Optimal. Leaf size=75 \[ \frac{(a+b x)^{12} (-2 a B e+A b e+b B d)}{12 b^3}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)}{11 b^3}+\frac{B e (a+b x)^{13}}{13 b^3} \]
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Rubi [A] time = 0.41249, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ \frac{(a+b x)^{12} (-2 a B e+A b e+b B d)}{12 b^3}+\frac{(a+b x)^{11} (A b-a B) (b d-a e)}{11 b^3}+\frac{B e (a+b x)^{13}}{13 b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int (a+b x)^{10} (A+B x) (d+e x) \, dx &=\int \left (\frac{(A b-a B) (b d-a e) (a+b x)^{10}}{b^2}+\frac{(b B d+A b e-2 a B e) (a+b x)^{11}}{b^2}+\frac{B e (a+b x)^{12}}{b^2}\right ) \, dx\\ &=\frac{(A b-a B) (b d-a e) (a+b x)^{11}}{11 b^3}+\frac{(b B d+A b e-2 a B e) (a+b x)^{12}}{12 b^3}+\frac{B e (a+b x)^{13}}{13 b^3}\\ \end{align*}
Mathematica [B] time = 0.172169, size = 383, normalized size = 5.11 \[ \frac{1}{22} a^2 b^8 x^9 \left (110 A d+99 A e x+99 B d x+90 B e x^2\right )+\frac{1}{3} a^3 b^7 x^8 (5 A (9 d+8 e x)+4 B x (10 d+9 e x))+\frac{5}{12} a^4 b^6 x^7 (9 A (8 d+7 e x)+7 B x (9 d+8 e x))+\frac{3}{2} a^5 b^5 x^6 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+a^6 b^4 x^5 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+2 a^7 b^3 x^4 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+\frac{3}{4} a^8 b^2 x^3 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+\frac{5}{6} a^9 b x^2 (A (6 d+4 e x)+B x (4 d+3 e x))+\frac{1}{6} a^{10} x (3 A (2 d+e x)+B x (3 d+2 e x))+\frac{1}{66} a b^9 x^{10} \left (66 A d+60 A e x+60 B d x+55 B e x^2\right )+\frac{b^{10} x^{11} (13 A (12 d+11 e x)+11 B x (13 d+12 e x))}{1716} \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 485, normalized size = 6.5 \begin{align*}{\frac{{b}^{10}Be{x}^{13}}{13}}+{\frac{ \left ( \left ({b}^{10}A+10\,a{b}^{9}B \right ) e+{b}^{10}Bd \right ){x}^{12}}{12}}+{\frac{ \left ( \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ) e+ \left ({b}^{10}A+10\,a{b}^{9}B \right ) d \right ){x}^{11}}{11}}+{\frac{ \left ( \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ) e+ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ) d \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ) e+ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ) d \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ) e+ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ) d \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ) e+ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ) d \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ) e+ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ) d \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ) e+ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ) d \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ) e+ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ) d \right ){x}^{4}}{4}}+{\frac{ \left ( \left ( 10\,{a}^{9}bA+{a}^{10}B \right ) e+ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ) d \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{10}Ae+ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ) d \right ){x}^{2}}{2}}+{a}^{10}Adx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11425, size = 666, normalized size = 8.88 \begin{align*} \frac{1}{13} \, B b^{10} e x^{13} + A a^{10} d x + \frac{1}{12} \,{\left (B b^{10} d +{\left (10 \, B a b^{9} + A b^{10}\right )} e\right )} x^{12} + \frac{1}{11} \,{\left ({\left (10 \, B a b^{9} + A b^{10}\right )} d + 5 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} e\right )} x^{11} + \frac{1}{2} \,{\left ({\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d + 3 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} e\right )} x^{10} + \frac{5}{3} \,{\left ({\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d + 2 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} e\right )} x^{9} + \frac{3}{4} \,{\left (5 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d + 7 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} e\right )} x^{8} + 6 \,{\left ({\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d +{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} e\right )} x^{7} +{\left (7 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d + 5 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} e\right )} x^{6} + 3 \,{\left (2 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d +{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} e\right )} x^{5} + \frac{5}{4} \,{\left (3 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d +{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (5 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d +{\left (B a^{10} + 10 \, A a^{9} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{10} e +{\left (B a^{10} + 10 \, A a^{9} b\right )} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.59776, size = 1256, normalized size = 16.75 \begin{align*} \frac{1}{13} x^{13} e b^{10} B + \frac{1}{12} x^{12} d b^{10} B + \frac{5}{6} x^{12} e b^{9} a B + \frac{1}{12} x^{12} e b^{10} A + \frac{10}{11} x^{11} d b^{9} a B + \frac{45}{11} x^{11} e b^{8} a^{2} B + \frac{1}{11} x^{11} d b^{10} A + \frac{10}{11} x^{11} e b^{9} a A + \frac{9}{2} x^{10} d b^{8} a^{2} B + 12 x^{10} e b^{7} a^{3} B + x^{10} d b^{9} a A + \frac{9}{2} x^{10} e b^{8} a^{2} A + \frac{40}{3} x^{9} d b^{7} a^{3} B + \frac{70}{3} x^{9} e b^{6} a^{4} B + 5 x^{9} d b^{8} a^{2} A + \frac{40}{3} x^{9} e b^{7} a^{3} A + \frac{105}{4} x^{8} d b^{6} a^{4} B + \frac{63}{2} x^{8} e b^{5} a^{5} B + 15 x^{8} d b^{7} a^{3} A + \frac{105}{4} x^{8} e b^{6} a^{4} A + 36 x^{7} d b^{5} a^{5} B + 30 x^{7} e b^{4} a^{6} B + 30 x^{7} d b^{6} a^{4} A + 36 x^{7} e b^{5} a^{5} A + 35 x^{6} d b^{4} a^{6} B + 20 x^{6} e b^{3} a^{7} B + 42 x^{6} d b^{5} a^{5} A + 35 x^{6} e b^{4} a^{6} A + 24 x^{5} d b^{3} a^{7} B + 9 x^{5} e b^{2} a^{8} B + 42 x^{5} d b^{4} a^{6} A + 24 x^{5} e b^{3} a^{7} A + \frac{45}{4} x^{4} d b^{2} a^{8} B + \frac{5}{2} x^{4} e b a^{9} B + 30 x^{4} d b^{3} a^{7} A + \frac{45}{4} x^{4} e b^{2} a^{8} A + \frac{10}{3} x^{3} d b a^{9} B + \frac{1}{3} x^{3} e a^{10} B + 15 x^{3} d b^{2} a^{8} A + \frac{10}{3} x^{3} e b a^{9} A + \frac{1}{2} x^{2} d a^{10} B + 5 x^{2} d b a^{9} A + \frac{1}{2} x^{2} e a^{10} A + x d