Optimal. Leaf size=226 \[ -\frac {n x \left (-c e (2 a e+3 b d)+b^2 e^2+6 c^2 d^2\right )}{3 c^2}-\frac {n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{6 c^3 e}+\frac {n \sqrt {b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{3 c^3}+\frac {(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac {e n x^2 (6 c d-b e)}{6 c}-\frac {2}{9} e^2 n x^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 226, 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} \[ -\frac {n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{6 c^3 e}-\frac {n x \left (-c e (2 a e+3 b d)+b^2 e^2+6 c^2 d^2\right )}{3 c^2}+\frac {n \sqrt {b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{3 c^3}+\frac {(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac {e n x^2 (6 c d-b e)}{6 c}-\frac {2}{9} e^2 n x^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 618
Rule 628
Rule 634
Rule 800
Rule 2525
Rubi steps
\begin {align*} \int (d+e x)^2 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac {(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac {n \int \frac {(b+2 c x) (d+e x)^3}{a+b x+c x^2} \, dx}{3 e}\\ &=\frac {(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac {n \int \left (\frac {e \left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right )}{c^2}+\frac {e^2 (6 c d-b e) x}{c}+2 e^3 x^2+\frac {-a b^2 e^3-2 a c e \left (3 c d^2-a e^2\right )+b c d \left (c d^2+3 a e^2\right )+(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{3 e}\\ &=-\frac {\left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) n x}{3 c^2}-\frac {e (6 c d-b e) n x^2}{6 c}-\frac {2}{9} e^2 n x^3+\frac {(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac {n \int \frac {-a b^2 e^3-2 a c e \left (3 c d^2-a e^2\right )+b c d \left (c d^2+3 a e^2\right )+(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{a+b x+c x^2} \, dx}{3 c^2 e}\\ &=-\frac {\left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) n x}{3 c^2}-\frac {e (6 c d-b e) n x^2}{6 c}-\frac {2}{9} e^2 n x^3+\frac {(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac {\left (\left (b^2-4 a c\right ) \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{6 c^3}-\frac {\left ((2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{6 c^3 e}\\ &=-\frac {\left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) n x}{3 c^2}-\frac {e (6 c d-b e) n x^2}{6 c}-\frac {2}{9} e^2 n x^3-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n \log \left (a+b x+c x^2\right )}{6 c^3 e}+\frac {(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}+\frac {\left (\left (b^2-4 a c\right ) \left (3 c^2 d^2+b^2 e^2-c e (3 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 )}{3 c^3}\\ &=-\frac {\left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) n x}{3 c^2}-\frac {e (6 c d-b e) n x^2}{6 c}-\frac {2}{9} e^2 n x^3+\frac {\sqrt {b^2-4 a c} \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{3 c^3}-\frac {(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n \log \left (a+b x+c x^2\right )}{6 c^3 e}+\frac {(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.45, size = 204, normalized size = 0.90 \[ \frac {(d+e x)^3 \log \left (d (a+x (b+c x))^n\right )-\frac {n \left (c e x \left (-3 c e (4 a e+6 b d+b e x)+6 b^2 e^2+2 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log (a+x (b+c x))-6 e \sqrt {b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )}{6 c^3}}{3 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 567, normalized size = 2.51 \[ \left [-\frac {4 \, c^{3} e^{2} n x^{3} + 3 \, {\left (6 \, c^{3} d e - b c^{2} e^{2}\right )} n x^{2} + 3 \, {\left (3 \, c^{2} d^{2} - 3 \, b c d e + {\left (b^{2} - a c\right )} e^{2}\right )} \sqrt {b^{2} - 4 \, a c} n \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 6 \, {\left (6 \, c^{3} d^{2} - 3 \, b c^{2} d e + {\left (b^{2} c - 2 \, a c^{2}\right )} e^{2}\right )} n x - 3 \, {\left (2 \, c^{3} e^{2} n x^{3} + 6 \, c^{3} d e n x^{2} + 6 \, c^{3} d^{2} n x + {\left (3 \, b c^{2} d^{2} - 3 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d e + {\left (b^{3} - 3 \, a b c\right )} e^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right ) - 6 \, {\left (c^{3} e^{2} x^{3} + 3 \, c^{3} d e x^{2} + 3 \, c^{3} d^{2} x\right )} \log \relax (d)}{18 \, c^{3}}, -\frac {4 \, c^{3} e^{2} n x^{3} + 3 \, {\left (6 \, c^{3} d e - b c^{2} e^{2}\right )} n x^{2} - 6 \, {\left (3 \, c^{2} d^{2} - 3 \, b c d e + {\left (b^{2} - a c\right )} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c} n \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 6 \, {\left (6 \, c^{3} d^{2} - 3 \, b c^{2} d e + {\left (b^{2} c - 2 \, a c^{2}\right )} e^{2}\right )} n x - 3 \, {\left (2 \, c^{3} e^{2} n x^{3} + 6 \, c^{3} d e n x^{2} + 6 \, c^{3} d^{2} n x + {\left (3 \, b c^{2} d^{2} - 3 \, {\left (b^{2} c - 2 \, a c^{2}\right )} d e + {\left (b^{3} - 3 \, a b c\right )} e^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right ) - 6 \, {\left (c^{3} e^{2} x^{3} + 3 \, c^{3} d e x^{2} + 3 \, c^{3} d^{2} x\right )} \log \relax (d)}{18 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 349, normalized size = 1.54 \[ \frac {6 \, c^{2} n x^{3} e^{2} \log \left (c x^{2} + b x + a\right ) + 18 \, c^{2} d n x^{2} e \log \left (c x^{2} + b x + a\right ) - 4 \, c^{2} n x^{3} e^{2} - 18 \, c^{2} d n x^{2} e + 18 \, c^{2} d^{2} n x \log \left (c x^{2} + b x + a\right ) + 6 \, c^{2} x^{3} e^{2} \log \relax (d) + 18 \, c^{2} d x^{2} e \log \relax (d) - 36 \, c^{2} d^{2} n x + 3 \, b c n x^{2} e^{2} + 18 \, b c d n x e + 18 \, c^{2} d^{2} x \log \relax (d) - 6 \, b^{2} n x e^{2} + 12 \, a c n x e^{2}}{18 \, c^{2}} + \frac {{\left (3 \, b c^{2} d^{2} n - 3 \, b^{2} c d n e + 6 \, a c^{2} d n e + b^{3} n e^{2} - 3 \, a b c n e^{2}\right )} \log \left (c x^{2} + b x + a\right )}{6 \, c^{3}} - \frac {{\left (3 \, b^{2} c^{2} d^{2} n - 12 \, a c^{3} d^{2} n - 3 \, b^{3} c d n e + 12 \, a b c^{2} d n e + b^{4} n e^{2} - 5 \, a b^{2} c n e^{2} + 4 \, a^{2} c^{2} n e^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt {-b^{2} + 4 \, a c} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.70, size = 7155, normalized size = 31.66 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.67, size = 457, normalized size = 2.02 \[ \ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {\frac {d^2\,n\,\sqrt {b^2-4\,a\,c}}{2}+\frac {b\,d^2\,n}{2}+a\,d\,e\,n}{c}-\frac {\frac {a\,b\,e^2\,n}{2}+\frac {b^2\,d\,e\,n}{2}+\frac {a\,e^2\,n\,\sqrt {b^2-4\,a\,c}}{6}+\frac {b\,d\,e\,n\,\sqrt {b^2-4\,a\,c}}{2}}{c^2}+\frac {b^3\,e^2\,n}{6\,c^3}+\frac {b^2\,e^2\,n\,\sqrt {b^2-4\,a\,c}}{6\,c^3}\right )+x\,\left (\frac {b\,\left (\frac {e\,n\,\left (b\,e+6\,c\,d\right )}{3\,c}-\frac {2\,b\,e^2\,n}{3\,c}\right )}{c}-\frac {d\,n\,\left (b\,e+2\,c\,d\right )}{c}+\frac {2\,a\,e^2\,n}{3\,c}\right )-\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {\frac {a\,b\,e^2\,n}{2}+\frac {b^2\,d\,e\,n}{2}-\frac {a\,e^2\,n\,\sqrt {b^2-4\,a\,c}}{6}-\frac {b\,d\,e\,n\,\sqrt {b^2-4\,a\,c}}{2}}{c^2}-\frac {\frac {b\,d^2\,n}{2}-\frac {d^2\,n\,\sqrt {b^2-4\,a\,c}}{2}+a\,d\,e\,n}{c}-\frac {b^3\,e^2\,n}{6\,c^3}+\frac {b^2\,e^2\,n\,\sqrt {b^2-4\,a\,c}}{6\,c^3}\right )+\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )\,\left (d^2\,x+d\,e\,x^2+\frac {e^2\,x^3}{3}\right )-x^2\,\left (\frac {e\,n\,\left (b\,e+6\,c\,d\right )}{6\,c}-\frac {b\,e^2\,n}{3\,c}\right )-\frac {2\,e^2\,n\,x^3}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________