Optimal. Leaf size=120 \[ -\frac {b (d+e x)^7 (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac {(d+e x)^6 (b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4}-\frac {(d+e x)^5 (b d-a e)^2 (B d-A e)}{5 e^4}+\frac {b^2 B (d+e x)^8}{8 e^4} \]
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Rubi [A] time = 0.23, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {b (d+e x)^7 (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac {(d+e x)^6 (b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4}-\frac {(d+e x)^5 (b d-a e)^2 (B d-A e)}{5 e^4}+\frac {b^2 B (d+e x)^8}{8 e^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x)^2 (A+B x) (d+e x)^4 \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^4}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^5}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^6}{e^3}+\frac {b^2 B (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac {(b d-a e)^2 (B d-A e) (d+e x)^5}{5 e^4}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^6}{6 e^4}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^7}{7 e^4}+\frac {b^2 B (d+e x)^8}{8 e^4}\\ \end {align*}
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Mathematica [B] time = 0.10, size = 283, normalized size = 2.36 \[ \frac {1}{5} e x^5 \left (a^2 e^2 (A e+4 B d)+4 a b d e (2 A e+3 B d)+2 b^2 d^2 (3 A e+2 B d)\right )+\frac {1}{4} d x^4 \left (2 a^2 e^2 (2 A e+3 B d)+4 a b d e (3 A e+2 B d)+b^2 d^2 (4 A e+B d)\right )+\frac {1}{3} d^2 x^3 \left (A \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+2 a B d (2 a e+b d)\right )+\frac {1}{6} e^2 x^6 \left (a^2 B e^2+2 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+a^2 A d^4 x+\frac {1}{2} a d^3 x^2 (4 a A e+a B d+2 A b d)+\frac {1}{7} b e^3 x^7 (2 a B e+A b e+4 b B d)+\frac {1}{8} b^2 B e^4 x^8 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 374, normalized size = 3.12 \[ \frac {1}{8} x^{8} e^{4} b^{2} B + \frac {4}{7} x^{7} e^{3} d b^{2} B + \frac {2}{7} x^{7} e^{4} b a B + \frac {1}{7} x^{7} e^{4} b^{2} A + x^{6} e^{2} d^{2} b^{2} B + \frac {4}{3} x^{6} e^{3} d b a B + \frac {1}{6} x^{6} e^{4} a^{2} B + \frac {2}{3} x^{6} e^{3} d b^{2} A + \frac {1}{3} x^{6} e^{4} b a A + \frac {4}{5} x^{5} e d^{3} b^{2} B + \frac {12}{5} x^{5} e^{2} d^{2} b a B + \frac {4}{5} x^{5} e^{3} d a^{2} B + \frac {6}{5} x^{5} e^{2} d^{2} b^{2} A + \frac {8}{5} x^{5} e^{3} d b a A + \frac {1}{5} x^{5} e^{4} a^{2} A + \frac {1}{4} x^{4} d^{4} b^{2} B + 2 x^{4} e d^{3} b a B + \frac {3}{2} x^{4} e^{2} d^{2} a^{2} B + x^{4} e d^{3} b^{2} A + 3 x^{4} e^{2} d^{2} b a A + x^{4} e^{3} d a^{2} A + \frac {2}{3} x^{3} d^{4} b a B + \frac {4}{3} x^{3} e d^{3} a^{2} B + \frac {1}{3} x^{3} d^{4} b^{2} A + \frac {8}{3} x^{3} e d^{3} b a A + 2 x^{3} e^{2} d^{2} a^{2} A + \frac {1}{2} x^{2} d^{4} a^{2} B + x^{2} d^{4} b a A + 2 x^{2} e d^{3} a^{2} A + x d^{4} a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.16, size = 362, normalized size = 3.02 \[ \frac {1}{8} \, B b^{2} x^{8} e^{4} + \frac {4}{7} \, B b^{2} d x^{7} e^{3} + B b^{2} d^{2} x^{6} e^{2} + \frac {4}{5} \, B b^{2} d^{3} x^{5} e + \frac {1}{4} \, B b^{2} d^{4} x^{4} + \frac {2}{7} \, B a b x^{7} e^{4} + \frac {1}{7} \, A b^{2} x^{7} e^{4} + \frac {4}{3} \, B a b d x^{6} e^{3} + \frac {2}{3} \, A b^{2} d x^{6} e^{3} + \frac {12}{5} \, B a b d^{2} x^{5} e^{2} + \frac {6}{5} \, A b^{2} d^{2} x^{5} e^{2} + 2 \, B a b d^{3} x^{4} e + A b^{2} d^{3} x^{4} e + \frac {2}{3} \, B a b d^{4} x^{3} + \frac {1}{3} \, A b^{2} d^{4} x^{3} + \frac {1}{6} \, B a^{2} x^{6} e^{4} + \frac {1}{3} \, A a b x^{6} e^{4} + \frac {4}{5} \, B a^{2} d x^{5} e^{3} + \frac {8}{5} \, A a b d x^{5} e^{3} + \frac {3}{2} \, B