Optimal. Leaf size=75 \[ \frac {(a+b x)^5 (-2 a B e+A b e+b B d)}{5 b^3}+\frac {(a+b x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac {B e (a+b x)^6}{6 b^3} \]
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Rubi [A] time = 0.10, antiderivative size = 75, normalized size of antiderivative = 1.00, 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} \[ \frac {(a+b x)^5 (-2 a B e+A b e+b B d)}{5 b^3}+\frac {(a+b x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac {B e (a+b x)^6}{6 b^3} \]
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) \, dx &=\int \left (\frac {(A b-a B) (b d-a e) (a+b x)^3}{b^2}+\frac {(b B d+A b e-2 a B e) (a+b x)^4}{b^2}+\frac {B e (a+b x)^5}{b^2}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e) (a+b x)^4}{4 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^5}{5 b^3}+\frac {B e (a+b x)^6}{6 b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 130, normalized size = 1.73 \[ a^3 A d x+\frac {1}{2} a^2 x^2 (a A e+a B d+3 A b d)+\frac {1}{5} b^2 x^5 (3 a B e+A b e+b B d)+\frac {1}{4} b x^4 (A b (3 a e+b d)+3 a B (a e+b d))+\frac {1}{3} a x^3 (3 A b (a e+b d)+a B (a e+3 b d))+\frac {1}{6} b^3 B e x^6 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 163, normalized size = 2.17 \[ \frac {1}{6} x^{6} e b^{3} B + \frac {1}{5} x^{5} d b^{3} B + \frac {3}{5} x^{5} e b^{2} a B + \frac {1}{5} x^{5} e b^{3} A + \frac {3}{4} x^{4} d b^{2} a B + \frac {3}{4} x^{4} e b a^{2} B + \frac {1}{4} x^{4} d b^{3} A + \frac {3}{4} x^{4} e b^{2} a A + x^{3} d b a^{2} B + \frac {1}{3} x^{3} e a^{3} B + x^{3} d b^{2} a A + x^{3} e b a^{2} A + \frac {1}{2} x^{2} d a^{3} B + \frac {3}{2} x^{2} d b a^{2} A + \frac {1}{2} x^{2} e a^{3} A + x d a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.24, size = 171, normalized size = 2.28 \[ \frac {1}{6} \, B b^{3} x^{6} e + \frac {1}{5} \, B b^{3} d x^{5} + \frac {3}{5} \, B a b^{2} x^{5} e + \frac {1}{5} \, A b^{3} x^{5} e + \frac {3}{4} \, B a b^{2} d x^{4} + \frac {1}{4} \, A b^{3} d x^{4} + \frac {3}{4} \, B a^{2} b x^{4} e + \frac {3}{4} \, A a b^{2} x^{4} e + B a^{2} b d x^{3} + A a b^{2} d x^{3} + \frac {1}{3} \, B a^{3} x^{3} e + A a^{2} b x^{3} e + \frac {1}{2} \, B a^{3} d x^{2} + \frac {3}{2} \, A a^{2} b d x^{2} + \frac {1}{2} \, A a^{3} x^{2} e + A a^{3} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 149, normalized size = 1.99 \[ \frac {B \,b^{3} e \,x^{6}}{6}+A \,a^{3} d x +\frac {\left (B \,b^{3} d +\left (b^{3} A +3 a \,b^{2} B \right ) e \right ) x^{5}}{5}+\frac {\left (\left (b^{3} A +3 a \,b^{2} B \right ) d +\left (3 a \,b^{2} A +3 a^{2} b B \right ) e \right ) x^{4}}{4}+\frac {\left (\left (3 a \,b^{2} A +3 a^{2} b B \right ) d +\left (3 A \,a^{2} b +B \,a^{3}\right ) e \right ) x^{3}}{3}+\frac {\left (A \,a^{3} e +\left (3 A \,a^{2} b +B \,a^{3}\right ) d \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 146, normalized size = 1.95 \[ \frac {1}{6} \, B b^{3} e x^{6} + A a^{3} d x + \frac {1}{5} \, {\left (B b^{3} d + {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} x^{5} + \frac {1}{4} \, {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (B a^{2} b + A a b^{2}\right )} d + {\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{3} e + {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 141, normalized size = 1.88 \[ x^2\,\left (\frac {A\,a^3\,e}{2}+\frac {B\,a^3\,d}{2}+\frac {3\,A\,a^2\,b\,d}{2}\right )+x^5\,\left (\frac {A\,b^3\,e}{5}+\frac {B\,b^3\,d}{5}+\frac {3\,B\,a\,b^2\,e}{5}\right )+x^3\,\left (\frac {B\,a^3\,e}{3}+A\,a\,b^2\,d+A\,a^2\,b\,e+B\,a^2\,b\,d\right )+x^4\,\left (\frac {A\,b^3\,d}{4}+\frac {3\,A\,a\,b^2\,e}{4}+\frac {3\,B\,a\,b^2\,d}{4}+\frac {3\,B\,a^2\,b\,e}{4}\right )+A\,a^3\,d\,x+\frac {B\,b^3\,e\,x^6}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 168, normalized size = 2.24 \[ A a^{3} d x + \frac {B b^{3} e x^{6}}{6} + x^{5} \left (\frac {A b^{3} e}{5} + \frac {3 B a b^{2} e}{5} + \frac {B b^{3} d}{5}\right ) + x^{4} \left (\frac {3 A a b^{2} e}{4} + \frac {A b^{3} d}{4} + \frac {3 B a^{2} b e}{4} + \frac {3 B a b^{2} d}{4}\right ) + x^{3} \left (A a^{2} b e + A a b^{2} d + \frac {B a^{3} e}{3} + B a^{2} b d\right ) + x^{2} \left (\frac {A a^{3} e}{2} + \frac {3 A a^{2} b d}{2} + \frac {B a^{3} d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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