Optimal. Leaf size=96 \[ \frac {a^3 A x^{1+m}}{1+m}+\frac {a^2 (3 A b+a B) x^{3+m}}{3+m}+\frac {3 a b (A b+a B) x^{5+m}}{5+m}+\frac {b^2 (A b+3 a B) x^{7+m}}{7+m}+\frac {b^3 B x^{9+m}}{9+m} \]
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Rubi [A]
time = 0.04, antiderivative size = 96, 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 = {459}
\begin {gather*} \frac {a^3 A x^{m+1}}{m+1}+\frac {a^2 x^{m+3} (a B+3 A b)}{m+3}+\frac {b^2 x^{m+7} (3 a B+A b)}{m+7}+\frac {3 a b x^{m+5} (a B+A b)}{m+5}+\frac {b^3 B x^{m+9}}{m+9} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx &=\int \left (a^3 A x^m+a^2 (3 A b+a B) x^{2+m}+3 a b (A b+a B) x^{4+m}+b^2 (A b+3 a B) x^{6+m}+b^3 B x^{8+m}\right ) \, dx\\ &=\frac {a^3 A x^{1+m}}{1+m}+\frac {a^2 (3 A b+a B) x^{3+m}}{3+m}+\frac {3 a b (A b+a B) x^{5+m}}{5+m}+\frac {b^2 (A b+3 a B) x^{7+m}}{7+m}+\frac {b^3 B x^{9+m}}{9+m}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 89, normalized size = 0.93 \begin {gather*} x^{1+m} \left (\frac {a^3 A}{1+m}+\frac {a^2 (3 A b+a B) x^2}{3+m}+\frac {3 a b (A b+a B) x^4}{5+m}+\frac {b^2 (A b+3 a B) x^6}{7+m}+\frac {b^3 B x^8}{9+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs.
\(2(96)=192\).
time = 0.07, size = 473, normalized size = 4.93
method | result | size |
risch | \(\frac {x \left (B \,b^{3} m^{4} x^{8}+16 B \,b^{3} m^{3} x^{8}+A \,b^{3} m^{4} x^{6}+3 B a \,b^{2} m^{4} x^{6}+86 B \,b^{3} m^{2} x^{8}+18 A \,b^{3} m^{3} x^{6}+54 B a \,b^{2} m^{3} x^{6}+176 m \,x^{8} B \,b^{3}+3 A a \,b^{2} m^{4} x^{4}+104 A \,b^{3} m^{2} x^{6}+3 B \,a^{2} b \,m^{4} x^{4}+312 B a \,b^{2} m^{2} x^{6}+105 B \,b^{3} x^{8}+60 A a \,b^{2} m^{3} x^{4}+222 A \,b^{3} x^{6} m +60 B \,a^{2} b \,m^{3} x^{4}+666 B a \,b^{2} x^{6} m +3 A \,a^{2} b \,m^{4} x^{2}+390 A a \,b^{2} m^{2} x^{4}+135 A \,b^{3} x^{6}+B \,a^{3} m^{4} x^{2}+390 B \,a^{2} b \,m^{2} x^{4}+405 B a \,b^{2} x^{6}+66 A \,a^{2} b \,m^{3} x^{2}+900 A a \,b^{2} x^{4} m +22 B \,a^{3} m^{3} x^{2}+900 B \,a^{2} b \,x^{4} m +A \,a^{3} m^{4}+492 A \,a^{2} b \,m^{2} x^{2}+567 A a \,b^{2} x^{4}+164 B \,a^{3} m^{2} x^{2}+567 B \,a^{2} b \,x^{4}+24 A \,a^{3} m^{3}+1374 A \,a^{2} b \,x^{2} m +458 B \,a^{3} x^{2} m +206 A \,a^{3} m^{2}+945 A \,a^{2} b \,x^{2}+315 B \,a^{3} x^{2}+744 A \,a^{3} m +945 A \,a^{3}\right ) x^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(473\) |
gosper | \(\frac {x^{1+m} \left (B \,b^{3} m^{4} x^{8}+16 B \,b^{3} m^{3} x^{8}+A \,b^{3} m^{4} x^{6}+3 B a \,b^{2} m^{4} x^{6}+86 B \,b^{3} m^{2} x^{8}+18 A \,b^{3} m^{3} x^{6}+54 B a \,b^{2} m^{3} x^{6}+176 m \,x^{8} B \,b^{3}+3 A a \,b^{2} m^{4} x^{4}+104 A \,b^{3} m^{2} x^{6}+3 B \,a^{2} b \,m^{4} x^{4}+312 B a \,b^{2} m^{2} x^{6}+105 B \,b^{3} x^{8}+60 A a \,b^{2} m^{3} x^{4}+222 A \,b^{3} x^{6} m +60 B \,a^{2} b \,m^{3} x^{4}+666 B a \,b^{2} x^{6} m +3 A \,a^{2} b \,m^{4} x^{2}+390 A a \,b^{2} m^{2} x^{4}+135 A \,b^{3} x^{6}+B \,a^{3} m^{4} x^{2}+390 B \,a^{2} b \,m^{2} x^{4}+405 B a \,b^{2} x^{6}+66 A \,a^{2} b \,m^{3} x^{2}+900 A a \,b^{2} x^{4} m +22 B \,a^{3} m^{3} x^{2}+900 B \,a^{2} b \,x^{4} m +A \,a^{3} m^{4}+492 A \,a^{2} b \,m^{2} x^{2}+567 A a \,b^{2} x^{4}+164 B \,a^{3} m^{2} x^{2}+567 B \,a^{2} b \,x^{4}+24 A \,a^{3} m^{3}+1374 A \,a^{2} b \,x^{2} m +458 B \,a^{3} x^{2} m +206 A \,a^{3} m^{2}+945 A \,a^{2} b \,x^{2}+315 B \,a^{3} x^{2}+744 A \,a^{3} m +945 A \,a^{3}\right )}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(474\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 129, normalized size = 1.34 \begin {gather*} \frac {B b^{3} x^{m + 9}}{m + 9} + \frac {3 \, B a b^{2} x^{m + 7}}{m + 7} + \frac {A b^{3} x^{m + 7}}{m + 7} + \frac {3 \, B a^{2} b x^{m + 5}}{m + 5} + \frac {3 \, A a b^{2} x^{m + 5}}{m + 5} + \frac {B a^{3} x^{m + 3}}{m + 3} + \frac {3 \, A a^{2} b x^{m + 3}}{m + 3} + \frac {A a^{3} x^{m + 1}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 379 vs.