a^{2} d^{2} x^{4} e^{2} + 3 \, A a b d^{2} x^{4} e^{2} + \frac {4}{3} \, B a^{2} d^{3} x^{3} e + \frac {8}{3} \, A a b d^{3} x^{3} e + \frac {1}{2} \, B a^{2} d^{4} x^{2} + A a b d^{4} x^{2} + \frac {1}{5} \, A a^{2} x^{5} e^{4} + A a^{2} d x^{4} e^{3} + 2 \, A a^{2} d^{2} x^{3} e^{2} + 2 \, A a^{2} d^{3} x^{2} e + A a^{2} d^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 305, normalized size = 2.54 \[ \frac {B \,b^{2} e^{4} x^{8}}{8}+A \,a^{2} d^{4} x +\frac {\left (4 B \,b^{2} d \,e^{3}+\left (A \,b^{2}+2 B a b \right ) e^{4}\right ) x^{7}}{7}+\frac {\left (6 B \,b^{2} d^{2} e^{2}+4 \left (A \,b^{2}+2 B a b \right ) d \,e^{3}+\left (2 A a b +B \,a^{2}\right ) e^{4}\right ) x^{6}}{6}+\frac {\left (A \,a^{2} e^{4}+4 B \,b^{2} d^{3} e +6 \left (A \,b^{2}+2 B a b \right ) d^{2} e^{2}+4 \left (2 A a b +B \,a^{2}\right ) d \,e^{3}\right ) x^{5}}{5}+\frac {\left (4 A \,a^{2} d \,e^{3}+B \,b^{2} d^{4}+4 \left (A \,b^{2}+2 B a b \right ) d^{3} e +6 \left (2 A a b +B \,a^{2}\right ) d^{2} e^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,a^{2} d^{2} e^{2}+\left (A \,b^{2}+2 B a b \right ) d^{4}+4 \left (2 A a b +B \,a^{2}\right ) d^{3} e \right ) x^{3}}{3}+\frac {\left (4 A \,a^{2} d^{3} e +\left (2 A a b +B \,a^{2}\right ) d^{4}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 304, normalized size = 2.53 \[ \frac {1}{8} \, B b^{2} e^{4} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B b^{2} d e^{3} + {\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, B b^{2} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (4 \, B b^{2} d^{3} e + A a^{2} e^{4} + 6 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{4} + 4 \, A a^{2} d e^{3} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 305, normalized size = 2.54 \[ x^4\,\left (\frac {3\,B\,a^2\,d^2\,e^2}{2}+A\,a^2\,d\,e^3+2\,B\,a\,b\,d^3\,e+3\,A\,a\,b\,d^2\,e^2+\frac {B\,b^2\,d^4}{4}+A\,b^2\,d^3\,e\right )+x^5\,\left (\frac {4\,B\,a^2\,d\,e^3}{5}+\frac {A\,a^2\,e^4}{5}+\frac {12\,B\,a\,b\,d^2\,e^2}{5}+\frac {8\,A\,a\,b\,d\,e^3}{5}+\frac {4\,B\,b^2\,d^3\,e}{5}+\frac {6\,A\,b^2\,d^2\,e^2}{5}\right )+x^3\,\left (\frac {4\,B\,a^2\,d^3\,e}{3}+2\,A\,a^2\,d^2\,e^2+\frac {2\,B\,a\,b\,d^4}{3}+\frac {8\,A\,a\,b\,d^3\,e}{3}+\frac {A\,b^2\,d^4}{3}\right )+x^6\,\left (\frac {B\,a^2\,e^4}{6}+\frac {4\,B\,a\,b\,d\,e^3}{3}+\frac {A\,a\,b\,e^4}{3}+B\,b^2\,d^2\,e^2+\frac {2\,A\,b^2\,d\,e^3}{3}\right )+A\,a^2\,d^4\,x+\frac {a\,d^3\,x^2\,\left (4\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e^3\,x^7\,\left (A\,b\,e+2\,B\,a\,e+4\,B\,b\,d\right )}{7}+\frac {B\,b^2\,e^4\,x^8}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.12, size = 384, normalized size = 3.20 \[ A a^{2} d^{4} x + \frac {B b^{2} e^{4} x^{8}}{8} + x^{7} \left (\frac {A b^{2} e^{4}}{7} + \frac {2 B a b e^{4}}{7} + \frac {4 B b^{2} d e^{3}}{7}\right ) + x^{6} \left (\frac {A a b e^{4}}{3} + \frac {2 A b^{2} d e^{3}}{3} + \frac {B a^{2} e^{4}}{6} + \frac {4 B a b d e^{3}}{3} + B b^{2} d^{2} e^{2}\right ) + x^{5} \left (\frac {A a^{2} e^{4}}{5} + \frac {8 A a b d e^{3}}{5} + \frac {6 A b^{2} d^{2} e^{2}}{5} + \frac {4 B a^{2} d e^{3}}{5} + \frac {12 B a b d^{2} e^{2}}{5} + \frac {4 B b^{2} d^{3} e}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 3 A a b d^{2} e^{2} + A b^{2} d^{3} e + \frac {3 B a^{2} d^{2} e^{2}}{2} + 2 B a b d^{3} e + \frac {B b^{2} d^{4}}{4}\right ) + x^{3} \left (2 A a^{2} d^{2} e^{2} + \frac {8 A a b d^{3} e}{3} + \frac {A b^{2} d^{4}}{3} + \frac {4 B a^{2} d^{3} e}{3} + \frac {2 B a b d^{4}}{3}\right ) + x^{2} \left (2 A a^{2} d^{3} e + A a b d^{4} + \frac {B a^{2} d^{4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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