Optimal. Leaf size=163 \[ -\frac{b^2 (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac{3 b (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5}-\frac{(d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5}+\frac{(d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5}+\frac{b^3 B (d+e x)^{10}}{10 e^5} \]
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Rubi [A] time = 0.409213, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{b^2 (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac{3 b (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5}-\frac{(d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5}+\frac{(d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5}+\frac{b^3 B (d+e x)^{10}}{10 e^5} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int (a+b x)^3 (A+B x) (d+e x)^5 \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e) (d+e x)^5}{e^4}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^6}{e^4}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^7}{e^4}+\frac{b^2 (-4 b B d+A b e+3 a B e) (d+e x)^8}{e^4}+\frac{b^3 B (d+e x)^9}{e^4}\right ) \, dx\\ &=\frac{(b d-a e)^3 (B d-A e) (d+e x)^6}{6 e^5}-\frac{(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^7}{7 e^5}+\frac{3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^8}{8 e^5}-\frac{b^2 (4 b B d-A b e-3 a B e) (d+e x)^9}{9 e^5}+\frac{b^3 B (d+e x)^{10}}{10 e^5}\\ \end{align*}
Mathematica [B] time = 0.158246, size = 471, normalized size = 2.89 \[ \frac{1}{7} e^2 x^7 \left (3 a^2 b e^2 (A e+5 B d)+a^3 B e^3+15 a b^2 d e (A e+2 B d)+10 b^3 d^2 (A e+B d)\right )+\frac{1}{6} e x^6 \left (15 a^2 b d e^2 (A e+2 B d)+a^3 e^3 (A e+5 B d)+30 a b^2 d^2 e (A e+B d)+5 b^3 d^3 (2 A e+B d)\right )+\frac{1}{5} d x^5 \left (30 a^2 b d e^2 (A e+B d)+5 a^3 e^3 (A e+2 B d)+15 a b^2 d^2 e (2 A e+B d)+b^3 d^3 (5 A e+B d)\right )+\frac{1}{4} d^2 x^4 \left (A \left (30 a^2 b d e^2+10 a^3 e^3+15 a b^2 d^2 e+b^3 d^3\right )+a B d \left (10 a^2 e^2+15 a b d e+3 b^2 d^2\right )\right )+\frac{1}{3} a d^3 x^3 \left (A \left (10 a^2 e^2+15 a b d e+3 b^2 d^2\right )+a B d (5 a e+3 b d)\right )+\frac{1}{8} b e^3 x^8 \left (3 a^2 B e^2+3 a b e (A e+5 B d)+5 b^2 d (A e+2 B d)\right )+\frac{1}{2} a^2 d^4 x^2 (5 a A e+a B d+3 A b d)+a^3 A d^5 x+\frac{1}{9} b^2 e^4 x^9 (3 a B e+A b e+5 b B d)+\frac{1}{10} b^3 B e^5 x^{10} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 529, normalized size = 3.3 \begin{align*}{\frac{{b}^{3}B{e}^{5}{x}^{10}}{10}}+{\frac{ \left ( \left ({b}^{3}A+3\,a{b}^{2}B \right ){e}^{5}+5\,{b}^{3}Bd{e}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){e}^{5}+5\, \left ({b}^{3}A+3\,a{b}^{2}B \right ) d{e}^{4}+10\,{b}^{3}B{d}^{2}{e}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){e}^{5}+5\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ) d{e}^{4}+10\, \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{2}{e}^{3}+10\,{b}^{3}B{d}^{3}{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{3}A{e}^{5}+5\, \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ) d{e}^{4}+10\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{2}{e}^{3}+10\, \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{3}{e}^{2}+5\,{b}^{3}B{d}^{4}e \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,{a}^{3}Ad{e}^{4}+10\, \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){d}^{2}{e}^{3}+10\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{3}{e}^{2}+5\, \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{4}e+{b}^{3}B{d}^{5} \right ){x}^{5}}{5}}+{\frac{ \left ( 10\,{a}^{3}A{d}^{2}{e}^{3}+10\, \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){d}^{3}{e}^{2}+5\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{4}e+ \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{5} \right ){x}^{4}}{4}}+{\frac{ \left ( 10\,{a}^{3}A{d}^{3}{e}^{2}+5\, \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){d}^{4}e+ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{5} \right ){x}^{3}}{3}}+{\frac{ \left ( 5\,{a}^{3}A{d}^{4}e+ \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){d}^{5} \right ){x}^{2}}{2}}+{a}^{3}A{d}^{5}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04126, size = 699, normalized size = 4.29 \begin{align*} \frac{1}{10} \, B b^{3} e^{5} x^{10} + A a^{3} d^{5} x + \frac{1}{9} \,{\left (5 \, B b^{3} d e^{4} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{9} + \frac{1}{8} \,{\left (10 \, B b^{3} d^{2} e^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, B b^{3} d^{3} e^{2} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{4} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (5 \, B b^{3} d^{4} e + A a^{3} e^{5} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{5} + 5 \, A a^{3} d e^{4} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (10 \, A a^{3} d^{2} e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, A a^{3} d^{3} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e\right )} x^{3} + \frac{1}{2} \,{\left (5 \, A a^{3} d^{4} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.57278, size = 1451, normalized size = 8.9 \begin{align*} \frac{1}{10} x^{10} e^{5} b^{3} B + \frac{5}{9} x^{9} e^{4} d b^{3} B + \frac{1}{3} x^{9} e^{5} b^{2} a B + \frac{1}{9} x^{9} e^{5} b^{3} A + \frac{5}{4} x^{8} e^{3} d^{2} b^{3} B + \frac{15}{8} x^{8} e^{4} d b^{2} a B + \frac{3}{8} x^{8} e^{5} b a^{2} B + \frac{5}{8} x^{8} e^{4} d b^{3} A + \frac{3}{8} x^{8} e^{5} b^{2} a A + \frac{10}{7} x^{7} e^{2} d^{3} b^{3} B + \frac{30}{7} x^{7} e^{3} d^{2} b^{2} a B + \frac{15}{7} x^{7} e^{4} d b a^{2} B + \frac{1}{7} x^{7} e^{5} a^{3} B + \frac{10}{7} x^{7} e^{3} d^{2} b^{3} A + \frac{15}{7} x^{7} e^{4} d b^{2} a A + \frac{3}{7} x^{7} e^{5} b a^{2} A + \frac{5}{6} x^{6} e d^{4} b^{3} B + 5 x^{6} e^{2} d^{3} b^{2} a B + 5 x^{6} e^{3} d^{2} b a^{2} B + \frac{5}{6} x^{6} e^{4} d a^{3} B + \frac{5}{3} x^{6} e^{2} d^{3} b^{3} A + 5 x^{6} e^{3} d^{2} b^{2} a A + \frac{5}{2} x^{6} e^{4} d b a^{2} A + \frac{1}{6} x^{6} e^{5} a^{3} A + \frac{1}{5} x^{5} d^{5} b^{3} B + 3 x^{5} e d^{4} b^{2} a B + 6 x^{5} e^{2} d^{3} b a^{2} B + 2 x^{5} e^{3} d^{2} a^{3} B + x^{5} e d^{4} b^{3} A + 6 x^{5} e^{2} d^{3} b^{2} a A + 6 x^{5} e^{3} d^{2} b a^{2} A + x^{5} e^{4} d a^{3} A + \frac{3}{4} x^{4} d^{5} b^{2} a B + \frac{15}{4} x^{4} e d^{4} b a^{2} B + \frac{5}{2} x^{4} e^{2} d^{3} a^{3} B + \frac{1}{4} x^{4} d^{5} b^{3} A + \frac{15}{4} x^{4} e d^{4} b^{2} a A + \frac{15}{2} x^{4} e^{2} d^{3} b a^{2} A + \frac{5}{2} x^{4} e^{3} d^{2} a^{3} A + x^{3} d^{5} b a^{2} B + \frac{5}{3} x^{3} e d^{4} a^{3} B + x^{3} d^{5} b^{2} a A + 5 x^{3} e d^{4} b a^{2} A + \frac{10}{3} x^{3} e^{2} d^{3} a^{3} A + \frac{1}{2} x^{2} d^{5} a^{3} B + \frac{3}{2} x^{2} d^{5} b a^{2} A + \frac{5}{2} x^{2} e d^{4} a^{3} A + x d^{5} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.139182, size = 678, normalized size = 4.16 \begin{align*} A a^{3} d^{5} x + \frac{B b^{3} e^{5} x^{10}}{10} + x^{9} \left (\frac{A