Integrand size = 22, antiderivative size = 103 \[ \int x^5 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {a^2 (A b-a B) \left (a+b x^2\right )^{5/2}}{5 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{9/2}}{9 b^4}+\frac {B \left (a+b x^2\right )^{11/2}}{11 b^4} \]
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Time = 0.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78} \[ \int x^5 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {a^2 \left (a+b x^2\right )^{5/2} (A b-a B)}{5 b^4}+\frac {\left (a+b x^2\right )^{9/2} (A b-3 a B)}{9 b^4}-\frac {a \left (a+b x^2\right )^{7/2} (2 A b-3 a B)}{7 b^4}+\frac {B \left (a+b x^2\right )^{11/2}}{11 b^4} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^2 (a+b x)^{3/2} (A+B x) \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^2 (-A b+a B) (a+b x)^{3/2}}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^{5/2}}{b^3}+\frac {(A b-3 a B) (a+b x)^{7/2}}{b^3}+\frac {B (a+b x)^{9/2}}{b^3}\right ) \, dx,x,x^2\right ) \\ & = \frac {a^2 (A b-a B) \left (a+b x^2\right )^{5/2}}{5 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{9/2}}{9 b^4}+\frac {B \left (a+b x^2\right )^{11/2}}{11 b^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int x^5 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {\left (a+b x^2\right )^{5/2} \left (88 a^2 A b-48 a^3 B-220 a A b^2 x^2+120 a^2 b B x^2+385 A b^3 x^4-210 a b^2 B x^4+315 b^3 B x^6\right )}{3465 b^4} \]
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Time = 2.78 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.66
method | result | size |
pseudoelliptic | \(\frac {8 \left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (\frac {35 x^{4} \left (\frac {9 x^{2} B}{11}+A \right ) b^{3}}{8}-\frac {5 x^{2} a \left (\frac {21 x^{2} B}{22}+A \right ) b^{2}}{2}+a^{2} \left (\frac {15 x^{2} B}{11}+A \right ) b -\frac {6 a^{3} B}{11}\right )}{315 b^{4}}\) | \(68\) |
gosper | \(\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (315 b^{3} B \,x^{6}+385 A \,b^{3} x^{4}-210 B a \,b^{2} x^{4}-220 a A \,b^{2} x^{2}+120 B \,a^{2} b \,x^{2}+88 a^{2} b A -48 a^{3} B \right )}{3465 b^{4}}\) | \(77\) |
trager | \(\frac {\left (315 b^{5} B \,x^{10}+385 b^{5} A \,x^{8}+420 a \,b^{4} B \,x^{8}+550 a \,b^{4} A \,x^{6}+15 a^{2} b^{3} B \,x^{6}+33 a^{2} b^{3} A \,x^{4}-18 a^{3} b^{2} B \,x^{4}-44 a^{3} A \,b^{2} x^{2}+24 B \,a^{4} b \,x^{2}+88 a^{4} b A -48 a^{5} B \right ) \sqrt {b \,x^{2}+a}}{3465 b^{4}}\) | \(125\) |
risch | \(\frac {\left (315 b^{5} B \,x^{10}+385 b^{5} A \,x^{8}+420 a \,b^{4} B \,x^{8}+550 a \,b^{4} A \,x^{6}+15 a^{2} b^{3} B \,x^{6}+33 a^{2} b^{3} A \,x^{4}-18 a^{3} b^{2} B \,x^{4}-44 a^{3} A \,b^{2} x^{2}+24 B \,a^{4} b \,x^{2}+88 a^{4} b A -48 a^{5} B \right ) \sqrt {b \,x^{2}+a}}{3465 b^{4}}\) | \(125\) |
default | \(B \left (\frac {x^{6} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{11 b}-\frac {6 a \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )}{9 b}\right )}{11 b}\right )+A \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )}{9 b}\right )\) | \(144\) |
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Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.20 \[ \int x^5 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {{\left (315 \, B b^{5} x^{10} + 35 \, {\left (12 \, B a b^{4} + 11 \, A b^{5}\right )} x^{8} + 5 \, {\left (3 \, B a^{2} b^{3} + 110 \, A a b^{4}\right )} x^{6} - 48 \, B a^{5} + 88 \, A a^{4} b - 3 \, {\left (6 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{4} + 4 \, {\left (6 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3465 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (92) = 184\).
Time = 0.44 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.52 \[ \int x^5 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\begin {cases} \frac {8 A a^{4} \sqrt {a + b x^{2}}}{315 b^{3}} - \frac {4 A a^{3} x^{2} \sqrt {a + b x^{2}}}{315 b^{2}} + \frac {A a^{2} x^{4} \sqrt {a + b x^{2}}}{105 b} + \frac {10 A a x^{6} \sqrt {a + b x^{2}}}{63} + \frac {A b x^{8} \sqrt {a + b x^{2}}}{9} - \frac {16 B a^{5} \sqrt {a + b x^{2}}}{1155 b^{4}} + \frac {8 B a^{4} x^{2} \sqrt {a + b x^{2}}}{1155 b^{3}} - \frac {2 B a^{3} x^{4} \sqrt {a + b x^{2}}}{385 b^{2}} + \frac {B a^{2} x^{6} \sqrt {a + b x^{2}}}{231 b} + \frac {4 B a x^{8} \sqrt {a + b x^{2}}}{33} + \frac {B b x^{10} \sqrt {a + b x^{2}}}{11} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{6}}{6} + \frac {B x^{8}}{8}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.28 \[ \int x^5 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B x^{6}}{11 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a x^{4}}{33 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A x^{4}}{9 \, b} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} x^{2}}{231 \, b^{3}} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a x^{2}}{63 \, b^{2}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{3}}{1155 \, b^{4}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a^{2}}{315 \, b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01 \[ \int x^5 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {315 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} B - 1155 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B a + 1485 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2} - 693 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{3} + 385 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} A b - 990 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a b + 693 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a^{2} b}{3465 \, b^{4}} \]
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Time = 5.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.14 \[ \int x^5 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\sqrt {b\,x^2+a}\,\left (\frac {x^8\,\left (385\,A\,b^5+420\,B\,a\,b^4\right )}{3465\,b^4}-\frac {48\,B\,a^5-88\,A\,a^4\,b}{3465\,b^4}+\frac {B\,b\,x^{10}}{11}+\frac {a^2\,x^4\,\left (11\,A\,b-6\,B\,a\right )}{1155\,b^2}-\frac {4\,a^3\,x^2\,\left (11\,A\,b-6\,B\,a\right )}{3465\,b^3}+\frac {a\,x^6\,\left (110\,A\,b+3\,B\,a\right )}{693\,b}\right ) \]
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