Optimal. Leaf size=154 \[ -\frac {n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{4 c^2 e}+\frac {n \sqrt {b^2-4 a c} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 c^2}+\frac {(d+e x)^2 \log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e}-\frac {1}{2} n x \left (4 d-\frac {b e}{c}\right )-\frac {1}{2} e n x^2 \]
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Rubi [A] time = 0.19, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2525, 800, 634, 618, 206, 628} \[ -\frac {n \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{4 c^2 e}+\frac {n \sqrt {b^2-4 a c} (2 c d-b e) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{2 c^2}+\frac {(d+e x)^2 \log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e}-\frac {1}{2} n x \left (4 d-\frac {b e}{c}\right )-\frac {1}{2} e n x^2 \]
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) \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac {(d+e x)^2 \log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e}-\frac {n \int \frac {(b+2 c x) (d+e x)^2}{a+b x+c x^2} \, dx}{2 e}\\ &=\frac {(d+e x)^2 \log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e}-\frac {n \int \left (e \left (4 d-\frac {b e}{c}\right )+2 e^2 x+\frac {b c d^2-4 a c d e+a b e^2+\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) x}{c \left (a+b x+c x^2\right )}\right ) \, dx}{2 e}\\ &=-\frac {1}{2} \left (4 d-\frac {b e}{c}\right ) n x-\frac {1}{2} e n x^2+\frac {(d+e x)^2 \log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e}-\frac {n \int \frac {b c d^2-4 a c d e+a b e^2+\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 c e}\\ &=-\frac {1}{2} \left (4 d-\frac {b e}{c}\right ) n x-\frac {1}{2} e n x^2+\frac {(d+e x)^2 \log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e}-\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 c^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 c^2 e}\\ &=-\frac {1}{2} \left (4 d-\frac {b e}{c}\right ) n x-\frac {1}{2} e n x^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 c^2 e}+\frac {(d+e x)^2 \log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e}+\frac {\left (\left (b^2-4 a c\right ) (2 c d-b e) n\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{2 c^2}\\ &=-\frac {1}{2} \left (4 d-\frac {b e}{c}\right ) n x-\frac {1}{2} e n x^2+\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 c^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 c^2 e}+\frac {(d+e x)^2 \log \left (d \left (a+b x+c x^2\right )^n\right )}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 123, normalized size = 0.80 \[ \frac {n \left (2 a c e+b^2 (-e)+2 b c d\right ) \log (a+x (b+c x))-2 n \sqrt {b^2-4 a c} (b e-2 c d) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+2 c x \left (c (2 d+e x) \log \left (d (a+x (b+c x))^n\right )+b e n-c n (4 d+e x)\right )}{4 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 336, normalized size = 2.18 \[ \left [-\frac {2 \, c^{2} e n x^{2} + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c d - b e\right )} n \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \, {\left (4 \, c^{2} d - b c e\right )} n x - {\left (2 \, c^{2} e n x^{2} + 4 \, c^{2} d n x + {\left (2 \, b c d - {\left (b^{2} - 2 \, a c\right )} e\right )} n\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )} \log \relax (d)}{4 \, c^{2}}, -\frac {2 \, c^{2} e n x^{2} - 2 \, \sqrt {-b^{2} + 4 \, a c} {\left (2 \, c d - b e\right )} n \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (4 \, c^{2} d - b c e\right )} n x - {\left (2 \, c^{2} e n x^{2} + 4 \, c^{2} d n x + {\left (2 \, b c d - {\left (b^{2} - 2 \, a c\right )} e\right )} n\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )} \log \relax (d)}{4 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 188, normalized size = 1.22 \[ \frac {c n x^{2} e \log \left (c x^{2} + b x + a\right ) - c n x^{2} e + 2 \, c d n x \log \left (c x^{2} + b x + a\right ) + c x^{2} e \log \relax (d) - 4 \, c d n x + b n x e + 2 \, c d x \log \relax (d)}{2 \, c} + \frac {{\left (2 \, b c d n - b^{2} n e + 2 \, a c n e\right )} \log \left (c x^{2} + b x + a\right )}{4 \, c^{2}} - \frac {{\left (2 \, b^{2} c d n - 8 \, a c^{2} d n - b^{3} n e + 4 \, a b c n e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.70, size = 1706, normalized size = 11.08 \[ \text {result too large to display} \]
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 = 0.59, size = 242, normalized size = 1.57 \[ \ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )\,\left (\frac {e\,x^2}{2}+d\,x\right )-x\,\left (\frac {n\,\left (b\,e+4\,c\,d\right )}{2\,c}-\frac {b\,e\,n}{c}\right )-\frac {e\,n\,x^2}{2}+\frac {\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (c\,\left (\frac {a\,e\,n}{2}+\frac {b\,d\,n}{2}-\frac {d\,n\,\sqrt {b^2-4\,a\,c}}{2}\right )-\frac {b^2\,e\,n}{4}+\frac {b\,e\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )}{c^2}-\frac {\ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {b^2\,e\,n}{4}-c\,\left (\frac {a\,e\,n}{2}+\frac {b\,d\,n}{2}+\frac {d\,n\,\sqrt {b^2-4\,a\,c}}{2}\right )+\frac {b\,e\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )}{c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 158.89, size = 394, normalized size = 2.56 \[ \begin {cases} \frac {a e n \log {\left (a + b x + c x^{2} \right )}}{2 c} - \frac {b^{2} e n \log {\left (a + b x + c x^{2} \right )}}{4 c^{2}} + \frac {b d n \log {\left (a + b x + c x^{2} \right )}}{2 c} + \frac {b e n x}{2 c} + \frac {b e n \sqrt {- 4 a c + b^{2}} \log {\left (a + b x + c x^{2} \right )}}{4 c^{2}} - \frac {b e n \sqrt {- 4 a c + b^{2}} \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{2 c^{2}} + d n x \log {\left (a + b x + c x^{2} \right )} - 2 d n x + d x \log {\relax (d )} + \frac {e n x^{2} \log {\left (a + b x + c x^{2} \right )}}{2} - \frac {e n x^{2}}{2} + \frac {e x^{2} \log {\relax (d )}}{2} - \frac {d n \sqrt {- 4 a c + b^{2}} \log {\left (a + b x + c x^{2} \right )}}{2 c} + \frac {d n \sqrt {- 4 a c + b^{2}} \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )}}{c} & \text {for}\: c \neq 0 \\- \frac {a^{2} e n \log {\left (a + b x \right )}}{2 b^{2}} + \frac {a d n \log {\left (a + b x \right )}}{b} + \frac {a e n x}{2 b} + d n x \log {\left (a + b x \right )} - d n x + d x \log {\relax (d )} + \frac {e n x^{2} \log {\left (a + b x \right )}}{2} - \frac {e n x^{2}}{4} + \frac {e x^{2} \log {\relax (d )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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