Optimal. Leaf size=45 \[ \frac {x^{m+3} (a B+A b)}{m+3}+\frac {a A x^{m+1}}{m+1}+\frac {b B x^{m+5}}{m+5} \]
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Rubi [A] time = 0.02, antiderivative size = 45, 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 = {448} \[ \frac {x^{m+3} (a B+A b)}{m+3}+\frac {a A x^{m+1}}{m+1}+\frac {b B x^{m+5}}{m+5} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int x^m \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx &=\int \left (a A x^m+(A b+a B) x^{2+m}+b B x^{4+m}\right ) \, dx\\ &=\frac {a A x^{1+m}}{1+m}+\frac {(A b+a B) x^{3+m}}{3+m}+\frac {b B x^{5+m}}{5+m}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.93 \[ x^{m+1} \left (\frac {x^2 (a B+A b)}{m+3}+\frac {a A}{m+1}+\frac {b B x^4}{m+5}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 92, normalized size = 2.04 \[ \frac {{\left ({\left (B b m^{2} + 4 \, B b m + 3 \, B b\right )} x^{5} + {\left ({\left (B a + A b\right )} m^{2} + 5 \, B a + 5 \, A b + 6 \, {\left (B a + A b\right )} m\right )} x^{3} + {\left (A a m^{2} + 8 \, A a m + 15 \, A a\right )} x\right )} x^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 143, normalized size = 3.18 \[ \frac {B b m^{2} x^{5} x^{m} + 4 \, B b m x^{5} x^{m} + B a m^{2} x^{3} x^{m} + A b m^{2} x^{3} x^{m} + 3 \, B b x^{5} x^{m} + 6 \, B a m x^{3} x^{m} + 6 \, A b m x^{3} x^{m} + A a m^{2} x x^{m} + 5 \, B a x^{3} x^{m} + 5 \, A b x^{3} x^{m} + 8 \, A a m x x^{m} + 15 \, A a x x^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 110, normalized size = 2.44 \[ \frac {\left (B b \,m^{2} x^{4}+4 B b m \,x^{4}+A b \,m^{2} x^{2}+B a \,m^{2} x^{2}+3 B b \,x^{4}+6 A b m \,x^{2}+6 B a m \,x^{2}+A a \,m^{2}+5 A b \,x^{2}+5 B a \,x^{2}+8 A a m +15 A a \right ) x^{m +1}}{\left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 53, normalized size = 1.18 \[ \frac {B b x^{m + 5}}{m + 5} + \frac {B a x^{m + 3}}{m + 3} + \frac {A b x^{m + 3}}{m + 3} + \frac {A a x^{m + 1}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 95, normalized size = 2.11 \[ x^m\,\left (\frac {x^3\,\left (A\,b+B\,a\right )\,\left (m^2+6\,m+5\right )}{m^3+9\,m^2+23\,m+15}+\frac {B\,b\,x^5\,\left (m^2+4\,m+3\right )}{m^3+9\,m^2+23\,m+15}+\frac {A\,a\,x\,\left (m^2+8\,m+15\right )}{m^3+9\,m^2+23\,m+15}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 410, normalized size = 9.11 \[ \begin {cases} - \frac {A a}{4 x^{4}} - \frac {A b}{2 x^{2}} - \frac {B a}{2 x^{2}} + B b \log {\relax (x )} & \text {for}\: m = -5 \\- \frac {A a}{2 x^{2}} + A b \log {\relax (x )} + B a \log {\relax (x )} + \frac {B b x^{2}}{2} & \text {for}\: m = -3 \\A a \log {\relax (x )} + \frac {A b x^{2}}{2} + \frac {B a x^{2}}{2} + \frac {B b x^{4}}{4} & \text {for}\: m = -1 \\\frac {A a m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {8 A a m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {15 A a x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {A b m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {6 A b m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {5 A b x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {B a m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {6 B a m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {5 B a x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {B b m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {4 B b m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {3 B b x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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