Optimal. Leaf size=62 \[ \frac {1}{3} x^3 (a B e+A c d)+\frac {1}{2} a x^2 (A e+B d)+a A d x+\frac {1}{4} c x^4 (A e+B d)+\frac {1}{5} B c e x^5 \]
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Rubi [A] time = 0.06, antiderivative size = 62, 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 = {772} \[ \frac {1}{3} x^3 (a B e+A c d)+\frac {1}{2} a x^2 (A e+B d)+a A d x+\frac {1}{4} c x^4 (A e+B d)+\frac {1}{5} B c e x^5 \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin {align*} \int (A+B x) (d+e x) \left (a+c x^2\right ) \, dx &=\int \left (a A d+a (B d+A e) x+(A c d+a B e) x^2+c (B d+A e) x^3+B c e x^4\right ) \, dx\\ &=a A d x+\frac {1}{2} a (B d+A e) x^2+\frac {1}{3} (A c d+a B e) x^3+\frac {1}{4} c (B d+A e) x^4+\frac {1}{5} B c e x^5\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 1.00 \[ \frac {1}{3} x^3 (a B e+A c d)+\frac {1}{2} a x^2 (A e+B d)+a A d x+\frac {1}{4} c x^4 (A e+B d)+\frac {1}{5} B c e x^5 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 62, normalized size = 1.00 \[ \frac {1}{5} x^{5} e c B + \frac {1}{4} x^{4} d c B + \frac {1}{4} x^{4} e c A + \frac {1}{3} x^{3} e a B + \frac {1}{3} x^{3} d c A + \frac {1}{2} x^{2} d a B + \frac {1}{2} x^{2} e a A + x d a A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 66, normalized size = 1.06 \[ \frac {1}{5} \, B c x^{5} e + \frac {1}{4} \, B c d x^{4} + \frac {1}{4} \, A c x^{4} e + \frac {1}{3} \, A c d x^{3} + \frac {1}{3} \, B a x^{3} e + \frac {1}{2} \, B a d x^{2} + \frac {1}{2} \, A a x^{2} e + A a d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 55, normalized size = 0.89 \[ \frac {B c e \,x^{5}}{5}+\frac {\left (A e +B d \right ) c \,x^{4}}{4}+A a d x +\frac {\left (A e +B d \right ) a \,x^{2}}{2}+\frac {\left (A c d +a B e \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 56, normalized size = 0.90 \[ \frac {1}{5} \, B c e x^{5} + \frac {1}{4} \, {\left (B c d + A c e\right )} x^{4} + A a d x + \frac {1}{3} \, {\left (A c d + B a e\right )} x^{3} + \frac {1}{2} \, {\left (B a d + A a e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.67, size = 55, normalized size = 0.89 \[ \frac {B\,c\,e\,x^5}{5}+\frac {c\,\left (A\,e+B\,d\right )\,x^4}{4}+\left (\frac {A\,c\,d}{3}+\frac {B\,a\,e}{3}\right )\,x^3+\frac {a\,\left (A\,e+B\,d\right )\,x^2}{2}+A\,a\,d\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 66, normalized size = 1.06 \[ A a d x + \frac {B c e x^{5}}{5} + x^{4} \left (\frac {A c e}{4} + \frac {B c d}{4}\right ) + x^{3} \left (\frac {A c d}{3} + \frac {B a e}{3}\right ) + x^{2} \left (\frac {A a e}{2} + \frac {B a d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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