Optimal. Leaf size=92 \[ \frac {e^2 (a+b x)^6 (b d-a e)}{2 b^4}+\frac {3 e (a+b x)^5 (b d-a e)^2}{5 b^4}+\frac {(a+b x)^4 (b d-a e)^3}{4 b^4}+\frac {e^3 (a+b x)^7}{7 b^4} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 43} \begin {gather*} \frac {e^2 (a+b x)^6 (b d-a e)}{2 b^4}+\frac {3 e (a+b x)^5 (b d-a e)^2}{5 b^4}+\frac {(a+b x)^4 (b d-a e)^3}{4 b^4}+\frac {e^3 (a+b x)^7}{7 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^3 \, dx\\ &=\int \left (\frac {(b d-a e)^3 (a+b x)^3}{b^3}+\frac {3 e (b d-a e)^2 (a+b x)^4}{b^3}+\frac {3 e^2 (b d-a e) (a+b x)^5}{b^3}+\frac {e^3 (a+b x)^6}{b^3}\right ) \, dx\\ &=\frac {(b d-a e)^3 (a+b x)^4}{4 b^4}+\frac {3 e (b d-a e)^2 (a+b x)^5}{5 b^4}+\frac {e^2 (b d-a e) (a+b x)^6}{2 b^4}+\frac {e^3 (a+b x)^7}{7 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 161, normalized size = 1.75 \begin {gather*} a^3 d^3 x+\frac {3}{5} b e x^5 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+a d x^3 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+\frac {3}{2} a^2 d^2 x^2 (a e+b d)+\frac {1}{4} x^4 \left (a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )+\frac {1}{2} b^2 e^2 x^6 (a e+b d)+\frac {1}{7} b^3 e^3 x^7 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.39, size = 188, normalized size = 2.04 \begin {gather*} \frac {1}{7} x^{7} e^{3} b^{3} + \frac {1}{2} x^{6} e^{2} d b^{3} + \frac {1}{2} x^{6} e^{3} b^{2} a + \frac {3}{5} x^{5} e d^{2} b^{3} + \frac {9}{5} x^{5} e^{2} d b^{2} a + \frac {3}{5} x^{5} e^{3} b a^{2} + \frac {1}{4} x^{4} d^{3} b^{3} + \frac {9}{4} x^{4} e d^{2} b^{2} a + \frac {9}{4} x^{4} e^{2} d b a^{2} + \frac {1}{4} x^{4} e^{3} a^{3} + x^{3} d^{3} b^{2} a + 3 x^{3} e d^{2} b a^{2} + x^{3} e^{2} d a^{3} + \frac {3}{2} x^{2} d^{3} b a^{2} + \frac {3}{2} x^{2} e d^{2} a^{3} + x d^{3} a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.15, size = 184, normalized size = 2.00 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} e^{3} + \frac {1}{2} \, b^{3} d x^{6} e^{2} + \frac {3}{5} \, b^{3} d^{2} x^{5} e + \frac {1}{4} \, b^{3} d^{3} x^{4} + \frac {1}{2} \, a b^{2} x^{6} e^{3} + \frac {9}{5} \, a b^{2} d x^{5} e^{2} + \frac {9}{4} \, a b^{2} d^{2} x^{4} e + a b^{2} d^{3} x^{3} + \frac {3}{5} \, a^{2} b x^{5} e^{3} + \frac {9}{4} \, a^{2} b d x^{4} e^{2} + 3 \, a^{2} b d^{2} x^{3} e + \frac {3}{2} \, a^{2} b d^{3} x^{2} + \frac {1}{4} \, a^{3} x^{4} e^{3} + a^{3} d x^{3} e^{2} + \frac {3}{2} \, a^{3} d^{2} x^{2} e + a^{3} d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 244, normalized size = 2.65 \begin {gather*} \frac {b^{3} e^{3} x^{7}}{7}+a^{3} d^{3} x +\frac {\left (2 a \,b^{2} e^{3}+\left (a \,e^{3}+3 b d \,e^{2}\right ) b^{2}\right ) x^{6}}{6}+\frac {\left (a^{2} b \,e^{3}+2 \left (a \,e^{3}+3 b d \,e^{2}\right ) a b +\left (3 a d \,e^{2}+3 b \,d^{2} e \right ) b^{2}\right ) x^{5}}{5}+\frac {\left (\left (a \,e^{3}+3 b d \,e^{2}\right ) a^{2}+2 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a b +\left (3 a \,d^{2} e +b \,d^{3}\right ) b^{2}\right ) x^{4}}{4}+\frac {\left (a \,b^{2} d^{3}+\left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{2}+2 \left (3 a \,d^{2} e +b \,d^{3}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 a^{2} b \,d^{3}+\left (3 a \,d^{2} e +b \,d^{3}\right ) a^{2}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.52, size = 167, normalized size = 1.82 \begin {gather*} \frac {1}{7} \, b^{3} e^{3} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (b^{3} d^{2} e + 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{3} + 9 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} x^{4} + {\left (a b^{2} d^{3} + 3 \, a^{2} b d^{2} e + a^{3} d e^{2}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 152, normalized size = 1.65 \begin {gather*} x^4\,\left (\frac {a^3\,e^3}{4}+\frac {9\,a^2\,b\,d\,e^2}{4}+\frac {9\,a\,b^2\,d^2\,e}{4}+\frac {b^3\,d^3}{4}\right )+a^3\,d^3\,x+\frac {b^3\,e^3\,x^7}{7}+a\,d\,x^3\,\left (a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right )+\frac {3\,b\,e\,x^5\,\left (a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right )}{5}+\frac {3\,a^2\,d^2\,x^2\,\left (a\,e+b\,d\right )}{2}+\frac {b^2\,e^2\,x^6\,\left (a\,e+b\,d\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.10, size = 190, normalized size = 2.07 \begin {gather*} a^{3} d^{3} x + \frac {b^{3} e^{3} x^{7}}{7} + x^{6} \left (\frac {a b^{2} e^{3}}{2} + \frac {b^{3} d e^{2}}{2}\right ) + x^{5} \left (\frac {3 a^{2} b e^{3}}{5} + \frac {9 a b^{2} d e^{2}}{5} + \frac {3 b^{3} d^{2} e}{5}\right ) + x^{4} \left (\frac {a^{3} e^{3}}{4} + \frac {9 a^{2} b d e^{2}}{4} + \frac {9 a b^{2} d^{2} e}{4} + \frac {b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} + 3 a^{2} b d^{2} e + a b^{2} d^{3}\right ) + x^{2} \left (\frac {3 a^{3} d^{2} e}{2} + \frac {3 a^{2} b d^{3}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________