Optimal. Leaf size=65 \[ -\frac{2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac{(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac{b^2 (d+e x)^6}{6 e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0679921, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 43} \[ -\frac{2 b (d+e x)^5 (b d-a e)}{5 e^3}+\frac{(d+e x)^4 (b d-a e)^2}{4 e^3}+\frac{b^2 (d+e x)^6}{6 e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (d+e x)^3 \, dx\\ &=\int \left (\frac{(-b d+a e)^2 (d+e x)^3}{e^2}-\frac{2 b (b d-a e) (d+e x)^4}{e^2}+\frac{b^2 (d+e x)^5}{e^2}\right ) \, dx\\ &=\frac{(b d-a e)^2 (d+e x)^4}{4 e^3}-\frac{2 b (b d-a e) (d+e x)^5}{5 e^3}+\frac{b^2 (d+e x)^6}{6 e^3}\\ \end{align*}
Mathematica [A] time = 0.0177003, size = 122, normalized size = 1.88 \[ \frac{1}{4} e x^4 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac{1}{3} d x^3 \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+a^2 d^3 x+\frac{1}{2} a d^2 x^2 (3 a e+2 b d)+\frac{1}{5} b e^2 x^5 (2 a e+3 b d)+\frac{1}{6} b^2 e^3 x^6 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.039, size = 125, normalized size = 1.9 \begin{align*}{\frac{{e}^{3}{b}^{2}{x}^{6}}{6}}+{\frac{ \left ( 2\,ab{e}^{3}+3\,d{e}^{2}{b}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{2}{e}^{3}+6\,d{e}^{2}ab+3\,{d}^{2}e{b}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{2}+6\,{d}^{2}eab+{d}^{3}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{2}+2\,{d}^{3}ab \right ){x}^{2}}{2}}+{d}^{3}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.13521, size = 167, normalized size = 2.57 \begin{align*} \frac{1}{6} \, b^{2} e^{3} x^{6} + a^{2} d^{3} x + \frac{1}{5} \,{\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.76083, size = 285, normalized size = 4.38 \begin{align*} \frac{1}{6} x^{6} e^{3} b^{2} + \frac{3}{5} x^{5} e^{2} d b^{2} + \frac{2}{5} x^{5} e^{3} b a + \frac{3}{4} x^{4} e d^{2} b^{2} + \frac{3}{2} x^{4} e^{2} d b a + \frac{1}{4} x^{4} e^{3} a^{2} + \frac{1}{3} x^{3} d^{3} b^{2} + 2 x^{3} e d^{2} b a + x^{3} e^{2} d a^{2} + x^{2} d^{3} b a + \frac{3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.081116, size = 133, normalized size = 2.05 \begin{align*} a^{2} d^{3} x + \frac{b^{2} e^{3} x^{6}}{6} + x^{5} \left (\frac{2 a b e^{3}}{5} + \frac{3 b^{2} d e^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} e^{3}}{4} + \frac{3 a b d e^{2}}{2} + \frac{3 b^{2} d^{2} e}{4}\right ) + x^{3} \left (a^{2} d e^{2} + 2 a b d^{2} e + \frac{b^{2} d^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{2} d^{2} e}{2} + a b d^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.15857, size = 171, normalized size = 2.63 \begin{align*} \frac{1}{6} \, b^{2} x^{6} e^{3} + \frac{3}{5} \, b^{2} d x^{5} e^{2} + \frac{3}{4} \, b^{2} d^{2} x^{4} e + \frac{1}{3} \, b^{2} d^{3} x^{3} + \frac{2}{5} \, a b x^{5} e^{3} + \frac{3}{2} \, a b d x^{4} e^{2} + 2 \, a b d^{2} x^{3} e + a b d^{3} x^{2} + \frac{1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac{3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]