Optimal. Leaf size=124 \[ \frac {(d+e x)^4 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{4 e^4}-\frac {(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac {3 c (d+e x)^5 (2 c d-b e)}{5 e^4}+\frac {c^2 (d+e x)^6}{3 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \[ \frac {(d+e x)^4 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{4 e^4}-\frac {(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac {3 c (d+e x)^5 (2 c d-b e)}{5 e^4}+\frac {c^2 (d+e x)^6}{3 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 771
Rubi steps
\begin {align*} \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^2}{e^3}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^3}{e^3}-\frac {3 c (2 c d-b e) (d+e x)^4}{e^3}+\frac {2 c^2 (d+e x)^5}{e^3}\right ) \, dx\\ &=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^3}{3 e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^4}{4 e^4}-\frac {3 c (2 c d-b e) (d+e x)^5}{5 e^4}+\frac {c^2 (d+e x)^6}{3 e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 133, normalized size = 1.07 \[ \frac {1}{4} x^4 \left (2 a c e^2+b^2 e^2+6 b c d e+2 c^2 d^2\right )+\frac {1}{3} x^3 \left (a b e^2+4 a c d e+2 b^2 d e+3 b c d^2\right )+\frac {1}{2} d x^2 \left (2 a b e+2 a c d+b^2 d\right )+a b d^2 x+\frac {1}{5} c e x^5 (3 b e+4 c d)+\frac {1}{3} c^2 e^2 x^6 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 146, normalized size = 1.18 \[ \frac {1}{3} x^{6} e^{2} c^{2} + \frac {4}{5} x^{5} e d c^{2} + \frac {3}{5} x^{5} e^{2} c b + \frac {1}{2} x^{4} d^{2} c^{2} + \frac {3}{2} x^{4} e d c b + \frac {1}{4} x^{4} e^{2} b^{2} + \frac {1}{2} x^{4} e^{2} c a + x^{3} d^{2} c b + \frac {2}{3} x^{3} e d b^{2} + \frac {4}{3} x^{3} e d c a + \frac {1}{3} x^{3} e^{2} b a + \frac {1}{2} x^{2} d^{2} b^{2} + x^{2} d^{2} c a + x^{2} e d b a + x d^{2} b a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 146, normalized size = 1.18 \[ \frac {1}{3} \, c^{2} x^{6} e^{2} + \frac {4}{5} \, c^{2} d x^{5} e + \frac {1}{2} \, c^{2} d^{2} x^{4} + \frac {3}{5} \, b c x^{5} e^{2} + \frac {3}{2} \, b c d x^{4} e + b c d^{2} x^{3} + \frac {1}{4} \, b^{2} x^{4} e^{2} + \frac {1}{2} \, a c x^{4} e^{2} + \frac {2}{3} \, b^{2} d x^{3} e + \frac {4}{3} \, a c d x^{3} e + \frac {1}{2} \, b^{2} d^{2} x^{2} + a c d^{2} x^{2} + \frac {1}{3} \, a b x^{3} e^{2} + a b d x^{2} e + a b d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 152, normalized size = 1.23 \[ \frac {c^{2} e^{2} x^{6}}{3}+a b \,d^{2} x +\frac {\left (2 b c \,e^{2}+\left (b \,e^{2}+4 c d e \right ) c \right ) x^{5}}{5}+\frac {\left (2 a c \,e^{2}+\left (b \,e^{2}+4 c d e \right ) b +\left (2 b d e +2 c \,d^{2}\right ) c \right ) x^{4}}{4}+\frac {\left (b c \,d^{2}+\left (b \,e^{2}+4 c d e \right ) a +\left (2 b d e +2 c \,d^{2}\right ) b \right ) x^{3}}{3}+\frac {\left (b^{2} d^{2}+\left (2 b d e +2 c \,d^{2}\right ) a \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.51, size = 126, normalized size = 1.02 \[ \frac {1}{3} \, c^{2} e^{2} x^{6} + \frac {1}{5} \, {\left (4 \, c^{2} d e + 3 \, b c e^{2}\right )} x^{5} + a b d^{2} x + \frac {1}{4} \, {\left (2 \, c^{2} d^{2} + 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, b c d^{2} + a b e^{2} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d e + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 124, normalized size = 1.00 \[ x^4\,\left (\frac {b^2\,e^2}{4}+\frac {3\,b\,c\,d\,e}{2}+\frac {c^2\,d^2}{2}+\frac {a\,c\,e^2}{2}\right )+x^3\,\left (\frac {2\,b^2\,d\,e}{3}+c\,b\,d^2+\frac {a\,b\,e^2}{3}+\frac {4\,a\,c\,d\,e}{3}\right )+x^2\,\left (\frac {b^2\,d^2}{2}+a\,e\,b\,d+a\,c\,d^2\right )+\frac {c^2\,e^2\,x^6}{3}+a\,b\,d^2\,x+\frac {c\,e\,x^5\,\left (3\,b\,e+4\,c\,d\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.09, size = 146, normalized size = 1.18 \[ a b d^{2} x + \frac {c^{2} e^{2} x^{6}}{3} + x^{5} \left (\frac {3 b c e^{2}}{5} + \frac {4 c^{2} d e}{5}\right ) + x^{4} \left (\frac {a c e^{2}}{2} + \frac {b^{2} e^{2}}{4} + \frac {3 b c d e}{2} + \frac {c^{2} d^{2}}{2}\right ) + x^{3} \left (\frac {a b e^{2}}{3} + \frac {4 a c d e}{3} + \frac {2 b^{2} d e}{3} + b c d^{2}\right ) + x^{2} \left (a b d e + a c d^{2} + \frac {b^{2} d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________