Optimal. Leaf size=65 \[ \frac{2 e (a+b x)^9 (b d-a e)}{9 b^3}+\frac{(a+b x)^8 (b d-a e)^2}{8 b^3}+\frac{e^2 (a+b x)^{10}}{10 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.154045, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ \frac{2 e (a+b x)^9 (b d-a e)}{9 b^3}+\frac{(a+b x)^8 (b d-a e)^2}{8 b^3}+\frac{e^2 (a+b x)^{10}}{10 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^2 \, dx\\ &=\int \left (\frac{(b d-a e)^2 (a+b x)^7}{b^2}+\frac{2 e (b d-a e) (a+b x)^8}{b^2}+\frac{e^2 (a+b x)^9}{b^2}\right ) \, dx\\ &=\frac{(b d-a e)^2 (a+b x)^8}{8 b^3}+\frac{2 e (b d-a e) (a+b x)^9}{9 b^3}+\frac{e^2 (a+b x)^{10}}{10 b^3}\\ \end{align*}
Mathematica [B] time = 0.0757656, size = 229, normalized size = 3.52 \[ \frac{1}{360} x \left (252 a^5 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+210 a^4 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+120 a^3 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+45 a^2 b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+210 a^6 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+120 a^7 \left (3 d^2+3 d e x+e^2 x^2\right )+10 a b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )+b^7 x^7 \left (45 d^2+80 d e x+36 e^2 x^2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.002, size = 433, normalized size = 6.7 \begin{align*}{\frac{{b}^{7}{e}^{2}{x}^{10}}{10}}+{\frac{ \left ( \left ( a{e}^{2}+2\,bde \right ){b}^{6}+6\,{b}^{6}{e}^{2}a \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 2\,ade+b{d}^{2} \right ){b}^{6}+6\, \left ( a{e}^{2}+2\,bde \right ) a{b}^{5}+15\,{b}^{5}{e}^{2}{a}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( a{d}^{2}{b}^{6}+6\, \left ( 2\,ade+b{d}^{2} \right ) a{b}^{5}+15\, \left ( a{e}^{2}+2\,bde \right ){a}^{2}{b}^{4}+20\,{b}^{4}{e}^{2}{a}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{a}^{2}{d}^{2}{b}^{5}+15\, \left ( 2\,ade+b{d}^{2} \right ){a}^{2}{b}^{4}+20\, \left ( a{e}^{2}+2\,bde \right ){a}^{3}{b}^{3}+15\,{b}^{3}{e}^{2}{a}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{a}^{3}{d}^{2}{b}^{4}+20\, \left ( 2\,ade+b{d}^{2} \right ){a}^{3}{b}^{3}+15\, \left ( a{e}^{2}+2\,bde \right ){a}^{4}{b}^{2}+6\,{b}^{2}{e}^{2}{a}^{5} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,{a}^{4}{d}^{2}{b}^{3}+15\, \left ( 2\,ade+b{d}^{2} \right ){a}^{4}{b}^{2}+6\, \left ( a{e}^{2}+2\,bde \right ){a}^{5}b+b{e}^{2}{a}^{6} \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,{a}^{5}{d}^{2}{b}^{2}+6\, \left ( 2\,ade+b{d}^{2} \right ){a}^{5}b+ \left ( a{e}^{2}+2\,bde \right ){a}^{6} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{a}^{6}{d}^{2}b+ \left ( 2\,ade+b{d}^{2} \right ){a}^{6} \right ){x}^{2}}{2}}+{a}^{7}{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.985894, size = 369, normalized size = 5.68 \begin{align*} \frac{1}{10} \, b^{7} e^{2} x^{10} + a^{7} d^{2} x + \frac{1}{9} \,{\left (2 \, b^{7} d e + 7 \, a b^{6} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (b^{7} d^{2} + 14 \, a b^{6} d e + 21 \, a^{2} b^{5} e^{2}\right )} x^{8} +{\left (a b^{6} d^{2} + 6 \, a^{2} b^{5} d e + 5 \, a^{3} b^{4} e^{2}\right )} x^{7} + \frac{7}{6} \,{\left (3 \, a^{2} b^{5} d^{2} + 10 \, a^{3} b^{4} d