Integrand size = 22, antiderivative size = 149 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {5 b^2 (A b+6 a B) \sqrt {a+b x^2}}{16 a}-\frac {5 b (A b+6 a B) \left (a+b x^2\right )^{3/2}}{48 a x^2}-\frac {(A b+6 a B) \left (a+b x^2\right )^{5/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{7/2}}{6 a x^6}-\frac {5 b^2 (A b+6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}} \]
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Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {457, 79, 43, 52, 65, 214} \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=-\frac {5 b^2 (6 a B+A b) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}}+\frac {5 b^2 \sqrt {a+b x^2} (6 a B+A b)}{16 a}-\frac {5 b \left (a+b x^2\right )^{3/2} (6 a B+A b)}{48 a x^2}-\frac {\left (a+b x^2\right )^{5/2} (6 a B+A b)}{24 a x^4}-\frac {A \left (a+b x^2\right )^{7/2}}{6 a x^6} \]
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Rule 43
Rule 52
Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{5/2} (A+B x)}{x^4} \, dx,x,x^2\right ) \\ & = -\frac {A \left (a+b x^2\right )^{7/2}}{6 a x^6}+\frac {(A b+6 a B) \text {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^3} \, dx,x,x^2\right )}{12 a} \\ & = -\frac {(A b+6 a B) \left (a+b x^2\right )^{5/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{7/2}}{6 a x^6}+\frac {(5 b (A b+6 a B)) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^2} \, dx,x,x^2\right )}{48 a} \\ & = -\frac {5 b (A b+6 a B) \left (a+b x^2\right )^{3/2}}{48 a x^2}-\frac {(A b+6 a B) \left (a+b x^2\right )^{5/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{7/2}}{6 a x^6}+\frac {\left (5 b^2 (A b+6 a B)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )}{32 a} \\ & = \frac {5 b^2 (A b+6 a B) \sqrt {a+b x^2}}{16 a}-\frac {5 b (A b+6 a B) \left (a+b x^2\right )^{3/2}}{48 a x^2}-\frac {(A b+6 a B) \left (a+b x^2\right )^{5/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{7/2}}{6 a x^6}+\frac {1}{32} \left (5 b^2 (A b+6 a B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right ) \\ & = \frac {5 b^2 (A b+6 a B) \sqrt {a+b x^2}}{16 a}-\frac {5 b (A b+6 a B) \left (a+b x^2\right )^{3/2}}{48 a x^2}-\frac {(A b+6 a B) \left (a+b x^2\right )^{5/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{7/2}}{6 a x^6}+\frac {1}{16} (5 b (A b+6 a B)) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right ) \\ & = \frac {5 b^2 (A b+6 a B) \sqrt {a+b x^2}}{16 a}-\frac {5 b (A b+6 a B) \left (a+b x^2\right )^{3/2}}{48 a x^2}-\frac {(A b+6 a B) \left (a+b x^2\right )^{5/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{7/2}}{6 a x^6}-\frac {5 b^2 (A b+6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {\sqrt {a+b x^2} \left (-8 a^2 A-26 a A b x^2-12 a^2 B x^2-33 A b^2 x^4-54 a b B x^4+48 b^2 B x^6\right )}{48 x^6}-\frac {5 b^2 (A b+6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}} \]
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Time = 2.85 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.67
method | result | size |
pseudoelliptic | \(-\frac {11 \left (\frac {5 b^{2} x^{6} \left (A b +6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{11}+\left (\frac {26 x^{2} \left (\frac {27 x^{2} B}{13}+A \right ) b \,a^{\frac {3}{2}}}{33}+\frac {4 \left (x^{2} B +\frac {2 A}{3}\right ) a^{\frac {5}{2}}}{11}+b^{2} x^{4} \sqrt {a}\, \left (-\frac {16 x^{2} B}{11}+A \right )\right ) \sqrt {b \,x^{2}+a}\right )}{16 \sqrt {a}\, x^{6}}\) | \(100\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (33 A \,b^{2} x^{4}+54 B a b \,x^{4}+26 a A b \,x^{2}+12 a^{2} B \,x^{2}+8 a^{2} A \right )}{48 x^{6}}+\frac {b^{2} \left (16 \sqrt {b \,x^{2}+a}\, B -\frac {\left (5 A b +30 B a \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}\right )}{16}\) | \(112\) |
default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{6 a \,x^{6}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )\) | \(258\) |
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Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=\left [\frac {15 \, {\left (6 \, B a b^{2} + A b^{3}\right )} \sqrt {a} x^{6} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (48 \, B a b^{2} x^{6} - 3 \, {\left (18 \, B a^{2} b + 11 \, A a b^{2}\right )} x^{4} - 8 \, A a^{3} - 2 \, {\left (6 \, B a^{3} + 13 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, a x^{6}}, \frac {15 \, {\left (6 \, B a b^{2} + A b^{3}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (48 \, B a b^{2} x^{6} - 3 \, {\left (18 \, B a^{2} b + 11 \, A a b^{2}\right )} x^{4} - 8 \, A a^{3} - 2 \, {\left (6 \, B a^{3} + 13 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, a x^{6}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (136) = 272\).
Time = 68.52 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=- \frac {A a^{3}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 A a^{2} \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 A a b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {3 A b^{\frac {5}{2}}}{16 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 \sqrt {a}} - \frac {15 B \sqrt {a} b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8} - \frac {B a^{3}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 B a^{2} \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{x} + \frac {7 B a b^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B b^{\frac {5}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} \]
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Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=-\frac {15}{8} \, B \sqrt {a} b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {5 \, A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, \sqrt {a}} + \frac {15}{8} \, \sqrt {b x^{2} + a} B b^{2} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{8 \, a^{2}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2}}{8 \, a} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3}}{16 \, a^{3}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{48 \, a^{2}} + \frac {5 \, \sqrt {b x^{2} + a} A b^{3}}{16 \, a} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B b}{8 \, a^{2} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{2}}{16 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B}{4 \, a x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{6 \, a x^{6}} \]
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Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {48 \, \sqrt {b x^{2} + a} B b^{3} + \frac {15 \, {\left (6 \, B a b^{3} + A b^{4}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {54 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a b^{3} - 96 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} b^{3} + 42 \, \sqrt {b x^{2} + a} B a^{3} b^{3} + 33 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{4} - 40 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b^{4} + 15 \, \sqrt {b x^{2} + a} A a^{2} b^{4}}{b^{3} x^{6}}}{48 \, b} \]
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Time = 7.38 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx=B\,b^2\,\sqrt {b\,x^2+a}-\frac {11\,A\,{\left (b\,x^2+a\right )}^{5/2}}{16\,x^6}+\frac {5\,A\,a\,{\left (b\,x^2+a\right )}^{3/2}}{6\,x^6}-\frac {9\,B\,a\,{\left (b\,x^2+a\right )}^{3/2}}{8\,x^4}-\frac {5\,A\,a^2\,\sqrt {b\,x^2+a}}{16\,x^6}+\frac {7\,B\,a^2\,\sqrt {b\,x^2+a}}{8\,x^4}+\frac {A\,b^3\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{16\,\sqrt {a}}+\frac {B\,\sqrt {a}\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,15{}\mathrm {i}}{8} \]
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