Optimal. Leaf size=118 \[ \frac{e (a+b x)^6 (-3 a B e+A b e+2 b B d)}{6 b^4}+\frac{(a+b x)^5 (b d-a e) (-3 a B e+2 A b e+b B d)}{5 b^4}+\frac{(a+b x)^4 (A b-a B) (b d-a e)^2}{4 b^4}+\frac{B e^2 (a+b x)^7}{7 b^4} \]
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Rubi [A] time = 0.157244, antiderivative size = 118, 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} \[ \frac{e (a+b x)^6 (-3 a B e+A b e+2 b B d)}{6 b^4}+\frac{(a+b x)^5 (b d-a e) (-3 a B e+2 A b e+b B d)}{5 b^4}+\frac{(a+b x)^4 (A b-a B) (b d-a e)^2}{4 b^4}+\frac{B e^2 (a+b x)^7}{7 b^4} \]
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)^2 \, dx &=\int \left (\frac{(A b-a B) (b d-a e)^2 (a+b x)^3}{b^3}+\frac{(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^4}{b^3}+\frac{e (2 b B d+A b e-3 a B e) (a+b x)^5}{b^3}+\frac{B e^2 (a+b x)^6}{b^3}\right ) \, dx\\ &=\frac{(A b-a B) (b d-a e)^2 (a+b x)^4}{4 b^4}+\frac{(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^5}{5 b^4}+\frac{e (2 b B d+A b e-3 a B e) (a+b x)^6}{6 b^4}+\frac{B e^2 (a+b x)^7}{7 b^4}\\ \end{align*}
Mathematica [A] time = 0.0737966, size = 224, normalized size = 1.9 \[ \frac{1}{4} x^4 \left (A b \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+a B \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )\right )+\frac{1}{3} a x^3 \left (A \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+a B d (2 a e+3 b d)\right )+\frac{1}{5} b x^5 \left (3 a^2 B e^2+3 a b e (A e+2 B d)+b^2 d (2 A e+B d)\right )+\frac{1}{2} a^2 d x^2 (2 a A e+a B d+3 A b d)+a^3 A d^2 x+\frac{1}{6} b^2 e x^6 (3 a B e+A b e+2 b B d)+\frac{1}{7} b^3 B e^2 x^7 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 244, normalized size = 2.1 \begin{align*}{\frac{{b}^{3}B{e}^{2}{x}^{7}}{7}}+{\frac{ \left ( \left ({b}^{3}A+3\,a{b}^{2}B \right ){e}^{2}+2\,{b}^{3}Bde \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){e}^{2}+2\, \left ({b}^{3}A+3\,a{b}^{2}B \right ) de+{b}^{3}B{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){e}^{2}+2\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ) de+ \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3}A{e}^{2}+2\, \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ) de+ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{3}Ade+ \left ( 3\,Ab{a}^{2}+B{a}^{3} \right ){d}^{2} \right ){x}^{2}}{2}}+{a}^{3}A{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.86736, size = 323, normalized size = 2.74 \begin{align*} \frac{1}{7} \, B b^{3} e^{2} x^{7} + A a^{3} d^{2} x + \frac{1}{6} \,{\left (2 \, B b^{3} d e +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{2} + 2 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (A a^{3} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a^{3} d e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84207, size = 652, normalized size = 5.53 \begin{align*} \frac{1}{7} x^{7} e^{2} b^{3} B + \frac{1}{3} x^{6} e d b^{3} B + \frac{1}{2} x^{6} e^{2} b^{2} a B + \frac{1}{6} x^{6} e^{2} b^{3} A + \frac{1}{5} x^{5} d^{2} b^{3} B + \frac{6}{5} x^{5} e d b^{2} a B + \frac{3}{5} x^{5} e^{2} b a^{2} B + \frac{2}{5} x^{5} e d b^{3} A + \frac{3}{5} x^{5} e^{2} b^{2} a A + \frac{3}{4} x^{4} d^{2} b^{2} a B + \frac{3}{2} x^{4} e d b a^{2} B + \frac{1}{4} x^{4} e^{2} a^{3} B + \frac{1}{4} x^{4} d^{2} b^{3} A + \frac{3}{2} x^{4} e d b^{2} a A + \frac{3}{4} x^{4} e^{2} b a^{2} A + x^{3} d^{2} b a^{2} B + \frac{2}{3} x^{3} e d a^{3} B + x^{3} d^{2} b^{2} a A + 2 x^{3} e d b a^{2} A + \frac{1}{3} x^{3} e^{2} a^{3} A + \frac{1}{2} x^{2} d^{2} a^{3} B + \frac{3}{2} x^{2} d^{2} b a^{2} A + x^{2} e d a^{3} A + x d^{2} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.148813, size = 296, normalized size = 2.51 \begin{align*} A a^{3} d^{2} x + \frac{B b^{3} e^{2} x^{7}}{7} + x^{6} \left (\frac{A b^{3} e^{2}}{6} + \frac{B a b^{2} e^{2}}{2} + \frac{B b^{3} d e}{3}\right ) + x^{5} \left (\frac{3 A a b^{2} e^{2}}{5} + \frac{2 A b^{3} d e}{5} + \frac{3 B a^{2} b e^{2}}{5} + \frac{6 B a b^{2} d e}{5} + \frac{B b^{3} d^{2}}{5}\right ) + x^{4} \left (\frac{3 A a^{2} b e^{2}}{4} + \frac{3 A a b^{2} d e}{2} + \frac{A b^{3} d^{2}}{4} + \frac{B a^{3} e^{2}}{4} + \frac{3 B a^{2} b d e}{2} + \frac{3 B a b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac{A a^{3} e^{2}}{3} + 2 A a^{2} b d e + A a b^{2} d^{2} + \frac{2 B a^{3} d e}{3} + B a^{2} b d^{2}\right ) + x^{2} \left (A a^{3} d e + \frac{3 A a^{2} b d^{2}}{2} + \frac{B a^{3} d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.8519, size = 387, normalized size = 3.28 \begin{align*} \frac{1}{7} \, B b^{3} x^{7} e^{2} + \frac{1}{3} \, B b^{3} d x^{6} e + \frac{1}{5} \, B b^{3} d^{2} x^{5} + \frac{1}{2} \, B a b^{2} x^{6} e^{2} + \frac{1}{6} \, A b^{3} x^{6} e^{2} + \frac{6}{5} \, B a b^{2} d x^{5} e + \frac{2}{5} \, A b^{3} d x^{5} e + \frac{3}{4} \, B a b^{2} d^{2} x^{4} + \frac{1}{4} \, A b^{3} d^{2} x^{4} + \frac{3}{5} \, B a^{2} b x^{5} e^{2} + \frac{3}{5} \, A a b^{2} x^{5} e^{2} + \frac{3}{2} \, B a^{2} b d x^{4} e + \frac{3}{2} \, A a b^{2} d x^{4} e + B a^{2} b d^{2} x^{3} + A a b^{2} d^{2} x^{3} + \frac{1}{4} \, B a^{3} x^{4} e^{2} + \frac{3}{4} \, A a^{2} b x^{4} e^{2} + \frac{2}{3} \, B a^{3} d x^{3} e + 2 \, A a^{2} b d x^{3} e + \frac{1}{2} \, B a^{3} d^{2} x^{2} + \frac{3}{2} \, A a^{2} b d^{2} x^{2} + \frac{1}{3} \, A a^{3} x^{3} e^{2} + A a^{3} d x^{2} e + A a^{3} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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