e + 5 \, a^{4} b^{3} e^{2}\right )} x^{6} + \frac{7}{5} \,{\left (5 \, a^{3} b^{4} d^{2} + 10 \, a^{4} b^{3} d e + 3 \, a^{5} b^{2} e^{2}\right )} x^{5} + \frac{7}{4} \,{\left (5 \, a^{4} b^{3} d^{2} + 6 \, a^{5} b^{2} d e + a^{6} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (21 \, a^{5} b^{2} d^{2} + 14 \, a^{6} b d e + a^{7} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (7 \, a^{6} b d^{2} + 2 \, a^{7} d e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.31513, size = 644, normalized size = 9.91 \begin{align*} \frac{1}{10} x^{10} e^{2} b^{7} + \frac{2}{9} x^{9} e d b^{7} + \frac{7}{9} x^{9} e^{2} b^{6} a + \frac{1}{8} x^{8} d^{2} b^{7} + \frac{7}{4} x^{8} e d b^{6} a + \frac{21}{8} x^{8} e^{2} b^{5} a^{2} + x^{7} d^{2} b^{6} a + 6 x^{7} e d b^{5} a^{2} + 5 x^{7} e^{2} b^{4} a^{3} + \frac{7}{2} x^{6} d^{2} b^{5} a^{2} + \frac{35}{3} x^{6} e d b^{4} a^{3} + \frac{35}{6} x^{6} e^{2} b^{3} a^{4} + 7 x^{5} d^{2} b^{4} a^{3} + 14 x^{5} e d b^{3} a^{4} + \frac{21}{5} x^{5} e^{2} b^{2} a^{5} + \frac{35}{4} x^{4} d^{2} b^{3} a^{4} + \frac{21}{2} x^{4} e d b^{2} a^{5} + \frac{7}{4} x^{4} e^{2} b a^{6} + 7 x^{3} d^{2} b^{2} a^{5} + \frac{14}{3} x^{3} e d b a^{6} + \frac{1}{3} x^{3} e^{2} a^{7} + \frac{7}{2} x^{2} d^{2} b a^{6} + x^{2} e d a^{7} + x d^{2} a^{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.110157, size = 303, normalized size = 4.66 \begin{align*} a^{7} d^{2} x + \frac{b^{7} e^{2} x^{10}}{10} + x^{9} \left (\frac{7 a b^{6} e^{2}}{9} + \frac{2 b^{7} d e}{9}\right ) + x^{8} \left (\frac{21 a^{2} b^{5} e^{2}}{8} + \frac{7 a b^{6} d e}{4} + \frac{b^{7} d^{2}}{8}\right ) + x^{7} \left (5 a^{3} b^{4} e^{2} + 6 a^{2} b^{5} d e + a b^{6} d^{2}\right ) + x^{6} \left (\frac{35 a^{4} b^{3} e^{2}}{6} + \frac{35 a^{3} b^{4} d e}{3} + \frac{7 a^{2} b^{5} d^{2}}{2}\right ) + x^{5} \left (\frac{21 a^{5} b^{2} e^{2}}{5} + 14 a^{4} b^{3} d e + 7 a^{3} b^{4} d^{2}\right ) + x^{4} \left (\frac{7 a^{6} b e^{2}}{4} + \frac{21 a^{5} b^{2} d e}{2} + \frac{35 a^{4} b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac{a^{7} e^{2}}{3} + \frac{14 a^{6} b d e}{3} + 7 a^{5} b^{2} d^{2}\right ) + x^{2} \left (a^{7} d e + \frac{7 a^{6} b d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.12323, size = 397, normalized size = 6.11 \begin{align*} \frac{1}{10} \, b^{7} x^{10} e^{2} + \frac{2}{9} \, b^{7} d x^{9} e + \frac{1}{8} \, b^{7} d^{2} x^{8} + \frac{7}{9} \, a b^{6} x^{9} e^{2} + \frac{7}{4} \, a b^{6} d x^{8} e + a b^{6} d^{2} x^{7} + \frac{21}{8} \, a^{2} b^{5} x^{8} e^{2} + 6 \, a^{2} b^{5} d x^{7} e + \frac{7}{2} \, a^{2} b^{5} d^{2} x^{6} + 5 \, a^{3} b^{4} x^{7} e^{2} + \frac{35}{3} \, a^{3} b^{4} d x^{6} e + 7 \, a^{3} b^{4} d^{2} x^{5} + \frac{35}{6} \, a^{4} b^{3} x^{6} e^{2} + 14 \, a^{4} b^{3} d x^{5} e + \frac{35}{4} \, a^{4} b^{3} d^{2} x^{4} + \frac{21}{5} \, a^{5} b^{2} x^{5} e^{2} + \frac{21}{2} \, a^{5} b^{2} d x^{4} e + 7 \, a^{5} b^{2} d^{2} x^{3} + \frac{7}{4} \, a^{6} b x^{4} e^{2} + \frac{14}{3} \, a^{6} b d x^{3} e + \frac{7}{2} \, a^{6} b d^{2} x^{2} + \frac{1}{3} \, a^{7} x^{3} e^{2} + a^{7} d x^{2} e + a^{7} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]