Integrand size = 22, antiderivative size = 156 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^9} \, dx=\frac {b (3 A b-8 a B) \sqrt {a+b x^2}}{64 a x^4}+\frac {b^2 (3 A b-8 a B) \sqrt {a+b x^2}}{128 a^2 x^2}+\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac {b^3 (3 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 156, 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, 44, 65, 214} \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^9} \, dx=-\frac {b^3 (3 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}}+\frac {b^2 \sqrt {a+b x^2} (3 A b-8 a B)}{128 a^2 x^2}+\frac {\left (a+b x^2\right )^{3/2} (3 A b-8 a B)}{48 a x^6}+\frac {b \sqrt {a+b x^2} (3 A b-8 a B)}{64 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8} \]
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Rule 43
Rule 44
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)^{3/2} (A+B x)}{x^5} \, dx,x,x^2\right ) \\ & = -\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac {\left (-\frac {3 A b}{2}+4 a B\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^4} \, dx,x,x^2\right )}{8 a} \\ & = \frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac {(b (3 A b-8 a B)) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^3} \, dx,x,x^2\right )}{32 a} \\ & = \frac {b (3 A b-8 a B) \sqrt {a+b x^2}}{64 a x^4}+\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac {\left (b^2 (3 A b-8 a B)\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{128 a} \\ & = \frac {b (3 A b-8 a B) \sqrt {a+b x^2}}{64 a x^4}+\frac {b^2 (3 A b-8 a B) \sqrt {a+b x^2}}{128 a^2 x^2}+\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac {\left (b^3 (3 A b-8 a B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{256 a^2} \\ & = \frac {b (3 A b-8 a B) \sqrt {a+b x^2}}{64 a x^4}+\frac {b^2 (3 A b-8 a B) \sqrt {a+b x^2}}{128 a^2 x^2}+\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}+\frac {\left (b^2 (3 A b-8 a B)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{128 a^2} \\ & = \frac {b (3 A b-8 a B) \sqrt {a+b x^2}}{64 a x^4}+\frac {b^2 (3 A b-8 a B) \sqrt {a+b x^2}}{128 a^2 x^2}+\frac {(3 A b-8 a B) \left (a+b x^2\right )^{3/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{5/2}}{8 a x^8}-\frac {b^3 (3 A b-8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^9} \, dx=\frac {\sqrt {a+b x^2} \left (-48 a^3 A-72 a^2 A b x^2-64 a^3 B x^2-6 a A b^2 x^4-112 a^2 b B x^4+9 A b^3 x^6-24 a b^2 B x^6\right )}{384 a^2 x^8}+\frac {b^3 (-3 A b+8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{5/2}} \]
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Time = 2.82 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.72
method | result | size |
pseudoelliptic | \(-\frac {3 \left (\frac {x^{8} b^{3} \left (A b -\frac {8 B a}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{8}+\sqrt {b \,x^{2}+a}\, \left (\frac {b^{2} x^{4} \left (4 x^{2} B +A \right ) a^{\frac {3}{2}}}{12}+b \,x^{2} \left (\frac {14 x^{2} B}{9}+A \right ) a^{\frac {5}{2}}+\left (\frac {8 x^{2} B}{9}+\frac {2 A}{3}\right ) a^{\frac {7}{2}}-\frac {A \sqrt {a}\, b^{3} x^{6}}{8}\right )\right )}{16 a^{\frac {5}{2}} x^{8}}\) | \(112\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-9 x^{6} b^{3} A +24 x^{6} a \,b^{2} B +6 A a \,b^{2} x^{4}+112 B \,a^{2} b \,x^{4}+72 A \,a^{2} b \,x^{2}+64 B \,a^{3} x^{2}+48 a^{3} A \right )}{384 x^{8} a^{2}}-\frac {\left (3 A b -8 B a \right ) b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{128 a^{\frac {5}{2}}}\) | \(124\) |
default | \(B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 a \,x^{8}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )\) | \(278\) |
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Time = 0.28 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.74 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^9} \, dx=\left [-\frac {3 \, {\left (8 \, B a b^{3} - 3 \, A b^{4}\right )} \sqrt {a} x^{8} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (8 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{6} + 48 \, A a^{4} + 2 \, {\left (56 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{4} + 8 \, {\left (8 \, B a^{4} + 9 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{768 \, a^{3} x^{8}}, -\frac {3 \, {\left (8 \, B a b^{3} - 3 \, A b^{4}\right )} \sqrt {-a} x^{8} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, {\left (8 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{6} + 48 \, A a^{4} + 2 \, {\left (56 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{4} + 8 \, {\left (8 \, B a^{4} + 9 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{384 \, a^{3} x^{8}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (141) = 282\).
Time = 115.17 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^9} \, dx=- \frac {A a^{2}}{8 \sqrt {b} x^{9} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 A a \sqrt {b}}{16 x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {13 A b^{\frac {3}{2}}}{64 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {5}{2}}}{128 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 A b^{\frac {7}{2}}}{128 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 A b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{128 a^{\frac {5}{2}}} - \frac {B a^{2}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {11 B a \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 B b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B b^{\frac {5}{2}}}{16 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {3}{2}}} \]
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Time = 0.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^9} \, dx=\frac {B b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} - \frac {3 \, A b^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {5}{2}}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{3}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} B b^{3}}{16 \, a^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{4}}{128 \, a^{4}} + \frac {3 \, \sqrt {b x^{2} + a} A b^{4}}{128 \, a^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{64 \, a^{3} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{6 \, a x^{6}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{16 \, a^{2} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{8 \, a x^{8}} \]
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Time = 0.31 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^9} \, dx=-\frac {\frac {3 \, {\left (8 \, B a b^{4} - 3 \, A b^{5}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {24 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a b^{4} + 40 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} b^{4} - 88 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{3} b^{4} + 24 \, \sqrt {b x^{2} + a} B a^{4} b^{4} - 9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{5} + 33 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a b^{5} + 33 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{2} b^{5} - 9 \, \sqrt {b x^{2} + a} A a^{3} b^{5}}{a^{2} b^{4} x^{8}}}{384 \, b} \]
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Time = 7.68 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^9} \, dx=\frac {3\,A\,a\,\sqrt {b\,x^2+a}}{128\,x^8}-\frac {B\,{\left (b\,x^2+a\right )}^{3/2}}{6\,x^6}-\frac {11\,A\,{\left (b\,x^2+a\right )}^{3/2}}{128\,x^8}+\frac {B\,a\,\sqrt {b\,x^2+a}}{16\,x^6}-\frac {11\,A\,{\left (b\,x^2+a\right )}^{5/2}}{128\,a\,x^8}+\frac {3\,A\,{\left (b\,x^2+a\right )}^{7/2}}{128\,a^2\,x^8}-\frac {B\,{\left (b\,x^2+a\right )}^{5/2}}{16\,a\,x^6}+\frac {A\,b^4\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,3{}\mathrm {i}}{128\,a^{5/2}}-\frac {B\,b^3\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{16\,a^{3/2}} \]
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