Integrand size = 22, antiderivative size = 146 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\frac {5}{8} b (4 A b+3 a B) x \sqrt {a+b x^2}+\frac {5 b (4 A b+3 a B) x \left (a+b x^2\right )^{3/2}}{12 a}-\frac {(4 A b+3 a B) \left (a+b x^2\right )^{5/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}+\frac {5}{8} a \sqrt {b} (4 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {464, 283, 201, 223, 212} \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\frac {5}{8} a \sqrt {b} (3 a B+4 A b) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {\left (a+b x^2\right )^{5/2} (3 a B+4 A b)}{3 a x}+\frac {5 b x \left (a+b x^2\right )^{3/2} (3 a B+4 A b)}{12 a}+\frac {5}{8} b x \sqrt {a+b x^2} (3 a B+4 A b)-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3} \]
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Rule 201
Rule 212
Rule 223
Rule 283
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}-\frac {(-4 A b-3 a B) \int \frac {\left (a+b x^2\right )^{5/2}}{x^2} \, dx}{3 a} \\ & = -\frac {(4 A b+3 a B) \left (a+b x^2\right )^{5/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}+\frac {(5 b (4 A b+3 a B)) \int \left (a+b x^2\right )^{3/2} \, dx}{3 a} \\ & = \frac {5 b (4 A b+3 a B) x \left (a+b x^2\right )^{3/2}}{12 a}-\frac {(4 A b+3 a B) \left (a+b x^2\right )^{5/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}+\frac {1}{4} (5 b (4 A b+3 a B)) \int \sqrt {a+b x^2} \, dx \\ & = \frac {5}{8} b (4 A b+3 a B) x \sqrt {a+b x^2}+\frac {5 b (4 A b+3 a B) x \left (a+b x^2\right )^{3/2}}{12 a}-\frac {(4 A b+3 a B) \left (a+b x^2\right )^{5/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}+\frac {1}{8} (5 a b (4 A b+3 a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx \\ & = \frac {5}{8} b (4 A b+3 a B) x \sqrt {a+b x^2}+\frac {5 b (4 A b+3 a B) x \left (a+b x^2\right )^{3/2}}{12 a}-\frac {(4 A b+3 a B) \left (a+b x^2\right )^{5/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}+\frac {1}{8} (5 a b (4 A b+3 a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right ) \\ & = \frac {5}{8} b (4 A b+3 a B) x \sqrt {a+b x^2}+\frac {5 b (4 A b+3 a B) x \left (a+b x^2\right )^{3/2}}{12 a}-\frac {(4 A b+3 a B) \left (a+b x^2\right )^{5/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}+\frac {5}{8} a \sqrt {b} (4 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\frac {\sqrt {a+b x^2} \left (-8 a^2 A-56 a A b x^2-24 a^2 B x^2+12 A b^2 x^4+27 a b B x^4+6 b^2 B x^6\right )}{24 x^3}+\frac {5}{4} a \sqrt {b} (4 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right ) \]
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Time = 2.85 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-6 b^{2} B \,x^{6}-12 A \,b^{2} x^{4}-27 B a b \,x^{4}+56 a A b \,x^{2}+24 a^{2} B \,x^{2}+8 a^{2} A \right )}{24 x^{3}}+\frac {5 a \sqrt {b}\, \left (4 A b +3 B a \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{8}\) | \(97\) |
pseudoelliptic | \(-\frac {-\frac {15 x^{3} \left (A b +\frac {3 B a}{4}\right ) b a \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{2}+\left (7 x^{2} \left (-\frac {27 x^{2} B}{56}+A \right ) a \,b^{\frac {3}{2}}-\frac {3 x^{4} \left (\frac {x^{2} B}{2}+A \right ) b^{\frac {5}{2}}}{2}+a^{2} \sqrt {b}\, \left (3 x^{2} B +A \right )\right ) \sqrt {b \,x^{2}+a}}{3 \sqrt {b}\, x^{3}}\) | \(101\) |
default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{3 a \,x^{3}}+\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{a x}+\frac {6 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{a}\right )}{3 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{a x}+\frac {6 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{a}\right )\) | \(212\) |
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Time = 0.29 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\left [\frac {15 \, {\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt {b} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (6 \, B b^{2} x^{6} + 3 \, {\left (9 \, B a b + 4 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} - 8 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, x^{3}}, -\frac {15 \, {\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (6 \, B b^{2} x^{6} + 3 \, {\left (9 \, B a b + 4 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} - 8 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{24 \, x^{3}}\right ] \]
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Time = 2.71 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.93 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=- \frac {2 A a^{\frac {3}{2}} b}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {2 A \sqrt {a} b^{2} x}{\sqrt {1 + \frac {b x^{2}}{a}}} - \frac {A a^{2} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {A a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3} + 2 A a b^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + A b^{2} \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) - \frac {B a^{\frac {5}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{\frac {3}{2}} b x}{\sqrt {1 + \frac {b x^{2}}{a}}} + B a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + 2 B a b \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + B b^{2} \left (\begin {cases} - \frac {a^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b} + \frac {a x \sqrt {a + b x^{2}}}{8 b} + \frac {x^{3} \sqrt {a + b x^{2}}}{4} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{3}}{3} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\frac {5}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b x + \frac {15}{8} \, \sqrt {b x^{2} + a} B a b x + \frac {5}{2} \, \sqrt {b x^{2} + a} A b^{2} x + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2} x}{3 \, a} + \frac {15}{8} \, B a^{2} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) + \frac {5}{2} \, A a b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{x} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{3 \, a x^{3}} \]
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Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\frac {1}{8} \, {\left (2 \, B b^{2} x^{2} + \frac {9 \, B a b^{3} + 4 \, A b^{4}}{b^{2}}\right )} \sqrt {b x^{2} + a} x - \frac {5}{16} \, {\left (3 \, B a^{2} \sqrt {b} + 4 \, A a b^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} \sqrt {b} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {3}{2}} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} \sqrt {b} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {3}{2}} + 3 \, B a^{5} \sqrt {b} + 7 \, A a^{4} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{5/2}}{x^4} \,d x \]
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