Integrand size = 22, antiderivative size = 100 \[ \int \frac {x^5 \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {a^2 (A b-a B) \sqrt {a+b x^2}}{b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{3 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac {B \left (a+b x^2\right )^{7/2}}{7 b^4} \]
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Time = 0.05 (sec) , antiderivative size = 100, 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} \[ \int \frac {x^5 \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {a^2 \sqrt {a+b x^2} (A b-a B)}{b^4}+\frac {\left (a+b x^2\right )^{5/2} (A b-3 a B)}{5 b^4}-\frac {a \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{3 b^4}+\frac {B \left (a+b x^2\right )^{7/2}}{7 b^4} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2 (A+B x)}{\sqrt {a+b x}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^2 (-A b+a B)}{b^3 \sqrt {a+b x}}+\frac {a (-2 A b+3 a B) \sqrt {a+b x}}{b^3}+\frac {(A b-3 a B) (a+b x)^{3/2}}{b^3}+\frac {B (a+b x)^{5/2}}{b^3}\right ) \, dx,x,x^2\right ) \\ & = \frac {a^2 (A b-a B) \sqrt {a+b x^2}}{b^4}-\frac {a (2 A b-3 a B) \left (a+b x^2\right )^{3/2}}{3 b^4}+\frac {(A b-3 a B) \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac {B \left (a+b x^2\right )^{7/2}}{7 b^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.80 \[ \int \frac {x^5 \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (56 a^2 A b-48 a^3 B-28 a A b^2 x^2+24 a^2 b B x^2+21 A b^3 x^4-18 a b^2 B x^4+15 b^3 B x^6\right )}{105 b^4} \]
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Time = 2.79 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {8 \left (\frac {3 x^{4} \left (\frac {5 x^{2} B}{7}+A \right ) b^{3}}{8}-\frac {x^{2} a \left (\frac {9 x^{2} B}{14}+A \right ) b^{2}}{2}+a^{2} \left (\frac {3 x^{2} B}{7}+A \right ) b -\frac {6 a^{3} B}{7}\right ) \sqrt {b \,x^{2}+a}}{15 b^{4}}\) | \(68\) |
gosper | \(\frac {\sqrt {b \,x^{2}+a}\, \left (15 b^{3} B \,x^{6}+21 A \,b^{3} x^{4}-18 B a \,b^{2} x^{4}-28 a A \,b^{2} x^{2}+24 B \,a^{2} b \,x^{2}+56 a^{2} b A -48 a^{3} B \right )}{105 b^{4}}\) | \(77\) |
trager | \(\frac {\sqrt {b \,x^{2}+a}\, \left (15 b^{3} B \,x^{6}+21 A \,b^{3} x^{4}-18 B a \,b^{2} x^{4}-28 a A \,b^{2} x^{2}+24 B \,a^{2} b \,x^{2}+56 a^{2} b A -48 a^{3} B \right )}{105 b^{4}}\) | \(77\) |
risch | \(\frac {\sqrt {b \,x^{2}+a}\, \left (15 b^{3} B \,x^{6}+21 A \,b^{3} x^{4}-18 B a \,b^{2} x^{4}-28 a A \,b^{2} x^{2}+24 B \,a^{2} b \,x^{2}+56 a^{2} b A -48 a^{3} B \right )}{105 b^{4}}\) | \(77\) |
default | \(B \left (\frac {x^{6} \sqrt {b \,x^{2}+a}}{7 b}-\frac {6 a \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )}{7 b}\right )+A \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )\) | \(144\) |
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Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.76 \[ \int \frac {x^5 \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {{\left (15 \, B b^{3} x^{6} - 3 \, {\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} - 48 \, B a^{3} + 56 \, A a^{2} b + 4 \, {\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, b^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.72 \[ \int \frac {x^5 \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\begin {cases} \frac {8 A a^{2} \sqrt {a + b x^{2}}}{15 b^{3}} - \frac {4 A a x^{2} \sqrt {a + b x^{2}}}{15 b^{2}} + \frac {A x^{4} \sqrt {a + b x^{2}}}{5 b} - \frac {16 B a^{3} \sqrt {a + b x^{2}}}{35 b^{4}} + \frac {8 B a^{2} x^{2} \sqrt {a + b x^{2}}}{35 b^{3}} - \frac {6 B a x^{4} \sqrt {a + b x^{2}}}{35 b^{2}} + \frac {B x^{6} \sqrt {a + b x^{2}}}{7 b} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{6}}{6} + \frac {B x^{8}}{8}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.32 \[ \int \frac {x^5 \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} B x^{6}}{7 \, b} - \frac {6 \, \sqrt {b x^{2} + a} B a x^{4}}{35 \, b^{2}} + \frac {\sqrt {b x^{2} + a} A x^{4}}{5 \, b} + \frac {8 \, \sqrt {b x^{2} + a} B a^{2} x^{2}}{35 \, b^{3}} - \frac {4 \, \sqrt {b x^{2} + a} A a x^{2}}{15 \, b^{2}} - \frac {16 \, \sqrt {b x^{2} + a} B a^{3}}{35 \, b^{4}} + \frac {8 \, \sqrt {b x^{2} + a} A a^{2}}{15 \, b^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.01 \[ \int \frac {x^5 \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=-\frac {{\left (B a^{3} - A a^{2} b\right )} \sqrt {b x^{2} + a}}{b^{4}} + \frac {15 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B - 63 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a + 105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} + 21 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b - 70 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b}{105 \, b^{4}} \]
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Time = 5.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.80 \[ \int \frac {x^5 \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx=-\sqrt {b\,x^2+a}\,\left (\frac {48\,B\,a^3-56\,A\,a^2\,b}{105\,b^4}-\frac {B\,x^6}{7\,b}-\frac {x^4\,\left (21\,A\,b^3-18\,B\,a\,b^2\right )}{105\,b^4}+\frac {4\,a\,x^2\,\left (7\,A\,b-6\,B\,a\right )}{105\,b^3}\right ) \]
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