Optimal. Leaf size=485 \[ -\frac {\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac {e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac {e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac {e^3 (10 c d-b e) n x^4}{20 c}-\frac {2}{25} e^4 n x^5+\frac {\sqrt {b^2-4 a c} \left (5 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+a e)-b^2 c e^3 (5 b d+3 a e)+c^2 e^2 \left (10 b^2 d^2+10 a b d e+a^2 e^2\right )\right ) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{5 c^5}-\frac {(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{10 c^5 e}+\frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e} \]
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Rubi [A]
time = 1.36, antiderivative size = 485, 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 x \left (c^2 e^2 \left (2 a^2 e^2+15 a b d e+10 b^2 d^2\right )-b^2 c e^3 (4 a e+5 b d)-10 c^3 d^2 e (2 a e+b d)+b^4 e^4+10 c^4 d^4\right )}{5 c^4}-\frac {n (2 c d-b e) \left (c^2 e^2 \left (5 a^2 e^2+10 a b d e+4 b^2 d^2\right )-b^2 c e^3 (5 a e+3 b d)-2 c^3 d^2 e (5 a e+b d)+b^4 e^4+c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{10 c^5 e}+\frac {n \sqrt {b^2-4 a c} \left (c^2 e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )-b^2 c e^3 (3 a e+5 b d)-10 c^3 d^2 e (a e+b d)+b^4 e^4+5 c^4 d^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{5 c^5}-\frac {e n x^2 \left (-10 c^2 d e (a e+b d)+b c e^2 (3 a e+5 b d)-b^3 e^3+20 c^3 d^3\right )}{10 c^3}-\frac {e^2 n x^3 \left (-c e (2 a e+5 b d)+b^2 e^2+20 c^2 d^2\right )}{15 c^2}+\frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac {e^3 n x^4 (10 c d-b e)}{20 c}-\frac {2}{25} e^4 n x^5 \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 (d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac {n \int \frac {(b+2 c x) (d+e x)^5}{a+b x+c x^2} \, dx}{5 e}\\ &=\frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac {n \int \left (\frac {e \left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right )}{c^4}+\frac {e^2 \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) x}{c^3}+\frac {e^3 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) x^2}{c^2}+\frac {e^4 (10 c d-b e) x^3}{c}+2 e^5 x^4+\frac {5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )+(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) x}{c^4 \left (a+b x+c x^2\right )}\right ) \, dx}{5 e}\\ &=-\frac {\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac {e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac {e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac {e^3 (10 c d-b e) n x^4}{20 c}-\frac {2}{25} e^4 n x^5+\frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac {n \int \frac {5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )+(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) x}{a+b x+c x^2} \, dx}{5 c^4 e}\\ &=-\frac {\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac {e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac {e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac {e^3 (10 c d-b e) n x^4}{20 c}-\frac {2}{25} e^4 n x^5+\frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac {\left ((2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{10 c^5 e}-\frac {\left (\left (-b (2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right )+2 c \left (5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )\right )\right ) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{10 c^5 e}\\ &=-\frac {\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac {e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac {e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac {e^3 (10 c d-b e) n x^4}{20 c}-\frac {2}{25} e^4 