqrt {b^2-4\,a\,c}}{2}-\frac {b\,c\,d\,n}{2}\right )-e^2\,\left (\frac {a\,c\,n}{2}-\frac {b^2\,n}{4}+\frac {b\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )+\frac {c^2\,d^2\,n}{2}\right )}{a^2\,e^5-2\,a\,b\,d\,e^4+2\,a\,c\,d^2\,e^3+b^2\,d^2\,e^3-2\,b\,c\,d^3\,e^2+c^2\,d^4\,e}-\frac {\ln \left (d+e\,x\right )\,\left (e^2\,\left (b^2\,n-2\,a\,c\,n\right )+2\,c^2\,d^2\,n-2\,b\,c\,d\,e\,n\right )}{2\,a^2\,e^5-4\,a\,b\,d\,e^4+4\,a\,c\,d^2\,e^3+2\,b^2\,d^2\,e^3-4\,b\,c\,d^3\,e^2+2\,c^2\,d^4\,e}+\frac {\ln \left (2\,a\,b^4\,e^4+12\,a\,c^4\,d^4+2\,b^5\,e^4\,x+12\,a^3\,c^2\,e^4-3\,b^2\,c^3\,d^4+2\,b^4\,e^4\,x\,\sqrt {b^2-4\,a\,c}+6\,c^4\,d^4\,x\,\sqrt {b^2-4\,a\,c}-11\,a^2\,b^2\,c\,e^4+2\,b^3\,c^2\,d^3\,e-b^4\,c\,d^2\,e^2-40\,a^2\,c^3\,d^2\,e^2+2\,a\,b^3\,e^4\,\sqrt {b^2-4\,a\,c}+3\,b\,c^3\,d^4\,\sqrt {b^2-4\,a\,c}-8\,a\,b\,c^3\,d^3\,e-6\,a\,b^3\,c\,d\,e^3-12\,a\,b^3\,c\,e^4\,x+32\,a\,c^4\,d^3\,e\,x-8\,b^4\,c\,d\,e^3\,x-5\,a^2\,b\,c\,e^4\,\sqrt {b^2-4\,a\,c}-16\,a\,c^3\,d^3\,e\,\sqrt {b^2-4\,a\,c}+24\,a^2\,b\,c^2\,d\,e^3+16\,a^2\,b\,c^2\,e^4\,x-32\,a^2\,c^3\,d\,e^3\,x-8\,b^2\,c^3\,d^3\,e\,x+16\,a^2\,c^2\,d\,e^3\,\sqrt {b^2-4\,a\,c}-2\,b^2\,c^2\,d^3\,e\,\sqrt {b^2-4\,a\,c}+b^3\,c\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}+6\,a^2\,c^2\,e^4\,x\,\sqrt {b^2-4\,a\,c}+14\,a\,b^2\,c^2\,d^2\,e^2+12\,b^3\,c^2\,d^2\,e^2\,x+14\,a\,b\,c^2\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}-20\,a\,c^3\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}+14\,b^2\,c^2\,d^2\,e^2\,x\,\sqrt {b^2-4\,a\,c}-10\,a\,b^2\,c\,d\,e^3\,\sqrt {b^2-4\,a\,c}-8\,a\,b^2\,c\,e^4\,x\,\sqrt {b^2-4\,a\,c}-12\,b\,c^3\,d^3\,e\,x\,\sqrt {b^2-4\,a\,c}-8\,b^3\,c\,d\,e^3\,x\,\sqrt {b^2-4\,a\,c}-48\,a\,b\,c^3\,d^2\,e^2\,x+40\,a\,b^2\,c^2\,d\,e^3\,x+20\,a\,b\,c^2\,d\,e^3\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (e^2\,\left (\frac {b^2\,n}{4}-\frac {a\,c\,n}{2}+\frac {b\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )-e\,\left (\frac {c\,d\,n\,\sqrt {b^2-4\,a\,c}}{2}+\frac {b\,c\,d\,n}{2}\right )+\frac {c^2\,d^2\,n}{2}\right )}{a^2\,e^5-2\,a\,b\,d\,e^4+2\,a\,c\,d^2\,e^3+b^2\,d^2\,e^3-2\,b\,c\,d^3\,e^2+c^2\,d^4\,e}-\frac {\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{2\,e\,\left (d^2+2\,d\,e\,x+e^2\,x^2\right )}-\frac {n\,\left (b\,e-2\,c\,d\right )}{\left (2\,x\,e^2+2\,d\,e\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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