Optimal. Leaf size=60 \[ \frac {1}{3} x^3 (a C+A b)+a A x+\frac {1}{4} x^4 (a D+b B)+\frac {1}{2} a B x^2+\frac {1}{5} b C x^5+\frac {1}{6} b D x^6 \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1810} \[ \frac {1}{3} x^3 (a C+A b)+a A x+\frac {1}{4} x^4 (a D+b B)+\frac {1}{2} a B x^2+\frac {1}{5} b C x^5+\frac {1}{6} b D x^6 \]
Antiderivative was successfully verified.
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Rule 1810
Rubi steps
\begin {align*} \int \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (a A+a B x+(A b+a C) x^2+(b B+a D) x^3+b C x^4+b D x^5\right ) \, dx\\ &=a A x+\frac {1}{2} a B x^2+\frac {1}{3} (A b+a C) x^3+\frac {1}{4} (b B+a D) x^4+\frac {1}{5} b C x^5+\frac {1}{6} b D x^6\\ \end {align*}
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Mathematica [A] time = 0.01, size = 60, normalized size = 1.00 \[ \frac {1}{3} x^3 (a C+A b)+a A x+\frac {1}{4} x^4 (a D+b B)+\frac {1}{2} a B x^2+\frac {1}{5} b C x^5+\frac {1}{6} b D x^6 \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 54, normalized size = 0.90 \[ \frac {1}{6} \, D b x^{6} + \frac {1}{5} \, C b x^{5} + \frac {1}{4} \, D a x^{4} + \frac {1}{4} \, B b x^{4} + \frac {1}{3} \, C a x^{3} + \frac {1}{3} \, A b x^{3} + \frac {1}{2} \, B a x^{2} + A a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 51, normalized size = 0.85 \[ \frac {D b \,x^{6}}{6}+\frac {C b \,x^{5}}{5}+\frac {B a \,x^{2}}{2}+\frac {\left (b B +a D\right ) x^{4}}{4}+A a x +\frac {\left (A b +a C \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 50, normalized size = 0.83 \[ \frac {1}{6} \, D b x^{6} + \frac {1}{5} \, C b x^{5} + \frac {1}{4} \, {\left (D a + B b\right )} x^{4} + \frac {1}{2} \, B a x^{2} + \frac {1}{3} \, {\left (C a + A b\right )} x^{3} + A a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 54, normalized size = 0.90 \[ \frac {a\,x^4\,D}{4}+\frac {b\,x^6\,D}{6}+A\,a\,x+\frac {B\,a\,x^2}{2}+\frac {A\,b\,x^3}{3}+\frac {C\,a\,x^3}{3}+\frac {B\,b\,x^4}{4}+\frac {C\,b\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 56, normalized size = 0.93 \[ A a x + \frac {B a x^{2}}{2} + \frac {C b x^{5}}{5} + \frac {D b x^{6}}{6} + x^{4} \left (\frac {B b}{4} + \frac {D a}{4}\right ) + x^{3} \left (\frac {A b}{3} + \frac {C a}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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