n x^5-\frac {(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{10 c^5 e}+\frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}+\frac {\left (\left (-b (2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right )+2 c \left (5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )\right )\right ) n\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{5 c^5 e}\\ &=-\frac {\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac {e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac {e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac {e^3 (10 c d-b e) n x^4}{20 c}-\frac {2}{25} e^4 n x^5+\frac {\sqrt {b^2-4 a c} \left (5 c^4 d^4-10 b c^3 d^3 e+10 b^2 c^2 d^2 e^2-10 a c^3 d^2 e^2-5 b^3 c d e^3+10 a b c^2 d e^3+b^4 e^4-3 a b^2 c e^4+a^2 c^2 e^4\right ) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{5 c^5}-\frac {(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{10 c^5 e}+\frac {(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}\\ \end {align*}
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Mathematica [A]
time = 1.30, size = 468, normalized size = 0.96 \begin {gather*} \frac {-\frac {n \left (60 c e \left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) x+30 c^2 e^2 \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) x^2+20 c^3 e^3 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) x^3+15 c^4 e^4 (10 c d-b e) x^4+24 c^5 e^5 x^5-60 \sqrt {b^2-4 a c} e \left (5 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+a e)-b^2 c e^3 (5 b d+3 a e)+c^2 e^2 \left (10 b^2 d^2+10 a b d e+a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+30 (2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) \log (a+x (b+c x))\right )}{60 c^5}+(d+e x)^5 \log \left (d (a+x (b+c x))^n\right )}{5 e} \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 = 0.23, size = 31895, normalized size = 65.76
method | result | size |
risch | \(\text {Expression too large to display}\) | \(31895\) |
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 [A]
time = 0.45, size = 1274, normalized size = 2.63 \begin {gather*} \left [-\frac {600 \, c^{5} d^{4} n x - 30 \, {\left (5 \, c^{4} d^{4} n - 10 \, b c^{3} d^{3} n e + 10 \, {\left (b^{2} c^{2} - a c^{3}\right )} d^{2} n e^{2} - 5 \, {\left (b^{3} c - 2 \, a b c^{2}\right )} d n e^{3} + {\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} n e^{4}\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 (24 \, c^{5} n x^{5} - 15 \, b c^{4} n x^{4} + 20 \, {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} n x^{3} - 30 \, {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} n x^{2} + 60 \, {\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} n x\right )} e^{4} + 50 \, {\left (3 \, c^{5} d n x^{4} - 2 \, b c^{4} d n x^{3} + 3 \, {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} d n x^{2} - 6 \, {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} d n x\right )} e^{3} + 100 \, {\left (4 \, c^{5} d^{2} n x^{3} - 3 \, b c^{4} d^{2} n x^{2} + 6 \, {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} d^{2} n x\right )} e^{2} + 600 \, {\left (c^{5} d^{3} n x^{2} - b c^{4} d^{3} n x\right )} e - 30 \, {\left (10 \, c^{5} d^{4} n x + 5 \, b c^{4} d^{4} n + {\left (2 \, c^{5} n x^{5} + {\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} n\right )} e^{4} + 5 \, {\left (2 \, c^{5} d n x^{4} - {\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d n\right )} e^{3} + 10 \, {\left (2 \, c^{5} d^{2} n x^{3} + {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} d^{2} n\right )} e^{2} + 10 \, {\left (2 \, c^{5} d^{3} n x^{2} - {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} d^{3} n\right )} e\right )} \log \left (c x^{2} + b x + a\right ) - 60 \, {\left (c^{5} x^{5} e^{4} + 5 \, c^{5} d x^{4} e^{3} + 10 \, c^{5} d^{2} x^{3} e^{2} + 10 \, c^{5} d^{3} x^{2} e + 5 \, c^{5} d^{4} x\right )} \log \left (d\right )}{300 \, c^{5}}, -\frac {600 \, c^{5} d^{4} n x - 60 \, {\left (5 \, c^{4} d^{4} n - 10 \, b c^{3} d^{3} n e + 10 \, {\left (b^{2} c^{2} - a c^{3}\right )} d^{2} n e^{2} - 5 \, {\left (b^{3} c - 2 \, a b c^{2}\right )} d n e^{3} + {\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} n e^{4}\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 (24 \, c^{5} n x^{5} - 15 \, b c^{4} n x^{4} + 20 \, {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} n x^{3} - 30 \, {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} n x^{2} + 60 \, {\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} n x\right )} e^{4} + 50 \, {\left (3 \, c^{5} d n x^{4} - 2 \, b c^{4} d n x^{3} + 3 \, {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} d n x^{2} - 6 \, {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} d n x\right )} e^{3} + 100 \, {\left (4 \, c^{5} d^{2} n x^{3} - 3 \, b c^{4} d^{2} n x^{2} + 6 \, {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} d^{2} n x\right )} e^{2} + 600 \, {\left (c^{5} d^{3} n x^{2} - b c^{4} d^{3} n x\right )} e - 30 \, {\left (10 \, c^{5} d^{4} n x + 5 \, b c^{4} d^{4} n + {\left (2 \, c^{5} n x^{5} + {\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} n\right )} e^{4} + 5 \, {\left (2 \, c^{5} d n x^{4} - {\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d n\right )} e^{3} + 10 \, {\left (2 \, c^{5} d^{2} n x^{3} + {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} d^{2} n\right )} e^{2} + 10 \, {\left (2 \, c^{5} d^{3} n x^{2} - {\left (b^{2} c^{3} - 2 \, a c^{4}\right )} d^{3} n\right )} e\right )} \log \left (c x^{2} + b x + a\right ) - 60 \, {\left (c^{5} x^{5} e^{4} + 5 \, c^{5} d x^{4} e^{3} + 10 \, c^{5} d^{2} x^{3} e^{2} + 10 \, c^{5} d^{3} x^{2} e + 5 \, c^{5} d^{4} x\right )} \log \left (d\right )}{300 \, c^{5}}\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 [A]
time = 4.80, size = 817, normalized size = 1.68 \begin {gather*} \frac {60 \, c^{4} n x^{5} e^{4} \log \left (c x^{2} + b x + a\right ) + 300 \, c^{4} d n x^{4} e^{3} \log \left (c x^{2} + b x + a\right ) + 600 \, c^{4} d^{2} n x^{3} e^{2} \log \left (c x^{2} + b x + a\right ) + 600 \, c^{4} d^{3} n x^{2} e \log \left (c x^{2} + b x + a\right ) - 24 \, c^{4} n x^{5} e^{4} - 150 \, c^{4} d n x^{4} e^{3} - 400 \, c^{4} d^{2} n x^{3} e^{2} - 600 \, c^{4} d^{3} n x^{2} e + 300 \, c^{4} d^{4} n x \log \left (c x^{2} + b x + a\right ) + 60 \, c^{4} x^{5} e^{4} \log \left (d\right ) + 300 \, c^{4} d x^{4} e^{3} \log \left (d\right ) + 600 \, c^{4} d^{2} x^{3} e^{2} \log \left (d\right ) + 600 \, c^{4} d^{3} x^{2} e \log \left (d\right ) - 600 \, c^{4} d^{4} n x + 15 \, b c^{3} n x^{4} e^{4} + 100 \, b c^{3} d n x^{3} e^{3} + 300 \, b c^{3} d^{2} n x^{2} e^{2} + 600 \, b c^{3} d^{3} n x e + 300 \, c^{4} d^{4} x \log \left (d\right ) - 20 \, b^{2} c^{2} n x^{3} e^{4} + 40 \, a c^{3} n x^{3} e^{4} - 150 \, b^{2} c^{2} d n x^{2} e^{3} + 300 \, a c^{3} d n x^{2} e^{3} - 600 \, b^{2} c^{2} d^{2} n x e^{2} + 1200 \, a c^{3} d^{2} n x e^{2} + 30 \, b^{3} c n x^{2} e^{4} - 90 \, a b c^{2} n x^{2} e^{4} + 300 \, b^{3} c d n x e^{3} - 900 \, a b c^{2} d n x e^{3} - 60 \, b^{4} n x e^{4} + 240 \, a b^{2} c n x e^{4} - 120 \, a^{2} c^{2} n x e^{4}}{300 \, c^{4}} + \frac {{\left (5 \, b c^{4} d^{4} n - 10 \, b^{2} c^{3} d^{3} n e + 20 \, a c^{4} d^{3} n e + 10 \, b^{3} c^{2} d^{2} n e^{2} - 30 \, a b c^{3} d^{2} n e^{2} - 5 \, b^{4} c d n e^{3} + 