Optimal. Leaf size=119 \[ -\frac{4 b^3 (d+e x)^9 (b d-a e)}{9 e^5}+\frac{3 b^2 (d+e x)^8 (b d-a e)^2}{4 e^5}-\frac{4 b (d+e x)^7 (b d-a e)^3}{7 e^5}+\frac{(d+e x)^6 (b d-a e)^4}{6 e^5}+\frac{b^4 (d+e x)^{10}}{10 e^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.208124, antiderivative size = 119, 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} \[ -\frac{4 b^3 (d+e x)^9 (b d-a e)}{9 e^5}+\frac{3 b^2 (d+e x)^8 (b d-a e)^2}{4 e^5}-\frac{4 b (d+e x)^7 (b d-a e)^3}{7 e^5}+\frac{(d+e x)^6 (b d-a e)^4}{6 e^5}+\frac{b^4 (d+e x)^{10}}{10 e^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^5 \, dx\\ &=\int \left (\frac{(-b d+a e)^4 (d+e x)^5}{e^4}-\frac{4 b (b d-a e)^3 (d+e x)^6}{e^4}+\frac{6 b^2 (b d-a e)^2 (d+e x)^7}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^8}{e^4}+\frac{b^4 (d+e x)^9}{e^4}\right ) \, dx\\ &=\frac{(b d-a e)^4 (d+e x)^6}{6 e^5}-\frac{4 b (b d-a e)^3 (d+e x)^7}{7 e^5}+\frac{3 b^2 (b d-a e)^2 (d+e x)^8}{4 e^5}-\frac{4 b^3 (b d-a e) (d+e x)^9}{9 e^5}+\frac{b^4 (d+e x)^{10}}{10 e^5}\\ \end{align*}
Mathematica [B] time = 0.048776, size = 350, normalized size = 2.94 \[ \frac{1}{4} b^2 e^3 x^8 \left (3 a^2 e^2+10 a b d e+5 b^2 d^2\right )+\frac{2}{7} b e^2 x^7 \left (15 a^2 b d e^2+2 a^3 e^3+20 a b^2 d^2 e+5 b^3 d^3\right )+\frac{1}{6} e x^6 \left (60 a^2 b^2 d^2 e^2+20 a^3 b d e^3+a^4 e^4+40 a b^3 d^3 e+5 b^4 d^4\right )+\frac{1}{5} d x^5 \left (60 a^2 b^2 d^2 e^2+40 a^3 b d e^3+5 a^4 e^4+20 a b^3 d^3 e+b^4 d^4\right )+\frac{1}{2} a d^2 x^4 \left (20 a^2 b d e^2+5 a^3 e^3+15 a b^2 d^2 e+2 b^3 d^3\right )+\frac{2}{3} a^2 d^3 x^3 \left (5 a^2 e^2+10 a b d e+3 b^2 d^2\right )+\frac{1}{2} a^3 d^4 x^2 (5 a e+4 b d)+a^4 d^5 x+\frac{1}{9} b^3 e^4 x^9 (4 a e+5 b d)+\frac{1}{10} b^4 e^5 x^{10} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.041, size = 361, normalized size = 3. \begin{align*}{\frac{{e}^{5}{b}^{4}{x}^{10}}{10}}+{\frac{ \left ( 4\,a{b}^{3}{e}^{5}+5\,d{e}^{4}{b}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 6\,{e}^{5}{b}^{2}{a}^{2}+20\,d{e}^{4}a{b}^{3}+10\,{d}^{2}{e}^{3}{b}^{4} \right ){x}^{8}}{8}}+{\frac{ \left ( 4\,{e}^{5}{a}^{3}b+30\,d{e}^{4}{b}^{2}{a}^{2}+40\,{d}^{2}{e}^{3}a{b}^{3}+10\,{d}^{3}{e}^{2}{b}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ({e}^{5}{a}^{4}+20\,d{e}^{4}{a}^{3}b+60\,{d}^{2}{e}^{3}{b}^{2}{a}^{2}+40\,{d}^{3}{e}^{2}a{b}^{3}+5\,{d}^{4}e{b}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,d{e}^{4}{a}^{4}+40\,{d}^{2}{e}^{3}{a}^{3}b+60\,{d}^{3}{e}^{2}{b}^{2}{a}^{2}+20\,{d}^{4}ea{b}^{3}+{d}^{5}{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{d}^{2}{e}^{3}{a}^{4}+40\,{d}^{3}{e}^{2}{a}^{3}b+30\,{d}^{4}e{b}^{2}{a}^{2}+4\,{d}^{5}a{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,{d}^{3}{e}^{2}{a}^{4}+20\,{d}^{4}e{a}^{3}b+6\,{d}^{5}{b}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 