\(2 (96) = 192\).
time = 1.00, size = 379, normalized size = 3.95 \begin {gather*} \frac {{\left ({\left (B b^{3} m^{4} + 16 \, B b^{3} m^{3} + 86 \, B b^{3} m^{2} + 176 \, B b^{3} m + 105 \, B b^{3}\right )} x^{9} + {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 405 \, B a b^{2} + 135 \, A b^{3} + 18 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 104 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 222 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{7} + 3 \, {\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 189 \, B a^{2} b + 189 \, A a b^{2} + 20 \, {\left (B a^{2} b + A a b^{2}\right )} m^{3} + 130 \, {\left (B a^{2} b + A a b^{2}\right )} m^{2} + 300 \, {\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{5} + {\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 315 \, B a^{3} + 945 \, A a^{2} b + 22 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 164 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 458 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{3} + {\left (A a^{3} m^{4} + 24 \, A a^{3} m^{3} + 206 \, A a^{3} m^{2} + 744 \, A a^{3} m + 945 \, A a^{3}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2069 vs.
\(2 (87) = 174\).
time = 0.61, size = 2069, normalized size = 21.55 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 593 vs.
\(2 (96) = 192\).
time = 0.58, size = 593, normalized size = 6.18 \begin {gather*} \frac {B b^{3} m^{4} x^{9} x^{m} + 16 \, B b^{3} m^{3} x^{9} x^{m} + 3 \, B a b^{2} m^{4} x^{7} x^{m} + A b^{3} m^{4} x^{7} x^{m} + 86 \, B b^{3} m^{2} x^{9} x^{m} + 54 \, B a b^{2} m^{3} x^{7} x^{m} + 18 \, A b^{3} m^{3} x^{7} x^{m} + 176 \, B b^{3} m x^{9} x^{m} + 3 \, B a^{2} b m^{4} x^{5} x^{m} + 3 \, A a b^{2} m^{4} x^{5} x^{m} + 312 \, B a b^{2} m^{2} x^{7} x^{m} + 104 \, A b^{3} m^{2} x^{7} x^{m} + 105 \, B b^{3} x^{9} x^{m} + 60 \, B a^{2} b m^{3} x^{5} x^{m} + 60 \, A a b^{2} m^{3} x^{5} x^{m} + 666 \, B a b^{2} m x^{7} x^{m} + 222 \, A b^{3} m x^{7} x^{m} + B a^{3} m^{4} x^{3} x^{m} + 3 \, A a^{2} b m^{4} x^{3} x^{m} + 390 \, B a^{2} b m^{2} x^{5} x^{m} + 390 \, A a b^{2} m^{2} x^{5} x^{m} + 405 \, B a b^{2} x^{7} x^{m} + 135 \, A b^{3} x^{7} x^{m} + 22 \, B a^{3} m^{3} x^{3} x^{m} + 66 \, A a^{2} b m^{3} x^{3} x^{m} + 900 \, B a^{2} b m x^{5} x^{m} + 900 \, A a b^{2} m x^{5} x^{m} + A a^{3} m^{4} x x^{m} + 164 \, B a^{3} m^{2} x^{3} x^{m} + 492 \, A a^{2} b m^{2} x^{3} x^{m} + 567 \, B a^{2} b x^{5} x^{m} + 567 \, A a b^{2} x^{5} x^{m} + 24 \, A a^{3} m^{3} x x^{m} + 458 \, B a^{3} m x^{3} x^{m} + 1374 \, A a^{2} b m x^{3} x^{m} + 206 \, A a^{3} m^{2} x x^{m} + 315 \, B a^{3} x^{3} x^{m} + 945 \, A a^{2} b x^{3} x^{m} + 744 \, A a^{3} m x x^{m} + 945 \, A a^{3} x x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 289, normalized size = 3.01 \begin {gather*} \frac {A\,a^3\,x\,x^m\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {B\,b^3\,x^m\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {a^2\,x^m\,x^3\,\left (3\,A\,b+B\,a\right )\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {b^2\,x^m\,x^7\,\left (A\,b+3\,B\,a\right )\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {3\,a\,b\,x^m\,x^5\,\left (A\,b+B\,a\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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