20 \, a b^{2} c^{2} d n e^{3} - 10 \, a^{2} c^{3} d n e^{3} + b^{5} n e^{4} - 5 \, a b^{3} c n e^{4} + 5 \, a^{2} b c^{2} n e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{10 \, c^{5}} - \frac {{\left (5 \, b^{2} c^{4} d^{4} n - 20 \, a c^{5} d^{4} n - 10 \, b^{3} c^{3} d^{3} n e + 40 \, a b c^{4} d^{3} n e + 10 \, b^{4} c^{2} d^{2} n e^{2} - 50 \, a b^{2} c^{3} d^{2} n e^{2} + 40 \, a^{2} c^{4} d^{2} n e^{2} - 5 \, b^{5} c d n e^{3} + 30 \, a b^{3} c^{2} d n e^{3} - 40 \, a^{2} b c^{3} d n e^{3} + b^{6} n e^{4} - 7 \, a b^{4} c n e^{4} + 13 \, a^{2} b^{2} c^{2} n e^{4} - 4 \, a^{3} c^{3} n e^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{5 \, \sqrt {-b^{2} + 4 \, a c} c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.03, size = 1240, normalized size = 2.56 \begin {gather*} x^3\,\left (\frac {b\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{3\,c}+\frac {2\,a\,e^4\,n}{15\,c}-\frac {d\,e^2\,n\,\left (b\,e+4\,c\,d\right )}{3\,c}\right )-x\,\left (\frac {a\,\left (\frac {b\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{c}+\frac {2\,a\,e^4\,n}{5\,c}-\frac {d\,e^2\,n\,\left (b\,e+4\,c\,d\right )}{c}\right )}{c}-\frac {b\,\left (\frac {b\,\left (\frac {b\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{c}+\frac {2\,a\,e^4\,n}{5\,c}-\frac {d\,e^2\,n\,\left (b\,e+4\,c\,d\right )}{c}\right )}{c}-\frac {a\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{c}+\frac {2\,d^2\,e\,n\,\left (b\,e+2\,c\,d\right )}{c}\right )}{c}+\frac {2\,d^3\,n\,\left (b\,e+c\,d\right )}{c}\right )-x^2\,\left (\frac {b\,\left (\frac {b\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{c}+\frac {2\,a\,e^4\,n}{5\,c}-\frac {d\,e^2\,n\,\left (b\,e+4\,c\,d\right )}{c}\right )}{2\,c}-\frac {a\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{5\,c}-\frac {2\,b\,e^4\,n}{5\,c}\right )}{2\,c}+\frac {d^2\,e\,n\,\left (b\,e+2\,c\,d\right )}{c}\right )-x^4\,\left (\frac {e^3\,n\,\left (b\,e+10\,c\,d\right )}{20\,c}-\frac {b\,e^4\,n}{10\,c}\right )+\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )\,\left (d^4\,x+2\,d^3\,e\,x^2+2\,d^2\,e^2\,x^3+d\,e^3\,x^4+\frac {e^4\,x^5}{5}\right )+\frac {\ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^5\,e^4\,n+5\,b\,c^4\,d^4\,n+b^4\,e^4\,n\,\sqrt {b^2-4\,a\,c}+5\,c^4\,d^4\,n\,\sqrt {b^2-4\,a\,c}-5\,a\,b^3\,c\,e^4\,n+20\,a\,c^4\,d^3\,e\,n-5\,b^4\,c\,d\,e^3\,n+5\,a^2\,b\,c^2\,e^4\,n-10\,a^2\,c^3\,d\,e^3\,n-10\,b^2\,c^3\,d^3\,e\,n+a^2\,c^2\,e^4\,n\,\sqrt {b^2-4\,a\,c}+10\,b^3\,c^2\,d^2\,e^2\,n-10\,a\,c^3\,d^2\,e^2\,n\,\sqrt {b^2-4\,a\,c}+10\,b^2\,c^2\,d^2\,e^2\,n\,\sqrt {b^2-4\,a\,c}-3\,a\,b^2\,c\,e^4\,n\,\sqrt {b^2-4\,a\,c}-10\,b\,c^3\,d^3\,e\,n\,\sqrt {b^2-4\,a\,c}-5\,b^3\,c\,d\,e^3\,n\,\sqrt {b^2-4\,a\,c}-30\,a\,b\,c^3\,d^2\,e^2\,n+20\,a\,b^2\,c^2\,d\,e^3\,n+10\,a\,b\,c^2\,d\,e^3\,n\,\sqrt {b^2-4\,a\,c}\right )}{10\,c^5}-\frac {2\,e^4\,n\,x^5}{25}+\frac {\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^5\,e^4\,n+5\,b\,c^4\,d^4\,n-b^4\,e^4\,n\,\sqrt {b^2-4\,a\,c}-5\,c^4\,d^4\,n\,\sqrt {b^2-4\,a\,c}-5\,a\,b^3\,c\,e^4\,n+20\,a\,c^4\,d^3\,e\,n-5\,b^4\,c\,d\,e^3\,n+5\,a^2\,b\,c^2\,e^4\,n-10\,a^2\,c^3\,d\,e^3\,n-10\,b^2\,c^3\,d^3\,e\,n-a^2\,c^2\,e^4\,n\,\sqrt {b^2-4\,a\,c}+10\,b^3\,c^2\,d^2\,e^2\,n+10\,a\,c^3\,d^2\,e^2\,n\,\sqrt {b^2-4\,a\,c}-10\,b^2\,c^2\,d^2\,e^2\,n\,\sqrt {b^2-4\,a\,c}+3\,a\,b^2\,c\,e^4\,n\,\sqrt {b^2-4\,a\,c}+10\,b\,c^3\,d^3\,e\,n\,\sqrt {b^2-4\,a\,c}+5\,b^3\,c\,d\,e^3\,n\,\sqrt {b^2-4\,a\,c}-30\,a\,b\,c^3\,d^2\,e^2\,n+20\,a\,b^2\,c^2\,d\,e^3\,n-10\,a\,b\,c^2\,d\,e^3\,n\,\sqrt {b^2-4\,a\,c}\right )}{10\,c^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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