Optimal. Leaf size=163 \[ -\frac{b^2 (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5}+\frac{3 b (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac{(d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5}+\frac{(d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5}+\frac{b^3 B (d+e x)^9}{9 e^5} \]
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Rubi [A] time = 0.291044, 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} \[ -\frac{b^2 (d+e x)^8 (-3 a B e-A b e+4 b B d)}{8 e^5}+\frac{3 b (d+e x)^7 (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac{(d+e x)^6 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5}+\frac{(d+e x)^5 (b d-a e)^3 (B d-A e)}{5 e^5}+\frac{b^3 B (d+e x)^9}{9 e^5} \]
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)^4 \, dx &=\int \left (\frac{(-b d+a e)^3 (-B d+A e) (d+e x)^4}{e^4}+\frac{(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^5}{e^4}-\frac{3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^6}{e^4}+\frac{b^2 (-4 b B d+A b e+3 a B e) (d+e x)^7}{e^4}+\frac{b^3 B (d+e x)^8}{e^4}\right ) \, dx\\ &=\frac{(b d-a e)^3 (B d-A e) (d+e x)^5}{5 e^5}-\frac{(b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^6}{6 e^5}+\frac{3 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^7}{7 e^5}-\frac{b^2 (4 b B d-A b e-3 a B e) (d+e x)^8}{8 e^5}+\frac{b^3 B (d+e x)^9}{9 e^5}\\ \end{align*}
Mathematica [B] time = 0.129816, size = 397, normalized size = 2.44 \[ \frac{1}{6} e x^6 \left (3 a^2 b e^2 (A e+4 B d)+a^3 B e^3+6 a b^2 d e (2 A e+3 B d)+2 b^3 d^2 (3 A e+2 B d)\right )+\frac{1}{5} x^5 \left (6 a^2 b d e^2 (2 A e+3 B d)+a^3 e^3 (A e+4 B d)+6 a b^2 d^2 e (3 A e+2 B d)+b^3 d^3 (4 A e+B d)\right )+\frac{1}{4} d x^4 \left (A \left (18 a^2 b d e^2+4 a^3 e^3+12 a b^2 d^2 e+b^3 d^3\right )+3 a B d \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )\right )+\frac{1}{3} a d^2 x^3 \left (3 A \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+a B d (4 a e+3 b d)\right )+\frac{1}{7} b e^2 x^7 \left (3 a^2 B e^2+3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+\frac{1}{2} a^2 d^3 x^2 (4 a A e+a B d+3 A b d)+a^3 A d^4 x+\frac{1}{8} b^2 e^3 x^8 (3 a B e+A b e+4 b B d)+\frac{1}{9} b^3 B e^4 x^9 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 434, normalized size = 2.7 \begin{align*}{\frac{{b}^{3}B{e}^{4}{x}^{9}}{9}}+{\frac{ \left ( \left ({b}^{3}A+3\,a{b}^{2}B \right ){e}^{4}+4\,{b}^{3}Bd{e}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){e}^{4}+4\, \left ({b}^{3}A+3\,a{b}^{2}B \right ) d{e}^{3}+6\,{b}^{3}B{d}^{2}{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){e}^{4}+4\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ) d{e}^{3}+6\, \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{2}{e}^{2}+4\,{b}^{3}B{d}^{3}e \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{3}A{e}^{4}+4\, \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ) d{e}^{3}+6\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{2}{e}^{2}+4\, \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{3}e+{b}^{3}B{d}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{3}Ad{e}^{3}+6\, \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){d}^{2}{e}^{2}+4\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{3}e+ \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{a}^{3}A{d}^{2}{e}^{2}+4\, \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){d}^{3}e+ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{3}A{d}^{3}e+ \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){d}^{4} \right ){x}^{2}}{2}}+{a}^{3}A{d}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05118, size = 574, normalized size = 3.52 \begin{align*} \frac{1}{9} \, B b^{3} e^{4} x^{9} + A a^{3} d^{4} x + \frac{1}{8} \,{\left (4 \, B b^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (6 \, B b^{3} d^{2} e^{2} + 4 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (4 \, B b^{3} d^{3} e + 6 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{4} + A a^{3} e^{4} + 4 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 18 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 4 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, A a^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e + 6 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, A a^{3} d^{2} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} + 4 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, A a^{3} d^{3} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.6111, size = 1175, normalized size = 7.21 \begin{align*} \frac{1}{9} x^{9} e^{4} b^{3} B + \frac{1}{2} x^{8} e^{3} d b^{3} B + \frac{3}{8} x^{8} e^{4} b^{2} a B + \frac{1}{8} x^{8} e^{4} b^{3} A + \frac{6}{7} x^{7} e^{2} d^{2} b^{3} B + \frac{12}{7} x^{7} e^{3} d b^{2} a B + \frac{3}{7} x^{7} e^{4} b a^{2} B + \frac{4}{7} x^{7} e^{3} d b^{3} A + \frac{3}{7} x^{7} e^{4} b^{2} a A + \frac{2}{3} x^{6} e d^{3} b^{3} B + 3 x^{6} e^{2} d^{2} b^{2} a B + 2 x^{6} e^{3} d b a^{2} B + \frac{1}{6} x^{6} e^{4} a^{3} B + x^{6} e^{2} d^{2} b^{3} A + 2 x^{6} e^{3} d b^{2} a A + \frac{1}{2} x^{6} e^{4} b a^{2} A + \frac{1}{5} x^{5} d^{4} b^{3} B + \frac{12}{5} x^{5} e d^{3} b^{2} a B + \frac{18}{5} x^{5} e^{2} d^{2} b a^{2} B + \frac{4}{5} x^{5} e^{3} d a^{3} B + \frac{4}{5} x^{5} e d^{3} b^{3} A + \frac{18}{5} x^{5} e^{2} d^{2} b^{2} a A + \frac{12}{5} x^{5} e^{3} d b a^{2} A + \frac{1}{5} x^{5} e^{4} a^{3} A + \frac{3}{4} x^{4} d^{4} b^{2} a B + 3 x^{4} e d^{3} b a^{2} B + \frac{3}{2} x^{4} e^{2} d^{2} a^{3} B + \frac{1}{4} x^{4} d^{4} b^{3} A + 3 x^{4} e d^{3} b^{2} a A + \frac{9}{2} x^{4} e^{2} d^{2} b a^{2} A + x^{4} e^{3} d a^{3} A + x^{3} d^{4} b a^{2} B + \frac{4}{3} x^{3} e d^{3} a^{3} B + x^{3} d^{4} b^{2} a A + 4 x^{3} e d^{3} b a^{2} A + 2 x^{3} e^{2} d^{2} a^{3} A + \frac{1}{2} x^{2} d^{4} a^{3} B + \frac{3}{2} x^{2} d^{4} b a^{2} A + 2 x^{2} e d^{3} a^{3} A + x d^{4} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.130912, size = 546, normalized size = 3.35 \begin{align*} A a^{3} d^{4} x + \frac{B b^{3} e^{4} x^{9}}{9} + x^{8} \left (\frac{A b^{3} e^{4}}{8} + \frac{3 B a b^{2} e^{4}}{8} + \frac{B b^{3} d e^{3}}{2}\right ) + x^{7} \left (\frac{3 A a b^{2} e^{4}}{7} + \frac{4 A b^{3} d e^{3}}{7} + \frac{3 B a^{2} b e^{4}}{7} + \frac{12 B a b^{2} d e^{3}}{7} + \frac{6 B b^{3} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac{A a^{2} b e^{4}}{2} + 2 A a b^{2} d e^{3} + A b^{3} d^{2} e^{2} + \frac{B a^{3} e^{4}}{6} + 2 B a^{2} b d e^{3} + 3 B a b^{2} d^{2} e^{2} + \frac{2 B b^{3} d^{3} e}{3}\right ) + x^{5} \left (\frac{A a^{3} e^{4}}{5} + \frac{12 A a^{2} b d e^{3}}{5} + \frac{18 A a b^{2} d^{2} e^{2}}{5} + \frac{4 A b^{3} d^{3} e}{5} + \frac{4 B a^{3} d e^{3}}{5} + \frac{18 B a^{2} b d^{2} e^{2}}{5} + \frac{12 B a b^{2} d^{3} e}{5} + \frac{B b^{3} d^{4}}{5}\right ) + x^{4} \left (A a^{3} d e^{3} + \frac{9 A a^{2} b d^{2} e^{2}}{2} + 3 A a b^{2} d^{3} e + \frac{A b^{3} d^{4}}{4} + \frac{3 B a^{3} d^{2} e^{2}}{2} + 3 B a^{2} b d^{3} e + \frac{3 B a b^{2} d^{4}}{4}\right ) + x^{3} \left (2 A a^{3} d^{2} e^{2} + 4 A a^{2} b d^{3} e + A a b^{2} d^{4} + \frac{4 B a^{3} d^{3} e}{3} + B a^{2} b d^{4}\right ) + x^{2} \left (2 A a^{3} d^{3} e + \frac{3 A a^{2} b d^{4}}{2} + \frac{B a^{3} d^{4}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.00447, size = 699, normalized size = 4.29 \begin{align*} \frac{1}{9} \, B b^{3} x^{9} e^{4} + \frac{1}{2} \, B b^{3} d x^{8} e^{3} + \frac{6}{7} \, B b^{3} d^{2} x^{7} e^{2} + \frac{2}{3} \, B b^{3} d^{3} x^{6} e + \frac{1}{5} \, B b^{3} d^{4} x^{5} + \frac{3}{8} \, B a b^{2} x^{8} e^{4} + \frac{1}{8} \, A b^{3} x^{8} e^{4} + \frac{12}{7} \, B a b^{2} d x^{7} e^{3} + \frac{4}{7} \, A b^{3} d x^{7} e^{3} + 3 \, B a b^{2} d^{2} x^{6} e^{2} + A b^{3} d^{2} x^{6} e^{2} + \frac{12}{5} \, B a b^{2} d^{3} x^{5} e + \frac{4}{5} \, A b^{3} d^{3} x^{5} e + \frac{3}{4} \, B a b^{2} d^{4} x^{4} + \frac{1}{4} \, A b^{3} d^{4} x^{4} + \frac{3}{7} \, B a^{2} b x^{7} e^{4} + \frac{3}{7} \, A a b^{2} x^{7} e^{4} + 2 \, B a^{2} b d x^{6} e^{3} + 2 \, A a b^{2} d x^{6} e^{3} + \frac{18}{5} \, B a^{2} b d^{2} x^{5} e^{2} + \frac{18}{5} \, A a b^{2} d^{2} x^{5} e^{2} + 3 \, B a^{2} b d^{3} x^{4} e + 3 \, A a b^{2} d^{3} x^{4} e + B a^{2} b d^{4} x^{3} + A a b^{2} d^{4} x^{3} + \frac{1}{6} \, B a^{3} x^{6} e^{4} + \frac{1}{2} \, A a^{2} b x^{6} e^{4} + \frac{4}{5} \, B a^{3} d x^{5} e^{3} + \frac{12}{5} \, A a^{2} b d x^{5} e^{3} + \frac{3}{2} \, B a^{3} d^{2} x^{4} e^{2} + \frac{9}{2} \, A a^{2} b d^{2} x^{4} e^{2} + \frac{4}{3} \, B a^{3} d^{3} x^{3} e + 4 \, A a^{2} b d^{3} x^{3} e + \frac{1}{2} \, B a^{3} d^{4} x^{2} + \frac{3}{2} \, A a^{2} b d^{4} x^{2} + \frac{1}{5} \, A a^{3} x^{5} e^{4} + A a^{3} d x^{4} e^{3} + 2 \, A a^{3} d^{2} x^{3} e^{2} + 2 \, A a^{3} d^{3} x^{2} e + A a^{3} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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