Optimal. Leaf size=65 \[ \frac{e (a+b x)^8 (b d-a e)}{4 b^3}+\frac{(a+b x)^7 (b d-a e)^2}{7 b^3}+\frac{e^2 (a+b x)^9}{9 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.135757, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac{e (a+b x)^8 (b d-a e)}{4 b^3}+\frac{(a+b x)^7 (b d-a e)^2}{7 b^3}+\frac{e^2 (a+b x)^9}{9 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^2 \, dx\\ &=\int \left (\frac{(b d-a e)^2 (a+b x)^6}{b^2}+\frac{2 e (b d-a e) (a+b x)^7}{b^2}+\frac{e^2 (a+b x)^8}{b^2}\right ) \, dx\\ &=\frac{(b d-a e)^2 (a+b x)^7}{7 b^3}+\frac{e (b d-a e) (a+b x)^8}{4 b^3}+\frac{e^2 (a+b x)^9}{9 b^3}\\ \end{align*}
Mathematica [B] time = 0.067054, size = 199, normalized size = 3.06 \[ \frac{1}{252} x \left (126 a^4 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+84 a^3 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+36 a^2 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+126 a^5 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+84 a^6 \left (3 d^2+3 d e x+e^2 x^2\right )+9 a b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.041, size = 239, normalized size = 3.7 \begin{align*}{\frac{{e}^{2}{b}^{6}{x}^{9}}{9}}+{\frac{ \left ( 6\,{e}^{2}a{b}^{5}+2\,de{b}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 15\,{e}^{2}{a}^{2}{b}^{4}+12\,dea{b}^{5}+{b}^{6}{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 20\,{e}^{2}{a}^{3}{b}^{3}+30\,de{a}^{2}{b}^{4}+6\,{d}^{2}a{b}^{5} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{e}^{2}{a}^{4}{b}^{2}+40\,de{a}^{3}{b}^{3}+15\,{d}^{2}{a}^{2}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,{e}^{2}{a}^{5}b+30\,de{a}^{4}{b}^{2}+20\,{d}^{2}{a}^{3}{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ({e}^{2}{a}^{6}+12\,de{a}^{5}b+15\,{d}^{2}{a}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{6}+6\,{d}^{2}{a}^{5}b \right ){x}^{2}}{2}}+{d}^{2}{a}^{6}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.09423, size = 316, normalized size = 4.86 \begin{align*} \frac{1}{9} \, b^{6} e^{2} x^{9} + a^{6} d^{2} x + \frac{1}{4} \,{\left (b^{6} d e + 3 \, a b^{5} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{2} + 12 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x^{7} + \frac{1}{3} \,{\left (3 \, a b^{5} d^{2} + 15 \, a^{2} b^{4} d e + 10 \, a^{3} b^{3} e^{2}\right )} x^{6} +{\left (3 \, a^{2} b^{4} d^{2} + 8 \, a^{3} b^{3} d e + 3 \, a^{4} b^{2} e^{2}\right )} x^{5} + \frac{1}{2} \,{\left (10 \, a^{3} b^{3} d^{2} + 15 \, a^{4} b^{2} d e + 3 \, a^{5} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (15 \, a^{4} b^{2} d^{2} + 12 \, a^{5} b d e + a^{6} e^{2}\right )} x^{3} +{\left (3 \, a^{5} b d^{2} + a^{6} d e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.55068, size = 539, normalized size = 8.29 \begin{align*} \frac{1}{9} x^{9} e^{2} b^{6} + \frac{1}{4} x^{8} e d b^{6} + \frac{3}{4} x^{8} e^{2} b^{5} a + \frac{1}{7} x^{7} d^{2} b^{6} + \frac{12}{7} x^{7} e d b^{5} a + \frac{15}{7} x^{7} e^{2} b^{4} a^{2} + x^{6} d^{2} b^{5} a + 5 x^{6} e d b^{4} a^{2} + \frac{10}{3} x^{6} e^{2} b^{3} a^{3} + 3 x^{5} d^{2} b^{4} a^{2} + 8 x^{5} e d b^{3} a^{3} + 3 x^{5} e^{2} b^{2} a^{4} + 5 x^{4} d^{2} b^{3} a^{3} + \frac{15}{2} x^{4} e d b^{2} a^{4} + \frac{3}{2} x^{4} e^{2} b a^{5} + 5 x^{3} d^{2} b^{2} a^{4} + 4 x^{3} e d b a^{5} + \frac{1}{3} x^{3} e^{2} a^{6} + 3 x^{2} d^{2} b a^{5} + x^{2} e d a^{6} + x d^{2} a^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.121035, size = 252, normalized size = 3.88 \begin{align*} a^{6} d^{2} x + \frac{b^{6} e^{2} x^{9}}{9} + x^{8} \left (\frac{3 a b^{5} e^{2}}{4} + \frac{b^{6} d e}{4}\right ) + x^{7} \left (\frac{15 a^{2} b^{4} e^{2}}{7} + \frac{12 a b^{5} d e}{7} + \frac{b^{6} d^{2}}{7}\right ) + x^{6} \left (\frac{10 a^{3} b^{3} e^{2}}{3} + 5 a^{2} b^{4} d e + a b^{5} d^{2}\right ) + x^{5} \left (3 a^{4} b^{2} e^{2} + 8 a^{3} b^{3} d e + 3 a^{2} b^{4} d^{2}\right ) + x^{4} \left (\frac{3 a^{5} b e^{2}}{2} + \frac{15 a^{4} b^{2} d e}{2} + 5 a^{3} b^{3} d^{2}\right ) + x^{3} \left (\frac{a^{6} e^{2}}{3} + 4 a^{5} b d e + 5 a^{4} b^{2} d^{2}\right ) + x^{2} \left (a^{6} d e + 3 a^{5} b d^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.17611, size = 342, normalized size = 5.26 \begin{align*} \frac{1}{9} \, b^{6} x^{9} e^{2} + \frac{1}{4} \, b^{6} d x^{8} e + \frac{1}{7} \, b^{6} d^{2} x^{7} + \frac{3}{4} \, a b^{5} x^{8} e^{2} + \frac{12}{7} \, a b^{5} d x^{7} e + a b^{5} d^{2} x^{6} + \frac{15}{7} \, a^{2} b^{4} x^{7} e^{2} + 5 \, a^{2} b^{4} d x^{6} e + 3 \, a^{2} b^{4} d^{2} x^{5} + \frac{10}{3} \, a^{3} b^{3} x^{6} e^{2} + 8 \, a^{3} b^{3} d x^{5} e + 5 \, a^{3} b^{3} d^{2} x^{4} + 3 \, a^{4} b^{2} x^{5} e^{2} + \frac{15}{2} \, a^{4} b^{2} d x^{4} e + 5 \, a^{4} b^{2} d^{2} x^{3} + \frac{3}{2} \, a^{5} b x^{4} e^{2} + 4 \, a^{5} b d x^{3} e + 3 \, a^{5} b d^{2} x^{2} + \frac{1}{3} \, a^{6} x^{3} e^{2} + a^{6} d x^{2} e + a^{6} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]