Optimal. Leaf size=338 \[ -\frac {n \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{8 c^4 e}-\frac {n x \left (-2 c^2 d e (4 a e+3 b d)+b c e^2 (3 a e+4 b d)-b^3 e^3+8 c^3 d^3\right )}{4 c^3}-\frac {e n x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )}{8 c^2}+\frac {n \sqrt {b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 c^4}+\frac {(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac {e^2 n x^3 (8 c d-b e)}{12 c}-\frac {1}{8} e^3 n x^4 \]
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Rubi [A] time = 0.52, antiderivative size = 338, 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 = {2525, 800, 634, 618, 206, 628} \[ -\frac {n \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{8 c^4 e}-\frac {e n x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )}{8 c^2}-\frac {n x \left (-2 c^2 d e (4 a e+3 b d)+b c e^2 (3 a e+4 b d)-b^3 e^3+8 c^3 d^3\right )}{4 c^3}+\frac {n \sqrt {b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 c^4}+\frac {(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac {e^2 n x^3 (8 c d-b e)}{12 c}-\frac {1}{8} e^3 n x^4 \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 800
Rule 2525
Rubi steps
\begin {align*} \int (d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac {(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac {n \int \frac {(b+2 c x) (d+e x)^4}{a+b x+c x^2} \, dx}{4 e}\\ &=\frac {(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac {n \int \left (\frac {e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right )}{c^3}+\frac {e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x}{c^2}+\frac {e^3 (8 c d-b e) x^2}{c}+2 e^4 x^3+\frac {-4 a b^2 c d e^3+a b^3 e^4-8 a c^2 d e \left (c d^2-a e^2\right )+b c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx}{4 e}\\ &=-\frac {\left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) n x}{4 c^3}-\frac {e \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) n x^2}{8 c^2}-\frac {e^2 (8 c d-b e) n x^3}{12 c}-\frac {1}{8} e^3 n x^4+\frac {(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac {n \int \frac {-4 a b^2 c d e^3+a b^3 e^4-8 a c^2 d e \left (c d^2-a e^2\right )+b c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{a+b x+c x^2} \, dx}{4 c^3 e}\\ &=-\frac {\left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) n x}{4 c^3}-\frac {e \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) n x^2}{8 c^2}-\frac {e^2 (8 c d-b e) n x^3}{12 c}-\frac {1}{8} e^3 n x^4+\frac {(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{8 c^4}-\frac {\left (\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{8 c^4 e}\\ &=-\frac {\left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) n x}{4 c^3}-\frac {e \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) n x^2}{8 c^2}-\frac {e^2 (8 c d-b e) n x^3}{12 c}-\frac {1}{8} e^3 n x^4-\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{8 c^4 e}+\frac {(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}+\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{4 c^4}\\ &=-\frac {\left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) n x}{4 c^3}-\frac {e \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) n x^2}{8 c^2}-\frac {e^2 (8 c d-b e) n x^3}{12 c}-\frac {1}{8} e^3 n x^4+\frac {\sqrt {b^2-4 a c} (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 