Optimal. Leaf size=71 \[ \frac {a^2 A x^{m+1}}{m+1}+\frac {a x^{m+4} (a B+2 A b)}{m+4}+\frac {b x^{m+7} (2 a B+A b)}{m+7}+\frac {b^2 B x^{m+10}}{m+10} \]
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Rubi [A] time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \[ \frac {a^2 A x^{m+1}}{m+1}+\frac {a x^{m+4} (a B+2 A b)}{m+4}+\frac {b x^{m+7} (2 a B+A b)}{m+7}+\frac {b^2 B x^{m+10}}{m+10} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int x^m \left (a+b x^3\right )^2 \left (A+B x^3\right ) \, dx &=\int \left (a^2 A x^m+a (2 A b+a B) x^{3+m}+b (A b+2 a B) x^{6+m}+b^2 B x^{9+m}\right ) \, dx\\ &=\frac {a^2 A x^{1+m}}{1+m}+\frac {a (2 A b+a B) x^{4+m}}{4+m}+\frac {b (A b+2 a B) x^{7+m}}{7+m}+\frac {b^2 B x^{10+m}}{10+m}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 66, normalized size = 0.93 \[ x^{m+1} \left (\frac {a^2 A}{m+1}+\frac {b x^6 (2 a B+A b)}{m+7}+\frac {a x^3 (a B+2 A b)}{m+4}+\frac {b^2 B x^9}{m+10}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 215, normalized size = 3.03 \[ \frac {{\left ({\left (B b^{2} m^{3} + 12 \, B b^{2} m^{2} + 39 \, B b^{2} m + 28 \, B b^{2}\right )} x^{10} + {\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 80 \, B a b + 40 \, A b^{2} + 15 \, {\left (2 \, B a b + A b^{2}\right )} m^{2} + 54 \, {\left (2 \, B a b + A b^{2}\right )} m\right )} x^{7} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 70 \, B a^{2} + 140 \, A a b + 18 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 87 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{4} + {\left (A a^{2} m^{3} + 21 \, A a^{2} m^{2} + 138 \, A a^{2} m + 280 \, A a^{2}\right )} x\right )} x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 332, normalized size = 4.68 \[ \frac {B b^{2} m^{3} x^{10} x^{m} + 12 \, B b^{2} m^{2} x^{10} x^{m} + 39 \, B b^{2} m x^{10} x^{m} + 2 \, B a b m^{3} x^{7} x^{m} + A b^{2} m^{3} x^{7} x^{m} + 28 \, B b^{2} x^{10} x^{m} + 30 \, B a b m^{2} x^{7} x^{m} + 15 \, A b^{2} m^{2} x^{7} x^{m} + 108 \, B a b m x^{7} x^{m} + 54 \, A b^{2} m x^{7} x^{m} + B a^{2} m^{3} x^{4} x^{m} + 2 \, A a b m^{3} x^{4} x^{m} + 80 \, B a b x^{7} x^{m} + 40 \, A b^{2} x^{7} x^{m} + 18 \, B a^{2} m^{2} x^{4} x^{m} + 36 \, A a b m^{2} x^{4} x^{m} + 87 \, B a^{2} m x^{4} x^{m} + 174 \, A a b m x^{4} x^{m} + A a^{2} m^{3} x x^{m} + 70 \, B a^{2} x^{4} x^{m} + 140 \, A a b x^{4} x^{m} + 21 \, A a^{2} m^{2} x x^{m} + 138 \, A a^{2} m x x^{m} + 280 \, A a^{2} x x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 262, normalized size = 3.69 \[ \frac {\left (B \,b^{2} m^{3} x^{9}+12 B \,b^{2} m^{2} x^{9}+39 B \,b^{2} m \,x^{9}+A \,b^{2} m^{3} x^{6}+2 B a b \,m^{3} x^{6}+28 b^{2} B \,x^{9}+15 A \,b^{2} m^{2} x^{6}+30 B a b \,m^{2} x^{6}+54 A \,b^{2} m \,x^{6}+108 B a b m \,x^{6}+2 A a b \,m^{3} x^{3}+40 A \,b^{2} x^{6}+B \,a^{2} m^{3} x^{3}+80 B a b \,x^{6}+36 A a b \,m^{2} x^{3}+18 B \,a^{2} m^{2} x^{3}+174 A a b m \,x^{3}+87 B \,a^{2} m \,x^{3}+A \,a^{2} m^{3}+140 A a b \,x^{3}+70 B \,a^{2} x^{3}+21 A \,a^{2} m^{2}+138 A \,a^{2} m +280 a^{2} A \right ) x^{m +1}}{\left (m +10\right ) \left (m +7\right ) \left (m +4\right ) \left (m +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 91, normalized size = 1.28 \[ \frac {B b^{2} x^{m + 10}}{m + 10} + \frac {2 \, B a b x^{m + 7}}{m + 7} + \frac {A b^{2} x^{m + 7}}{m + 7} + \frac {B a^{2} x^{m + 4}}{m + 4} + \frac {2 \, A a b x^{m + 4}}{m + 4} + \frac {A a^{2} x^{m + 1}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.72, size = 177, normalized size = 2.49 \[ x^m\,\left (\frac {B\,b^2\,x^{10}\,\left (m^3+12\,m^2+39\,m+28\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}+\frac {A\,a^2\,x\,\left (m^3+21\,m^2+138\,m+280\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}+\frac {a\,x^4\,\left (2\,A\,b+B\,a\right )\,\left (m^3+18\,m^2+87\,m+70\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}+\frac {b\,x^7\,\left (A\,b+2\,B\,a\right )\,\left (m^3+15\,m^2+54\,m+40\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.31, size = 1057, normalized size = 14.89 \[ \begin {cases} - \frac {A a^{2}}{9 x^{9}} - \frac {A a b}{3 x^{6}} - \frac {A b^{2}}{3 x^{3}} - \frac {B a^{2}}{6 x^{6}} - \frac {2 B a b}{3 x^{3}} + B b^{2} \log {\relax (x )} & \text {for}\: m = -10 \\- \frac {A a^{2}}{6 x^{6}} - \frac {2 A a b}{3 x^{3}} + A b^{2} \log {\relax (x )} - \frac {B a^{2}}{3 x^{3}} + 2 B a b \log {\relax (x )} + \frac {B b^{2} x^{3}}{3} & \text {for}\: m = -7 \\- \frac {A a^{2}}{3 x^{3}} + 2 A a b \log {\relax (x )} + \frac {A b^{2} x^{3}}{3} + B a^{2} \log {\relax (x )} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{6}}{6} & \text {for}\: m = -4 \\A a^{2} \log {\relax (x )} + \frac {2 A a b x^{3}}{3} + \frac {A b^{2} x^{6}}{6} + \frac {B a^{2} x^{3}}{3} + \frac {B a b x^{6}}{3} + \frac {B b^{2} x^{9}}{9} & \text {for}\: m = -1 \\\frac {A a^{2} m^{3} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {21 A a^{2} m^{2} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {138 A a^{2} m x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {280 A a^{2} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {2 A a b m^{3} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {36 A a b m^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {174 A a b m x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {140 A a b x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {A b^{2} m^{3} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {15 A b^{2} m^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {54 A b^{2} m x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {40 A b^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {B a^{2} m^{3} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {18 B a^{2} m^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {87 B a^{2} m x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {70 B a^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {2 B a b m^{3} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {30 B a b m^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {108 B a b m x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {80 B a b x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {B b^{2} m^{3} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {12 B b^{2} m^{2} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {39 B b^{2} m x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {28 B b^{2} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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