a^{10} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.13055, size = 549, normalized size = 7.32 \begin{align*} A a^{10} d x + \frac{B b^{10} e x^{13}}{13} + x^{12} \left (\frac{A b^{10} e}{12} + \frac{5 B a b^{9} e}{6} + \frac{B b^{10} d}{12}\right ) + x^{11} \left (\frac{10 A a b^{9} e}{11} + \frac{A b^{10} d}{11} + \frac{45 B a^{2} b^{8} e}{11} + \frac{10 B a b^{9} d}{11}\right ) + x^{10} \left (\frac{9 A a^{2} b^{8} e}{2} + A a b^{9} d + 12 B a^{3} b^{7} e + \frac{9 B a^{2} b^{8} d}{2}\right ) + x^{9} \left (\frac{40 A a^{3} b^{7} e}{3} + 5 A a^{2} b^{8} d + \frac{70 B a^{4} b^{6} e}{3} + \frac{40 B a^{3} b^{7} d}{3}\right ) + x^{8} \left (\frac{105 A a^{4} b^{6} e}{4} + 15 A a^{3} b^{7} d + \frac{63 B a^{5} b^{5} e}{2} + \frac{105 B a^{4} b^{6} d}{4}\right ) + x^{7} \left (36 A a^{5} b^{5} e + 30 A a^{4} b^{6} d + 30 B a^{6} b^{4} e + 36 B a^{5} b^{5} d\right ) + x^{6} \left (35 A a^{6} b^{4} e + 42 A a^{5} b^{5} d + 20 B a^{7} b^{3} e + 35 B a^{6} b^{4} d\right ) + x^{5} \left (24 A a^{7} b^{3} e + 42 A a^{6} b^{4} d + 9 B a^{8} b^{2} e + 24 B a^{7} b^{3} d\right ) + x^{4} \left (\frac{45 A a^{8} b^{2} e}{4} + 30 A a^{7} b^{3} d + \frac{5 B a^{9} b e}{2} + \frac{45 B a^{8} b^{2} d}{4}\right ) + x^{3} \left (\frac{10 A a^{9} b e}{3} + 15 A a^{8} b^{2} d + \frac{B a^{10} e}{3} + \frac{10 B a^{9} b d}{3}\right ) + x^{2} \left (\frac{A a^{10} e}{2} + 5 A a^{9} b d + \frac{B a^{10} d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21433, size = 744, normalized size = 9.92 \begin{align*} \frac{1}{13} \, B b^{10} x^{13} e + \frac{1}{12} \, B b^{10} d x^{12} + \frac{5}{6} \, B a b^{9} x^{12} e + \frac{1}{12} \, A b^{10} x^{12} e + \frac{10}{11} \, B a b^{9} d x^{11} + \frac{1}{11} \, A b^{10} d x^{11} + \frac{45}{11} \, B a^{2} b^{8} x^{11} e + \frac{10}{11} \, A a b^{9} x^{11} e + \frac{9}{2} \, B a^{2} b^{8} d x^{10} + A a b^{9} d x^{10} + 12 \, B a^{3} b^{7} x^{10} e + \frac{9}{2} \, A a^{2} b^{8} x^{10} e + \frac{40}{3} \, B a^{3} b^{7} d x^{9} + 5 \, A a^{2} b^{8} d x^{9} + \frac{70}{3} \, B a^{4} b^{6} x^{9} e + \frac{40}{3} \, A a^{3} b^{7} x^{9} e + \frac{105}{4} \, B a^{4} b^{6} d x^{8} + 15 \, A a^{3} b^{7} d x^{8} + \frac{63}{2} \, B a^{5} b^{5} x^{8} e + \frac{105}{4} \, A a^{4} b^{6} x^{8} e + 36 \, B a^{5} b^{5} d x^{7} + 30 \, A a^{4} b^{6} d x^{7} + 30 \, B a^{6} b^{4} x^{7} e + 36 \, A a^{5} b^{5} x^{7} e + 35 \, B a^{6} b^{4} d x^{6} + 42 \, A a^{5} b^{5} d x^{6} + 20 \, B a^{7} b^{3} x^{6} e + 35 \, A a^{6} b^{4} x^{6} e + 24 \, B a^{7} b^{3} d x^{5} + 42 \, A a^{6} b^{4} d x^{5} + 9 \, B a^{8} b^{2} x^{5} e + 24 \, A a^{7} b^{3} x^{5} e + \frac{45}{4} \, B a^{8} b^{2} d x^{4} + 30 \, A a^{7} b^{3} d x^{4} + \frac{5}{2} \, B a^{9} b x^{4} e + \frac{45}{4} \, A a^{8} b^{2} x^{4} e + \frac{10}{3} \, B a^{9} b d x^{3} + 15 \, A a^{8} b^{2} d x^{3} + \frac{1}{3} \, B a^{10} x^{3} e + \frac{10}{3} \, A a^{9} b x^{3} e + \frac{1}{2} \, B a^{10} d x^{2} + 5 \, A a^{9} b d x^{2} + \frac{1}{2} \, A a^{10} x^{2} e + A a^{10} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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