b^{3} e^{5}}{9} + \frac{B a b^{2} e^{5}}{3} + \frac{5 B b^{3} d e^{4}}{9}\right ) + x^{8} \left (\frac{3 A a b^{2} e^{5}}{8} + \frac{5 A b^{3} d e^{4}}{8} + \frac{3 B a^{2} b e^{5}}{8} + \frac{15 B a b^{2} d e^{4}}{8} + \frac{5 B b^{3} d^{2} e^{3}}{4}\right ) + x^{7} \left (\frac{3 A a^{2} b e^{5}}{7} + \frac{15 A a b^{2} d e^{4}}{7} + \frac{10 A b^{3} d^{2} e^{3}}{7} + \frac{B a^{3} e^{5}}{7} + \frac{15 B a^{2} b d e^{4}}{7} + \frac{30 B a b^{2} d^{2} e^{3}}{7} + \frac{10 B b^{3} d^{3} e^{2}}{7}\right ) + x^{6} \left (\frac{A a^{3} e^{5}}{6} + \frac{5 A a^{2} b d e^{4}}{2} + 5 A a b^{2} d^{2} e^{3} + \frac{5 A b^{3} d^{3} e^{2}}{3} + \frac{5 B a^{3} d e^{4}}{6} + 5 B a^{2} b d^{2} e^{3} + 5 B a b^{2} d^{3} e^{2} + \frac{5 B b^{3} d^{4} e}{6}\right ) + x^{5} \left (A a^{3} d e^{4} + 6 A a^{2} b d^{2} e^{3} + 6 A a b^{2} d^{3} e^{2} + A b^{3} d^{4} e + 2 B a^{3} d^{2} e^{3} + 6 B a^{2} b d^{3} e^{2} + 3 B a b^{2} d^{4} e + \frac{B b^{3} d^{5}}{5}\right ) + x^{4} \left (\frac{5 A a^{3} d^{2} e^{3}}{2} + \frac{15 A a^{2} b d^{3} e^{2}}{2} + \frac{15 A a b^{2} d^{4} e}{4} + \frac{A b^{3} d^{5}}{4} + \frac{5 B a^{3} d^{3} e^{2}}{2} + \frac{15 B a^{2} b d^{4} e}{4} + \frac{3 B a b^{2} d^{5}}{4}\right ) + x^{3} \left (\frac{10 A a^{3} d^{3} e^{2}}{3} + 5 A a^{2} b d^{4} e + A a b^{2} d^{5} + \frac{5 B a^{3} d^{4} e}{3} + B a^{2} b d^{5}\right ) + x^{2} \left (\frac{5 A a^{3} d^{4} e}{2} + \frac{3 A a^{2} b d^{5}}{2} + \frac{B a^{3} d^{5}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.10309, size = 856, normalized size = 5.25 \begin{align*} \frac{1}{10} \, B b^{3} x^{10} e^{5} + \frac{5}{9} \, B b^{3} d x^{9} e^{4} + \frac{5}{4} \, B b^{3} d^{2} x^{8} e^{3} + \frac{10}{7} \, B b^{3} d^{3} x^{7} e^{2} + \frac{5}{6} \, B b^{3} d^{4} x^{6} e + \frac{1}{5} \, B b^{3} d^{5} x^{5} + \frac{1}{3} \, B a b^{2} x^{9} e^{5} + \frac{1}{9} \, A b^{3} x^{9} e^{5} + \frac{15}{8} \, B a b^{2} d x^{8} e^{4} + \frac{5}{8} \, A b^{3} d x^{8} e^{4} + \frac{30}{7} \, B a b^{2} d^{2} x^{7} e^{3} + \frac{10}{7} \, A b^{3} d^{2} x^{7} e^{3} + 5 \, B a b^{2} d^{3} x^{6} e^{2} + \frac{5}{3} \, A b^{3} d^{3} x^{6} e^{2} + 3 \, B a b^{2} d^{4} x^{5} e + A b^{3} d^{4} x^{5} e + \frac{3}{4} \, B a b^{2} d^{5} x^{4} + \frac{1}{4} \, A b^{3} d^{5} x^{4} + \frac{3}{8} \, B a^{2} b x^{8} e^{5} + \frac{3}{8} \, A a b^{2} x^{8} e^{5} + \frac{15}{7} \, B a^{2} b d x^{7} e^{4} + \frac{15}{7} \, A a b^{2} d x^{7} e^{4} + 5 \, B a^{2} b d^{2} x^{6} e^{3} + 5 \, A a b^{2} d^{2} x^{6} e^{3} + 6 \, B a^{2} b d^{3} x^{5} e^{2} + 6 \, A a b^{2} d^{3} x^{5} e^{2} + \frac{15}{4} \, B a^{2} b d^{4} x^{4} e + \frac{15}{4} \, A a b^{2} d^{4} x^{4} e + B a^{2} b d^{5} x^{3} + A a b^{2} d^{5} x^{3} + \frac{1}{7} \, B a^{3} x^{7} e^{5} + \frac{3}{7} \, A a^{2} b x^{7} e^{5} + \frac{5}{6} \, B a^{3} d x^{6} e^{4} + \frac{5}{2} \, A a^{2} b d x^{6} e^{4} + 2 \, B a^{3} d^{2} x^{5} e^{3} + 6 \, A a^{2} b d^{2} x^{5} e^{3} + \frac{5}{2} \, B a^{3} d^{3} x^{4} e^{2} + \frac{15}{2} \, A a^{2} b d^{3} x^{4} e^{2} + \frac{5}{3} \, B a^{3} d^{4} x^{3} e + 5 \, A a^{2} b d^{4} x^{3} e + \frac{1}{2} \, B a^{3} d^{5} x^{2} + \frac{3}{2} \, A a^{2} b d^{5} x^{2} + \frac{1}{6} \, A a^{3} x^{6} e^{5} + A a^{3} d x^{5} e^{4} + \frac{5}{2} \, A a^{3} d^{2} x^{4} e^{3} + \frac{10}{3} \, A a^{3} d^{3} x^{3} e^{2} + \frac{5}{2} \, A a^{3} d^{4} x^{2} e + A a^{3} d^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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