Optimal. Leaf size=77 \[ -\frac {(d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3}+\frac {(d+e x)^3 (b d-a e) (B d-A e)}{3 e^3}+\frac {b B (d+e x)^5}{5 e^3} \]
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Rubi [A] time = 0.07, antiderivative size = 77, 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} \[ -\frac {(d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3}+\frac {(d+e x)^3 (b d-a e) (B d-A e)}{3 e^3}+\frac {b B (d+e x)^5}{5 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x) (A+B x) (d+e x)^2 \, dx &=\int \left (\frac {(-b d+a e) (-B d+A e) (d+e x)^2}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^3}{e^2}+\frac {b B (d+e x)^4}{e^2}\right ) \, dx\\ &=\frac {(b d-a e) (B d-A e) (d+e x)^3}{3 e^3}-\frac {(2 b B d-A b e-a B e) (d+e x)^4}{4 e^3}+\frac {b B (d+e x)^5}{5 e^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 96, normalized size = 1.25 \[ \frac {1}{3} x^3 \left (a A e^2+2 a B d e+2 A b d e+b B d^2\right )+\frac {1}{4} e x^4 (a B e+A b e+2 b B d)+\frac {1}{2} d x^2 (2 a A e+a B d+A b d)+a A d^2 x+\frac {1}{5} b B e^2 x^5 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 113, normalized size = 1.47 \[ \frac {1}{5} x^{5} e^{2} b B + \frac {1}{2} x^{4} e d b B + \frac {1}{4} x^{4} e^{2} a B + \frac {1}{4} x^{4} e^{2} b A + \frac {1}{3} x^{3} d^{2} b B + \frac {2}{3} x^{3} e d a B + \frac {2}{3} x^{3} e d b A + \frac {1}{3} x^{3} e^{2} a A + \frac {1}{2} x^{2} d^{2} a B + \frac {1}{2} x^{2} d^{2} b A + x^{2} e d a A + x d^{2} a A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 113, normalized size = 1.47 \[ \frac {1}{5} \, B b x^{5} e^{2} + \frac {1}{2} \, B b d x^{4} e + \frac {1}{3} \, B b d^{2} x^{3} + \frac {1}{4} \, B a x^{4} e^{2} + \frac {1}{4} \, A b x^{4} e^{2} + \frac {2}{3} \, B a d x^{3} e + \frac {2}{3} \, A b d x^{3} e + \frac {1}{2} \, B a d^{2} x^{2} + \frac {1}{2} \, A b d^{2} x^{2} + \frac {1}{3} \, A a x^{3} e^{2} + A a d x^{2} e + A a d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 94, normalized size = 1.22 \[ \frac {B b \,e^{2} x^{5}}{5}+A a \,d^{2} x +\frac {\left (2 B b d e +\left (A b +B a \right ) e^{2}\right ) x^{4}}{4}+\frac {\left (A a \,e^{2}+B b \,d^{2}+2 \left (A b +B a \right ) d e \right ) x^{3}}{3}+\frac {\left (2 A a d e +\left (A b +B a \right ) d^{2}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 93, normalized size = 1.21 \[ \frac {1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac {1}{4} \, {\left (2 \, B b d e + {\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b d^{2} + A a e^{2} + 2 \, {\left (B a + A b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a d e + {\left (B a + A b\right )} d^{2}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 98, normalized size = 1.27 \[ x^3\,\left (\frac {A\,a\,e^2}{3}+\frac {B\,b\,d^2}{3}+\frac {2\,A\,b\,d\,e}{3}+\frac {2\,B\,a\,d\,e}{3}\right )+x^2\,\left (\frac {A\,b\,d^2}{2}+\frac {B\,a\,d^2}{2}+A\,a\,d\,e\right )+x^4\,\left (\frac {A\,b\,e^2}{4}+\frac {B\,a\,e^2}{4}+\frac {B\,b\,d\,e}{2}\right )+A\,a\,d^2\,x+\frac {B\,b\,e^2\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 116, normalized size = 1.51 \[ A a d^{2} x + \frac {B b e^{2} x^{5}}{5} + x^{4} \left (\frac {A b e^{2}}{4} + \frac {B a e^{2}}{4} + \frac {B b d e}{2}\right ) + x^{3} \left (\frac {A a e^{2}}{3} + \frac {2 A b d e}{3} + \frac {2 B a d e}{3} + \frac {B b d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac {A b d^{2}}{2} + \frac {B a d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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