Optimal. Leaf size=159 \[ \frac{e^2 (a+b x)^7 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac{e (a+b x)^6 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac{(a+b x)^5 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac{(a+b x)^4 (A b-a B) (b d-a e)^3}{4 b^5}+\frac{B e^3 (a+b x)^8}{8 b^5} \]
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Rubi [A] time = 0.248493, antiderivative size = 159, 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} \[ \frac{e^2 (a+b x)^7 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac{e (a+b x)^6 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac{(a+b x)^5 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac{(a+b x)^4 (A b-a B) (b d-a e)^3}{4 b^5}+\frac{B e^3 (a+b x)^8}{8 b^5} \]
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)^3 \, dx &=\int \left (\frac{(A b-a B) (b d-a e)^3 (a+b x)^3}{b^4}+\frac{(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^4}{b^4}+\frac{3 e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^5}{b^4}+\frac{e^2 (3 b B d+A b e-4 a B e) (a+b x)^6}{b^4}+\frac{B e^3 (a+b x)^7}{b^4}\right ) \, dx\\ &=\frac{(A b-a B) (b d-a e)^3 (a+b x)^4}{4 b^5}+\frac{(b d-a e)^2 (b B d+3 A b e-4 a B e) (a+b x)^5}{5 b^5}+\frac{e (b d-a e) (b B d+A b e-2 a B e) (a+b x)^6}{2 b^5}+\frac{e^2 (3 b B d+A b e-4 a B e) (a+b x)^7}{7 b^5}+\frac{B e^3 (a+b x)^8}{8 b^5}\\ \end{align*}
Mathematica [A] time = 0.101098, size = 297, normalized size = 1.87 \[ \frac{1}{5} x^5 \left (3 a^2 b e^2 (A e+3 B d)+a^3 B e^3+9 a b^2 d e (A e+B d)+b^3 d^2 (3 A e+B d)\right )+\frac{1}{4} x^4 \left (A \left (9 a^2 b d e^2+a^3 e^3+9 a b^2 d^2 e+b^3 d^3\right )+3 a B d \left (a^2 e^2+3 a b d e+b^2 d^2\right )\right )+a d x^3 \left (A \left (a^2 e^2+3 a b d e+b^2 d^2\right )+a B d (a e+b d)\right )+\frac{1}{2} b e x^6 \left (a^2 B e^2+a b e (A e+3 B d)+b^2 d (A e+B d)\right )+\frac{1}{2} a^2 d^2 x^2 (3 A (a e+b d)+a B d)+a^3 A d^3 x+\frac{1}{7} b^2 e^2 x^7 (3 a B e+A b e+3 b B d)+\frac{1}{8} b^3 B e^3 x^8 \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 339, normalized size = 2.1 \begin{align*}{\frac{{b}^{3}B{e}^{3}{x}^{8}}{8}}+{\frac{ \left ( \left ({b}^{3}A+3\,a{b}^{2}B \right ){e}^{3}+3\,{b}^{3}Bd{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){e}^{3}+3\, \left ({b}^{3}A+3\,a{b}^{2}B \right ) d{e}^{2}+3\,{b}^{3}B{d}^{2}e \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){e}^{3}+3\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ) d{e}^{2}+3\, \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{2}e+{b}^{3}B{d}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{3}A{e}^{3}+3\, \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ) d{e}^{2}+3\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{2}e+ \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{3}Ad{e}^{2}+3\, \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){d}^{2}e+ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{3}A{d}^{2}e+ \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){d}^{3} \right ){x}^{2}}{2}}+{a}^{3}A{d}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22377, size = 439, normalized size = 2.76 \begin{align*} \frac{1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac{1}{7} \,{\left (3 \, B b^{3} d e^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (B b^{3} d^{2} e +{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} +{\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (A a^{3} e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} +{\left (A a^{3} d e^{2} +{\left (B a^{2} b + A a b^{2}\right )} d^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a^{3} d^{2} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75621, size = 917, normalized size = 5.77 \begin{align*} \frac{1}{8} x^{8} e^{3} b^{3} B + \frac{3}{7} x^{7} e^{2} d b^{3} B + \frac{3}{7} x^{7} e^{3} b^{2} a B + \frac{1}{7} x^{7} e^{3} b^{3} A + \frac{1}{2} x^{6} e d^{2} b^{3} B + \frac{3}{2} x^{6} e^{2} d b^{2} a B + \frac{1}{2} x^{6} e^{3} b a^{2} B + \frac{1}{2} x^{6} e^{2} d b^{3} A + \frac{1}{2} x^{6} e^{3} b^{2} a A + \frac{1}{5} x^{5} d^{3} b^{3} B + \frac{9}{5} x^{5} e d^{2} b^{2} a B + \frac{9}{5} x^{5} e^{2} d b a^{2} B + \frac{1}{5} x^{5} e^{3} a^{3} B + \frac{3}{5} x^{5} e d^{2} b^{3} A + \frac{9}{5} x^{5} e^{2} d b^{2} a A + \frac{3}{5} x^{5} e^{3} b a^{2} A + \frac{3}{4} x^{4} d^{3} b^{2} a B + \frac{9}{4} x^{4} e d^{2} b a^{2} B + \frac{3}{4} x^{4} e^{2} d a^{3} B + \frac{1}{4} x^{4} d^{3} b^{3} A + \frac{9}{4} x^{4} e d^{2} b^{2} a A + \frac{9}{4} x^{4} e^{2} d b a^{2} A + \frac{1}{4} x^{4} e^{3} a^{3} A + x^{3} d^{3} b a^{2} B + x^{3} e d^{2} a^{3} B + x^{3} d^{3} b^{2} a A + 3 x^{3} e d^{2} b a^{2} A + x^{3} e^{2} d a^{3} A + \frac{1}{2} x^{2} d^{3} a^{3} B + \frac{3}{2} x^{2} d^{3} b a^{2} A + \frac{3}{2} x^{2} e d^{2} a^{3} A + x d^{3} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.147295, size = 422, normalized size = 2.65 \begin{align*} A a^{3} d^{3} x + \frac{B b^{3} e^{3} x^{8}}{8} + x^{7} \left (\frac{A b^{3} e^{3}}{7} + \frac{3 B a b^{2} e^{3}}{7} + \frac{3 B b^{3} d e^{2}}{7}\right ) + x^{6} \left (\frac{A a b^{2} e^{3}}{2} + \frac{A b^{3} d e^{2}}{2} + \frac{B a^{2} b e^{3}}{2} + \frac{3 B a b^{2} d e^{2}}{2} + \frac{B b^{3} d^{2} e}{2}\right ) + x^{5} \left (\frac{3 A a^{2} b e^{3}}{5} + \frac{9 A a b^{2} d e^{2}}{5} + \frac{3 A b^{3} d^{2} e}{5} + \frac{B a^{3} e^{3}}{5} + \frac{9 B a^{2} b d e^{2}}{5} + \frac{9 B a b^{2} d^{2} e}{5} + \frac{B b^{3} d^{3}}{5}\right ) + x^{4} \left (\frac{A a^{3} e^{3}}{4} + \frac{9 A a^{2} b d e^{2}}{4} + \frac{9 A a b^{2} d^{2} e}{4} + \frac{A b^{3} d^{3}}{4} + \frac{3 B a^{3} d e^{2}}{4} + \frac{9 B a^{2} b d^{2} e}{4} + \frac{3 B a b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{3} d e^{2} + 3 A a^{2} b d^{2} e + A a b^{2} d^{3} + B a^{3} d^{2} e + B a^{2} b d^{3}\right ) + x^{2} \left (\frac{3 A a^{3} d^{2} e}{2} + \frac{3 A a^{2} b d^{3}}{2} + \frac{B a^{3} d^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.12407, size = 543, normalized size = 3.42 \begin{align*} \frac{1}{8} \, B b^{3} x^{8} e^{3} + \frac{3}{7} \, B b^{3} d x^{7} e^{2} + \frac{1}{2} \, B b^{3} d^{2} x^{6} e + \frac{1}{5} \, B b^{3} d^{3} x^{5} + \frac{3}{7} \, B a b^{2} x^{7} e^{3} + \frac{1}{7} \, A b^{3} x^{7} e^{3} + \frac{3}{2} \, B a b^{2} d x^{6} e^{2} + \frac{1}{2} \, A b^{3} d x^{6} e^{2} + \frac{9}{5} \, B a b^{2} d^{2} x^{5} e + \frac{3}{5} \, A b^{3} d^{2} x^{5} e + \frac{3}{4} \, B a b^{2} d^{3} x^{4} + \frac{1}{4} \, A b^{3} d^{3} x^{4} + \frac{1}{2} \, B a^{2} b x^{6} e^{3} + \frac{1}{2} \, A a b^{2} x^{6} e^{3} + \frac{9}{5} \, B a^{2} b d x^{5} e^{2} + \frac{9}{5} \, A a b^{2} d x^{5} e^{2} + \frac{9}{4} \, B a^{2} b d^{2} x^{4} e + \frac{9}{4} \, A a b^{2} d^{2} x^{4} e + B a^{2} b d^{3} x^{3} + A a b^{2} d^{3} x^{3} + \frac{1}{5} \, B a^{3} x^{5} e^{3} + \frac{3}{5} \, A a^{2} b x^{5} e^{3} + \frac{3}{4} \, B a^{3} d x^{4} e^{2} + \frac{9}{4} \, A a^{2} b d x^{4} e^{2} + B a^{3} d^{2} x^{3} e + 3 \, A a^{2} b d^{2} x^{3} e + \frac{1}{2} \, B a^{3} d^{3} x^{2} + \frac{3}{2} \, A a^{2} b d^{3} x^{2} + \frac{1}{4} \, A a^{3} x^{4} e^{3} + A a^{3} d x^{3} e^{2} + \frac{3}{2} \, A a^{3} d^{2} x^{2} e + A a^{3} d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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