c^4}-\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{8 c^4 e}+\frac {(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 324, normalized size = 0.96 \[ \frac {(d+e x)^4 \log \left (d (a+x (b+c x))^n\right )-\frac {n \left (3 \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log (a+x (b+c x))+6 c e x \left (-2 c^2 d e (4 a e+3 b d)+b c e^2 (3 a e+4 b d)-b^3 e^3+8 c^3 d^3\right )+3 c^2 e^2 x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )-6 e \sqrt {b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+2 c^3 e^3 x^3 (8 c d-b e)+3 c^4 e^4 x^4\right )}{6 c^4}}{4 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 880, normalized size = 2.60 \[ \left [-\frac {3 \, c^{4} e^{3} n x^{4} + 2 \, {\left (8 \, c^{4} d e^{2} - b c^{3} e^{3}\right )} n x^{3} + 3 \, {\left (12 \, c^{4} d^{2} e - 4 \, b c^{3} d e^{2} + {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} e^{3}\right )} n x^{2} - 3 \, {\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 4 \, {\left (b^{2} c - a c^{2}\right )} d e^{2} - {\left (b^{3} - 2 \, a b c\right )} e^{3}\right )} \sqrt {b^{2} - 4 \, a c} n \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 ) + 6 \, {\left (8 \, c^{4} d^{3} - 6 \, b c^{3} d^{2} e + 4 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c - 3 \, a b c^{2}\right )} e^{3}\right )} n x - 3 \, {\left (2 \, c^{4} e^{3} n x^{4} + 8 \, c^{4} d e^{2} n x^{3} + 12 \, c^{4} d^{2} e n x^{2} + 8 \, c^{4} d^{3} n x + {\left (4 \, b c^{3} d^{3} - 6 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2} e + 4 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} d e^{2} - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e^{3}\right )} n\right )} \log \left (c x^{2} + b x + a\right ) - 6 \, {\left (c^{4} e^{3} x^{4} + 4 \, c^{4} d e^{2} x^{3} + 6 \, c^{4} d^{2} e x^{2} + 4 \, c^{4} d^{3} x\right )} \log \relax (d)}{24 \, c^{4}}, -\frac {3 \, c^{4} e^{3} n x^{4} + 2 \, {\left (8 \, c^{4} d e^{2} - b c^{3} e^{3}\right )} n x^{3} + 3 \, {\left (12 \, c^{4} d^{2} e - 4 \, b c^{3} d e^{2} + {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} e^{3}\right )} n x^{2} - 6 \, {\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 4 \, {\left (b^{2} c - a c^{2}\right )} d e^{2} - {\left (b^{3} - 2 \, a b c\right )} e^{3}\right )} \sqrt {-b^{2} + 4 \, a c} n \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 6 \, {\left (8 \, c^{4} d^{3} - 6 \, b c^{3} d^{2} e + 4 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c - 3 \, a b c^{2}\right )} e^{3}\right )} n x - 3 \, {\left (2 \, c^{4} e^{3} n x^{4} + 8 \, c^{4} d e^{2} n x^{3} + 12 \, c^{4} d^{2} e n x^{2} + 8 \, c^{4} d^{3} n x + {\left (4 \, b c^{3} d^{3} - 6 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2} e + 4 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} d e^{2} - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e^{3}\right )} n\right )} \log \left (c x^{2} + b x + a\right ) - 6 \, {\left (c^{4} e^{3} x^{4} + 4 \, c^{4} d e^{2} x^{3} + 6 \, c^{4} d^{2} e x^{2} + 4 \, c^{4} d^{3} x\right )} \log \relax (d)}{24 \, c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 553, normalized size = 1.64 \[ \frac {6 \, c^{3} n x^{4} e^{3} \log \left (c x^{2} + b x + a\right ) + 24 \, c^{3} d n x^{3} e^{2} \log \left (c x^{2} + b x + a\right ) + 36 \, c^{3} d^{2} n x^{2} e \log \left (c x^{2} + b x + a\right ) - 3 \, c^{3} n x^{4} e^{3} - 16 \, c^{3} d n x^{3} e^{2} - 36 \, c^{3} d^{2} n x^{2} e + 24 \, c^{3} d^{3} n x \log \left (c x^{2} + b x + a\right ) + 6 \, c^{3} x^{4} e^{3} \log \relax (d) + 24 \, c^{3} d x^{3} e^{2} \log \relax (d) + 36 \, c^{3} d^{2} x^{2} e \log \relax (d) - 48 \, c^{3} d^{3} n x + 2 \, b c^{2} n x^{3} e^{3} + 12 \, b c^{2} d n x^{2} e^{2} + 36 \, b c^{2} d^{2} n x e + 24 \, c^{3} d^{3} x \log \relax (d) - 3 \, b^{2} c n x^{2} e^{3} + 6 \, a c^{2} n x^{2} e^{3} - 24 \, b^{2} c d n x e^{2} + 48 \, a c^{2} d n x e^{2} + 6 \, b^{3} n x e^{3} - 18 \, a b c n x e^{3}}{24 \, c^{3}} + \frac {{\left (4 \, b c^{3} d^{3} n - 6 \, b^{2} c^{2} d^{2} n e + 12 \, a c^{3} d^{2} n e + 4 \, b^{3} c d n e^{2} - 12 \, a b c^{2} d n e^{2} - b^{4} n e^{3} + 4 \, a b^{2} c n e^{3} - 2 \, a^{2} c^{2} n e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{8 \, c^{4}} - \frac {{\left (4 \, b^{2} c^{3} d^{3} n - 16 \, a c^{4} d^{3} n - 6 \, b^{3} c^{2} d^{2} n e + 24 \, a b c^{3} d^{2} n e + 4 \, b^{4} c d n e^{2} - 20 \, a b^{2} c^{2} d n e^{2} + 16 \, a^{2} c^{3} d n e^{2} - b^{5} n e^{3} + 6 \, a b^{3} c n e^{3} - 8 \, a^{2} b c^{2} n e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt {-b^{2} + 4 \, a c} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.87, size = 16059, normalized size = 47.51 \[ \text {output too large to display} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 775, normalized size = 2.29 \[ \ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )\,\left (d^3\,x+\frac {3\,d^2\,e\,x^2}{2}+d\,e^2\,x^3+\frac {e^3\,x^4}{4}\right )-x^3\,\left (\frac {e^2\,n\,\left (b\,e+8\,c\,d\right )}{12\,c}-\frac {b\,e^3\,n}{6\,c}\right )-x\,\left (\frac {b\,\left (\frac {b\,\left (\frac {e^2\,n\,\left (b\,e+8\,c\,d\right )}{4\,c}-\frac {b\,e^3\,n}{2\,c}\right )}{c}+\frac {a\,e^3\,n}{2\,c}-\frac {d\,e\,n\,\left (b\,e+3\,c\,d\right )}{c}\right )}{c}-\frac {a\,\left (\frac {e^2\,n\,\left (b\,e+8\,c\,d\right )}{4\,c}-\frac {b\,e^3\,n}{2\,c}\right )}{c}+\frac {d^2\,n\,\left (3\,b\,e+4\,c\,d\right )}{2\,c}\right )+x^2\,\left (\frac {b\,\left (\frac {e^2\,n\,\left (b\,e+8\,c\,d\right )}{4\,c}-\frac {b\,e^3\,n}{2\,c}\right )}{2\,c}+\frac {a\,e^3\,n}{4\,c}-\frac {d\,e\,n\,\left (b\,e+3\,c\,d\right )}{2\,c}\right )-\frac {\ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^4\,e^3\,n+2\,a^2\,c^2\,e^3\,n-4\,b\,c^3\,d^3\,n+b^3\,e^3\,n\,\sqrt {b^2-4\,a\,c}-4\,c^3\,d^3\,n\,\sqrt {b^2-4\,a\,c}-4\,a\,b^2\,c\,e^3\,n-12\,a\,c^3\,d^2\,e\,n-4\,b^3\,c\,d\,e^2\,n+6\,b^2\,c^2\,d^2\,e\,n-2\,a\,b\,c\,e^3\,n\,\sqrt {b^2-4\,a\,c}+12\,a\,b\,c^2\,d\,e^2\,n+4\,a\,c^2\,d\,e^2\,n\,\sqrt {b^2-4\,a\,c}+6\,b\,c^2\,d^2\,e\,n\,\sqrt {b^2-4\,a\,c}-4\,b^2\,c\,d\,e^2\,n\,\sqrt {b^2-4\,a\,c}\right )}{8\,c^4}-\frac {e^3\,n\,x^4}{8}-\frac {\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^4\,e^3\,n+2\,a^2\,c^2\,e^3\,n-4\,b\,c^3\,d^3\,n-b^3\,e^3\,n\,\sqrt {b^2-4\,a\,c}+4\,c^3\,d^3\,n\,\sqrt {b^2-4\,a\,c}-4\,a\,b^2\,c\,e^3\,n-12\,a\,c^3\,d^2\,e\,n-4\,b^3\,c\,d\,e^2\,n+6\,b^2\,c^2\,d^2\,e\,n+2\,a\,b\,c\,e^3\,n\,\sqrt {b^2-4\,a\,c}+12\,a\,b\,c^2\,d\,e^2\,n-4\,a\,c^2\,d\,e^2\,n\,\sqrt {b^2-4\,a\,c}-6\,b\,c^2\,d^2\,e\,n\,\sqrt {b^2-4\,a\,c}+4\,b^2\,c\,d\,e^2\,n\,\sqrt {b^2-4\,a\,c}\right )}{8\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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