Optimal. Leaf size=38 \[ \frac{(a+b x)^4 (b d-a e)}{4 b^2}+\frac{e (a+b x)^5}{5 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0166393, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 43} \[ \frac{(a+b x)^4 (b d-a e)}{4 b^2}+\frac{e (a+b x)^5}{5 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x) \, dx\\ &=\int \left (\frac{(b d-a e) (a+b x)^3}{b}+\frac{e (a+b x)^4}{b}\right ) \, dx\\ &=\frac{(b d-a e) (a+b x)^4}{4 b^2}+\frac{e (a+b x)^5}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0110288, size = 67, normalized size = 1.76 \[ \frac{1}{2} a^2 x^2 (a e+3 b d)+a^3 d x+\frac{1}{4} b^2 x^4 (3 a e+b d)+a b x^3 (a e+b d)+\frac{1}{5} b^3 e x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.002, size = 94, normalized size = 2.5 \begin{align*}{\frac{{b}^{3}e{x}^{5}}{5}}+{\frac{ \left ( \left ( ae+bd \right ){b}^{2}+2\,{b}^{2}ea \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{2}da+2\, \left ( ae+bd \right ) ab+be{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}db+ \left ( ae+bd \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.963005, size = 93, normalized size = 2.45 \begin{align*} \frac{1}{5} \, b^{3} e x^{5} + a^{3} d x + \frac{1}{4} \,{\left (b^{3} d + 3 \, a b^{2} e\right )} x^{4} +{\left (a b^{2} d + a^{2} b e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.29428, size = 163, normalized size = 4.29 \begin{align*} \frac{1}{5} x^{5} e b^{3} + \frac{1}{4} x^{4} d b^{3} + \frac{3}{4} x^{4} e b^{2} a + x^{3} d b^{2} a + x^{3} e b a^{2} + \frac{3}{2} x^{2} d b a^{2} + \frac{1}{2} x^{2} e a^{3} + x d a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.070982, size = 73, normalized size = 1.92 \begin{align*} a^{3} d x + \frac{b^{3} e x^{5}}{5} + x^{4} \left (\frac{3 a b^{2} e}{4} + \frac{b^{3} d}{4}\right ) + x^{3} \left (a^{2} b e + a b^{2} d\right ) + x^{2} \left (\frac{a^{3} e}{2} + \frac{3 a^{2} b d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.22442, size = 103, normalized size = 2.71 \begin{align*} \frac{1}{5} \, b^{3} x^{5} e + \frac{1}{4} \, b^{3} d x^{4} + \frac{3}{4} \, a b^{2} x^{4} e + a b^{2} d x^{3} + a^{2} b x^{3} e + \frac{3}{2} \, a^{2} b d x^{2} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]