5\,{d}^{4}e{a}^{4}+4\,{d}^{5}{a}^{3}b \right ){x}^{2}}{2}}+{d}^{5}{a}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.14052, size = 486, normalized size = 4.08 \begin{align*} \frac{1}{10} \, b^{4} e^{5} x^{10} + a^{4} d^{5} x + \frac{1}{9} \,{\left (5 \, b^{4} d e^{4} + 4 \, a b^{3} e^{5}\right )} x^{9} + \frac{1}{4} \,{\left (5 \, b^{4} d^{2} e^{3} + 10 \, a b^{3} d e^{4} + 3 \, a^{2} b^{2} e^{5}\right )} x^{8} + \frac{2}{7} \,{\left (5 \, b^{4} d^{3} e^{2} + 20 \, a b^{3} d^{2} e^{3} + 15 \, a^{2} b^{2} d e^{4} + 2 \, a^{3} b e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (5 \, b^{4} d^{4} e + 40 \, a b^{3} d^{3} e^{2} + 60 \, a^{2} b^{2} d^{2} e^{3} + 20 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} d^{5} + 20 \, a b^{3} d^{4} e + 60 \, a^{2} b^{2} d^{3} e^{2} + 40 \, a^{3} b d^{2} e^{3} + 5 \, a^{4} d e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, a b^{3} d^{5} + 15 \, a^{2} b^{2} d^{4} e + 20 \, a^{3} b d^{3} e^{2} + 5 \, a^{4} d^{2} e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} d^{5} + 10 \, a^{3} b d^{4} e + 5 \, a^{4} d^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d^{5} + 5 \, a^{4} d^{4} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.48742, size = 853, normalized size = 7.17 \begin{align*} \frac{1}{10} x^{10} e^{5} b^{4} + \frac{5}{9} x^{9} e^{4} d b^{4} + \frac{4}{9} x^{9} e^{5} b^{3} a + \frac{5}{4} x^{8} e^{3} d^{2} b^{4} + \frac{5}{2} x^{8} e^{4} d b^{3} a + \frac{3}{4} x^{8} e^{5} b^{2} a^{2} + \frac{10}{7} x^{7} e^{2} d^{3} b^{4} + \frac{40}{7} x^{7} e^{3} d^{2} b^{3} a + \frac{30}{7} x^{7} e^{4} d b^{2} a^{2} + \frac{4}{7} x^{7} e^{5} b a^{3} + \frac{5}{6} x^{6} e d^{4} b^{4} + \frac{20}{3} x^{6} e^{2} d^{3} b^{3} a + 10 x^{6} e^{3} d^{2} b^{2} a^{2} + \frac{10}{3} x^{6} e^{4} d b a^{3} + \frac{1}{6} x^{6} e^{5} a^{4} + \frac{1}{5} x^{5} d^{5} b^{4} + 4 x^{5} e d^{4} b^{3} a + 12 x^{5} e^{2} d^{3} b^{2} a^{2} + 8 x^{5} e^{3} d^{2} b a^{3} + x^{5} e^{4} d a^{4} + x^{4} d^{5} b^{3} a + \frac{15}{2} x^{4} e d^{4} b^{2} a^{2} + 10 x^{4} e^{2} d^{3} b a^{3} + \frac{5}{2} x^{4} e^{3} d^{2} a^{4} + 2 x^{3} d^{5} b^{2} a^{2} + \frac{20}{3} x^{3} e