Optimal. Leaf size=120 \[ -\frac {b (d+e x)^8 (-2 a B e-A b e+3 b B d)}{8 e^4}+\frac {(d+e x)^7 (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac {(d+e x)^6 (b d-a e)^2 (B d-A e)}{6 e^4}+\frac {b^2 B (d+e x)^9}{9 e^4} \]
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Rubi [A] time = 0.29, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 77} \begin {gather*} -\frac {b (d+e x)^8 (-2 a B e-A b e+3 b B d)}{8 e^4}+\frac {(d+e x)^7 (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac {(d+e x)^6 (b d-a e)^2 (B d-A e)}{6 e^4}+\frac {b^2 B (d+e x)^9}{9 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (A+B x) (d+e x)^5 \, dx\\ &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^5}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^6}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^7}{e^3}+\frac {b^2 B (d+e x)^8}{e^3}\right ) \, dx\\ &=-\frac {(b d-a e)^2 (B d-A e) (d+e x)^6}{6 e^4}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^7}{7 e^4}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^8}{8 e^4}+\frac {b^2 B (d+e x)^9}{9 e^4}\\ \end {align*}
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Mathematica [B] time = 0.11, size = 330, normalized size = 2.75 \begin {gather*} \frac {1}{6} e^2 x^6 \left (a^2 e^2 (A e+5 B d)+10 a b d e (A e+2 B d)+10 b^2 d^2 (A e+B d)\right )+d e x^5 \left (a^2 e^2 (A e+2 B d)+4 a b d e (A e+B d)+b^2 d^2 (2 A e+B d)\right )+\frac {1}{4} d^2 x^4 \left (10 a^2 e^2 (A e+B d)+10 a b d e (2 A e+B d)+b^2 d^2 (5 A e+B d)\right )+\frac {1}{3} d^3 x^3 \left (A \left (10 a^2 e^2+10 a b d e+b^2 d^2\right )+a B d (5 a e+2 b d)\right )+\frac {1}{7} e^3 x^7 \left (a^2 B e^2+2 a b e (A e+5 B d)+5 b^2 d (A e+2 B d)\right )+a^2 A d^5 x+\frac {1}{2} a d^4 x^2 (5 a A e+a B d+2 A b d)+\frac {1}{8} b e^4 x^8 (2 a B e+A b e+5 b B d)+\frac {1}{9} b^2 B e^5 x^9 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.36, size = 463, normalized size = 3.86 \begin {gather*} \frac {1}{9} x^{9} e^{5} b^{2} B + \frac {5}{8} x^{8} e^{4} d b^{2} B + \frac {1}{4} x^{8} e^{5} b a B + \frac {1}{8} x^{8} e^{5} b^{2} A + \frac {10}{7} x^{7} e^{3} d^{2} b^{2} B + \frac {10}{7} x^{7} e^{4} d b a B + \frac {1}{7} x^{7} e^{5} a^{2} B + \frac {5}{7} x^{7} e^{4} d b^{2} A + \frac {2}{7} x^{7} e^{5} b a A + \frac {5}{3} x^{6} e^{2} d^{3} b^{2} B + \frac {10}{3} x^{6} e^{3} d^{2} b a B + \frac {5}{6} x^{6} e^{4} d a^{2} B + \frac {5}{3} x^{6} e^{3} d^{2} b^{2} A + \frac {5}{3} x^{6} e^{4} d b a A + \frac {1}{6} x^{6} e^{5} a^{2} A + x^{5} e d^{4} b^{2} B + 4 x^{5} e^{2} d^{3} b a B + 2 x^{5} e^{3} d^{2} a^{2} B + 2 x^{5} e^{2} d^{3} b^{2} A + 4 x^{5} e^{3} d^{2} b a A + x^{5} e^{4} d a^{2} A + \frac {1}{4} x^{4} d^{5} b^{2} B + \frac {5}{2} x^{4} e d^{4} b a B + \frac {5}{2} x^{4} e^{2} d^{3} a^{2} B + \frac {5}{4} x^{4} e d^{4} b^{2} A + 5 x^{4} e^{2} d^{3} b a A + \frac {5}{2} x^{4} e^{3} d^{2} a^{2} A + \frac {2}{3} x^{3} d^{5} b a B + \frac {5}{3} x^{3} e d^{4} a^{2} B + \frac {1}{3} x^{3} d^{5} b^{2} A + \frac {10}{3} x^{3} e d^{4} b a A + \frac {10}{3} x^{3} e^{2} d^{3} a^{2} A + \frac {1}{2} x^{2} d^{5} a^{2} B + x^{2} d^{5} b a A + \frac {5}{2} x^{2} e d^{4} a^{2} A + x d^{5} a^{2} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 445, normalized size = 3.71 \begin {gather*} \frac {1}{9} \, B b^{2} x^{9} e^{5} + \frac {5}{8} \, B b^{2} d x^{8} e^{4} + \frac {10}{7} \, B b^{2} d^{2} x^{7} e^{3} + \frac {5}{3} \, B b^{2} d^{3} x^{6} e^{2} + B b^{2} d^{4} x^{5} e + \frac {1}{4} \, B b^{2} d^{5} x^{4} + \frac {1}{4} \, B a b x^{8} e^{5} + \frac {1}{8} \, A b^{2} x^{8} e^{5} + \frac {10}{7} \, B a b d x^{7} e^{4} + \frac {5}{7} \, A b^{2} d x^{7} e^{4} + \frac {10}{3} \, B a b d^{2} x^{6} e^{3} + \frac {5}{3} \, A b^{2} d^{2} x^{6} e^{3} + 4 \, B a b d^{3} x^{5} e^{2} + 2 \, A b^{2} d^{3} x^{5} e^{2} + \frac {5}{2} \, B a b d^{4} x^{4} e + \frac {5}{4} \, A b^{2} d^{4} x^{4} e + \frac {2}{3} \, B a b d^{5} x^{3} + \frac {1}{3} \, A b^{2} d^{5} x^{3} + \frac {1}{7} \, B a^{2} x^{7} e^{5} + \frac {2}{7} \, A a b x^{7} e^{5} + \frac {5}{6} \, B a^{2} d x^{6} e^{4} + \frac {5}{3} \, A a b d x^{6} e^{4} + 2 \, B a^{2} d^{2} x^{5} e^{3} + 4 \, A a b d^{2} x^{5} e^{3} + \frac {5}{2} \, B a^{2} d^{3} x^{4} e^{2} + 5 \, A a b d^{3} x^{4} e^{2} + \frac {5}{3} \, B a^{2} d^{4} x^{3} e + \frac {10}{3} \, A a b d^{4} x^{3} e + \frac {1}{2} \, B a^{2} d^{5} x^{2} + A a b d^{5} x^{2} + \frac {1}{6} \, A a^{2} x^{6} e^{5} + A a^{2} d x^{5} e^{4} + \frac {5}{2} \, A a^{2} d^{2} x^{4} e^{3} + \frac {10}{3} \, A a^{2} d^{3} x^{3} e^{2} + \frac {5}{2} \, A a^{2} d^{4} x^{2} e + A a^{2} d^{5} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 394, normalized size = 3.28 \begin {gather*} \frac {B \,b^{2} e^{5} x^{9}}{9}+A \,a^{2} d^{5} x +\frac {\left (2 B a b \,e^{5}+\left (A \,e^{5}+5 B d \,e^{4}\right ) b^{2}\right ) x^{8}}{8}+\frac {\left (B \,a^{2} e^{5}+2 \left (A \,e^{5}+5 B d \,e^{4}\right ) a b +\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) b^{2}\right ) x^{7}}{7}+\frac {\left (\left (A \,e^{5}+5 B d \,e^{4}\right ) a^{2}+2 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a b +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) b^{2}\right ) x^{6}}{6}+\frac {\left (\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{2}+2 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a b +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) b^{2}\right ) x^{5}}{5}+\frac {\left (\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{2}+2 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a b +\left (5 A \,d^{4} e +B \,d^{5}\right ) b^{2}\right ) x^{4}}{4}+\frac {\left (A \,b^{2} d^{5}+\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a^{2}+2 \left (5 A \,d^{4} e +B \,d^{5}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{5}+\left (5 A \,d^{4} e +B \,d^{5}\right ) a^{2}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 369, normalized size = 3.08 \begin {gather*} \frac {1}{9} \, B b^{2} e^{5} x^{9} + A a^{2} d^{5} x + \frac {1}{8} \, {\left (5 \, B b^{2} d e^{4} + {\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, B