Optimal. Leaf size=69 \[ \frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^3}-\frac {(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac {c (d+e x)^7}{7 e^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {698} \begin {gather*} \frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^3}-\frac {(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac {c (d+e x)^7}{7 e^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin {align*} \int (d+e x)^4 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^4}{e^2}+\frac {(-2 c d+b e) (d+e x)^5}{e^2}+\frac {c (d+e x)^6}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^5}{5 e^3}-\frac {(2 c d-b e) (d+e x)^6}{6 e^3}+\frac {c (d+e x)^7}{7 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 135, normalized size = 1.96 \begin {gather*} \frac {1}{5} e^2 x^5 \left (a e^2+4 b d e+6 c d^2\right )+\frac {1}{2} d e x^4 \left (2 a e^2+3 b d e+2 c d^2\right )+\frac {1}{3} d^2 x^3 \left (6 a e^2+4 b d e+c d^2\right )+\frac {1}{2} d^3 x^2 (4 a e+b d)+a d^4 x+\frac {1}{6} e^3 x^6 (b e+4 c d)+\frac {1}{7} c e^4 x^7 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^4 \left (a+b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.34, size = 146, normalized size = 2.12 \begin {gather*} \frac {1}{7} x^{7} e^{4} c + \frac {2}{3} x^{6} e^{3} d c + \frac {1}{6} x^{6} e^{4} b + \frac {6}{5} x^{5} e^{2} d^{2} c + \frac {4}{5} x^{5} e^{3} d b + \frac {1}{5} x^{5} e^{4} a + x^{4} e d^{3} c + \frac {3}{2} x^{4} e^{2} d^{2} b + x^{4} e^{3} d a + \frac {1}{3} x^{3} d^{4} c + \frac {4}{3} x^{3} e d^{3} b + 2 x^{3} e^{2} d^{2} a + \frac {1}{2} x^{2} d^{4} b + 2 x^{2} e d^{3} a + x d^{4} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.15, size = 140, normalized size = 2.03 \begin {gather*} \frac {1}{7} \, c x^{7} e^{4} + \frac {2}{3} \, c d x^{6} e^{3} + \frac {6}{5} \, c d^{2} x^{5} e^{2} + c d^{3} x^{4} e + \frac {1}{3} \, c d^{4} x^{3} + \frac {1}{6} \, b x^{6} e^{4} + \frac {4}{5} \, b d x^{5} e^{3} + \frac {3}{2} \, b d^{2} x^{4} e^{2} + \frac {4}{3} \, b d^{3} x^{3} e + \frac {1}{2} \, b d^{4} x^{2} + \frac {1}{5} \, a x^{5} e^{4} + a d x^{4} e^{3} + 2 \, a d^{2} x^{3} e^{2} + 2 \, a d^{3} x^{2} e + a d^{4} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 136, normalized size = 1.97 \begin {gather*} \frac {c \,e^{4} x^{7}}{7}+a \,d^{4} x +\frac {\left (e^{4} b +4 c d \,e^{3}\right ) x^{6}}{6}+\frac {\left (e^{4} a +4 d \,e^{3} b +6 c \,d^{2} e^{2}\right ) x^{5}}{5}+\frac {\left (4 a d \,e^{3}+6 d^{2} e^{2} b +4 c \,d^{3} e \right ) x^{4}}{4}+\frac {\left (6 a \,d^{2} e^{2}+4 d^{3} e b +c \,d^{4}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e a +b \,d^{4}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.99, size = 135, normalized size = 1.96 \begin {gather*} \frac {1}{7} \, c e^{4} x^{7} + \frac {1}{6} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{6} + a d^{4} x + \frac {1}{5} \, {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.65, size = 131, normalized size = 1.90 \begin {gather*} x^2\,\left (\frac {b\,d^4}{2}+2\,a\,e\,d^3\right )+x^6\,\left (\frac {b\,e^4}{6}+\frac {2\,c\,d\,e^3}{3}\right )+x^3\,\left (\frac {c\,d^4}{3}+\frac {4\,b\,d^3\,e}{3}+2\,a\,d^2\,e^2\right )+x^5\,\left (\frac {6\,c\,d^2\,e^2}{5}+\frac {4\,b\,d\,e^3}{5}+\frac {a\,e^4}{5}\right )+\frac {c\,e^4\,x^7}{7}+a\,d^4\,x+\frac {d\,e\,x^4\,\left (2\,c\,d^2+3\,b\,d\,e+2\,a\,e^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.10, size = 146, normalized size = 2.12 \begin {gather*} a d^{4} x + \frac {c e^{4} x^{7}}{7} + x^{6} \left (\frac {b e^{4}}{6} + \frac {2 c d e^{3}}{3}\right ) + x^{5} \left (\frac {a e^{4}}{5} + \frac {4 b d e^{3}}{5} + \frac {6 c d^{2} e^{2}}{5}\right ) + x^{4} \left (a d e^{3} + \frac {3 b d^{2} e^{2}}{2} + c d^{3} e\right ) + x^{3} \left (2 a d^{2} e^{2} + \frac {4 b d^{3} e}{3} + \frac {c d^{4}}{3}\right ) + x^{2} \left (2 a d^{3} e + \frac {b d^{4}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________