d^{4} b a^{3} + \frac{10}{3} x^{3} e^{2} d^{3} a^{4} + 2 x^{2} d^{5} b a^{3} + \frac{5}{2} x^{2} e d^{4} a^{4} + x d^{5} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.120092, size = 401, normalized size = 3.37 \begin{align*} a^{4} d^{5} x + \frac{b^{4} e^{5} x^{10}}{10} + x^{9} \left (\frac{4 a b^{3} e^{5}}{9} + \frac{5 b^{4} d e^{4}}{9}\right ) + x^{8} \left (\frac{3 a^{2} b^{2} e^{5}}{4} + \frac{5 a b^{3} d e^{4}}{2} + \frac{5 b^{4} d^{2} e^{3}}{4}\right ) + x^{7} \left (\frac{4 a^{3} b e^{5}}{7} + \frac{30 a^{2} b^{2} d e^{4}}{7} + \frac{40 a b^{3} d^{2} e^{3}}{7} + \frac{10 b^{4} d^{3} e^{2}}{7}\right ) + x^{6} \left (\frac{a^{4} e^{5}}{6} + \frac{10 a^{3} b d e^{4}}{3} + 10 a^{2} b^{2} d^{2} e^{3} + \frac{20 a b^{3} d^{3} e^{2}}{3} + \frac{5 b^{4} d^{4} e}{6}\right ) + x^{5} \left (a^{4} d e^{4} + 8 a^{3} b d^{2} e^{3} + 12 a^{2} b^{2} d^{3} e^{2} + 4 a b^{3} d^{4} e + \frac{b^{4} d^{5}}{5}\right ) + x^{4} \left (\frac{5 a^{4} d^{2} e^{3}}{2} + 10 a^{3} b d^{3} e^{2} + \frac{15 a^{2} b^{2} d^{4} e}{2} + a b^{3} d^{5}\right ) + x^{3} \left (\frac{10 a^{4} d^{3} e^{2}}{3} + \frac{20 a^{3} b d^{4} e}{3} + 2 a^{2} b^{2} d^{5}\right ) + x^{2} \left (\frac{5 a^{4} d^{4} e}{2} + 2 a^{3} b d^{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.10441, size = 514, normalized size = 4.32 \begin{align*} \frac{1}{10} \, b^{4} x^{10} e^{5} + \frac{5}{9} \, b^{4} d x^{9} e^{4} + \frac{5}{4} \, b^{4} d^{2} x^{8} e^{3} + \frac{10}{7} \, b^{4} d^{3} x^{7} e^{2} + \frac{5}{6} \, b^{4} d^{4} x^{6} e + \frac{1}{5} \, b^{4} d^{5} x^{5} + \frac{4}{9} \, a b^{3} x^{9} e^{5} + \frac{5}{2} \, a b^{3} d x^{8} e^{4} + \frac{40}{7} \, a b^{3} d^{2} x^{7} e^{3} + \frac{20}{3} \, a b^{3} d^{3} x^{6} e^{2} + 4 \, a b^{3} d^{4} x^{5} e + a b^{3} d^{5} x^{4} + \frac{3}{4} \, a^{2} b^{2} x^{8} e^{5} + \frac{30}{7} \, a^{2} b^{2} d x^{7} e^{4} + 10 \, a^{2} b^{2} d^{2} x^{6} e^{3} + 12 \, a^{2} b^{2} d^{3} x^{5} e^{2} + \frac{15}{2} \, a^{2} b^{2} d^{4} x^{4} e + 2 \, a^{2} b^{2} d^{5} x^{3} + \frac{4}{7} \, a^{3} b x^{7} e^{5} + \frac{10}{3} \, a^{3} b d x^{6} e^{4} + 8 \, a^{3} b d^{2} x^{5} e^{3} + 10 \, a^{3} b d^{3} x^{4} e^{2} + \frac{20}{3} \, a^{3} b d^{4} x^{3} e + 2 \, a^{3} b d^{5} x^{2} + \frac{1}{6} \, a^{4} x^{6} e^{5} + a^{4} d x^{5} e^{4} + \frac{5}{2} \, a^{4} d^{2} x^{4} e^{3} + \frac{10}{3} \, a^{4} d^{3} x^{3} e^{2} + \frac{5}{2} \, a^{4} d^{4} x^{2} e + a^{4} d^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]