b^{2} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2}\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, B b^{2} d^{3} e^{2} + A a^{2} e^{5} + 10 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{6} + {\left (B b^{2} d^{4} e + A a^{2} d e^{4} + 2 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{5} + 10 \, A a^{2} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, A a^{2} d^{3} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{5} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4} e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{2} d^{4} e + {\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 381, normalized size = 3.18 \begin {gather*} x^5\,\left (2\,B\,a^2\,d^2\,e^3+A\,a^2\,d\,e^4+4\,B\,a\,b\,d^3\,e^2+4\,A\,a\,b\,d^2\,e^3+B\,b^2\,d^4\,e+2\,A\,b^2\,d^3\,e^2\right )+x^4\,\left (\frac {5\,B\,a^2\,d^3\,e^2}{2}+\frac {5\,A\,a^2\,d^2\,e^3}{2}+\frac {5\,B\,a\,b\,d^4\,e}{2}+5\,A\,a\,b\,d^3\,e^2+\frac {B\,b^2\,d^5}{4}+\frac {5\,A\,b^2\,d^4\,e}{4}\right )+x^6\,\left (\frac {5\,B\,a^2\,d\,e^4}{6}+\frac {A\,a^2\,e^5}{6}+\frac {10\,B\,a\,b\,d^2\,e^3}{3}+\frac {5\,A\,a\,b\,d\,e^4}{3}+\frac {5\,B\,b^2\,d^3\,e^2}{3}+\frac {5\,A\,b^2\,d^2\,e^3}{3}\right )+x^3\,\left (\frac {5\,B\,a^2\,d^4\,e}{3}+\frac {10\,A\,a^2\,d^3\,e^2}{3}+\frac {2\,B\,a\,b\,d^5}{3}+\frac {10\,A\,a\,b\,d^4\,e}{3}+\frac {A\,b^2\,d^5}{3}\right )+x^7\,\left (\frac {B\,a^2\,e^5}{7}+\frac {10\,B\,a\,b\,d\,e^4}{7}+\frac {2\,A\,a\,b\,e^5}{7}+\frac {10\,B\,b^2\,d^2\,e^3}{7}+\frac {5\,A\,b^2\,d\,e^4}{7}\right )+A\,a^2\,d^5\,x+\frac {a\,d^4\,x^2\,\left (5\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e^4\,x^8\,\left (A\,b\,e+2\,B\,a\,e+5\,B\,b\,d\right )}{8}+\frac {B\,b^2\,e^5\,x^9}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.13, size = 481, normalized size = 4.01 \begin {gather*} A a^{2} d^{5} x + \frac {B b^{2} e^{5} x^{9}}{9} + x^{8} \left (\frac {A b^{2} e^{5}}{8} + \frac {B a b e^{5}}{4} + \frac {5 B b^{2} d e^{4}}{8}\right ) + x^{7} \left (\frac {2 A a b e^{5}}{7} + \frac {5 A b^{2} d e^{4}}{7} + \frac {B a^{2} e^{5}}{7} + \frac {10 B a b d e^{4}}{7} + \frac {10 B b^{2} d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac {A a^{2} e^{5}}{6} + \frac {5 A a b d e^{4}}{3} + \frac {5 A b^{2} d^{2} e^{3}}{3} + \frac {5 B a^{2} d e^{4}}{6} + \frac {10 B a b d^{2} e^{3}}{3} + \frac {5 B b^{2} d^{3} e^{2}}{3}\right ) + x^{5} \left (A a^{2} d e^{4} + 4 A a b d^{2} e^{3} + 2 A b^{2} d^{3} e^{2} + 2 B a^{2} d^{2} e^{3} + 4 B a b d^{3} e^{2} + B b^{2} d^{4} e\right ) + x^{4} \left (\frac {5 A a^{2} d^{2} e^{3}}{2} + 5 A a b d^{3} e^{2} + \frac {5 A b^{2} d^{4} e}{4} + \frac {5 B a^{2} d^{3} e^{2}}{2} + \frac {5 B a b d^{4} e}{2} + \frac {B b^{2} d^{5}}{4}\right ) + x^{3} \left (\frac {10 A a^{2} d^{3} e^{2}}{3} + \frac {10 A a b d^{4} e}{3} + \frac {A b^{2} d^{5}}{3} + \frac {5 B a^{2} d^{4} e}{3} + \frac {2 B a b d^{5}}{3}\right ) + x^{2} \left (\frac {5 A a^{2} d^{4} e}{2} + A a b d^{5} + \frac {B a^{2} d^{5}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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