Integrand size = 22, antiderivative size = 153 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {5 b^2 (7 A b-6 a B)}{16 a^4 \sqrt {a+b x^2}}-\frac {A}{6 a x^6 \sqrt {a+b x^2}}+\frac {7 A b-6 a B}{24 a^2 x^4 \sqrt {a+b x^2}}-\frac {5 b (7 A b-6 a B)}{48 a^3 x^2 \sqrt {a+b x^2}}+\frac {5 b^2 (7 A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{9/2}} \]
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Time = 0.09 (sec) , antiderivative size = 153, 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, 44, 53, 65, 214} \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=\frac {5 b^2 (7 A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{9/2}}-\frac {5 b^2 (7 A b-6 a B)}{16 a^4 \sqrt {a+b x^2}}-\frac {5 b (7 A b-6 a B)}{48 a^3 x^2 \sqrt {a+b x^2}}+\frac {7 A b-6 a B}{24 a^2 x^4 \sqrt {a+b x^2}}-\frac {A}{6 a x^6 \sqrt {a+b x^2}} \]
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Rule 44
Rule 53
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}{x^4 (a+b x)^{3/2}} \, dx,x,x^2\right ) \\ & = -\frac {A}{6 a x^6 \sqrt {a+b x^2}}+\frac {\left (-\frac {7 A b}{2}+3 a B\right ) \text {Subst}\left (\int \frac {1}{x^3 (a+b x)^{3/2}} \, dx,x,x^2\right )}{6 a} \\ & = -\frac {A}{6 a x^6 \sqrt {a+b x^2}}+\frac {7 A b-6 a B}{24 a^2 x^4 \sqrt {a+b x^2}}+\frac {(5 b (7 A b-6 a B)) \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )}{48 a^2} \\ & = -\frac {A}{6 a x^6 \sqrt {a+b x^2}}+\frac {7 A b-6 a B}{24 a^2 x^4 \sqrt {a+b x^2}}-\frac {5 b (7 A b-6 a B)}{48 a^3 x^2 \sqrt {a+b x^2}}-\frac {\left (5 b^2 (7 A b-6 a B)\right ) \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^2\right )}{32 a^3} \\ & = -\frac {5 b^2 (7 A b-6 a B)}{16 a^4 \sqrt {a+b x^2}}-\frac {A}{6 a x^6 \sqrt {a+b x^2}}+\frac {7 A b-6 a B}{24 a^2 x^4 \sqrt {a+b x^2}}-\frac {5 b (7 A b-6 a B)}{48 a^3 x^2 \sqrt {a+b x^2}}-\frac {\left (5 b^2 (7 A b-6 a B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{32 a^4} \\ & = -\frac {5 b^2 (7 A b-6 a B)}{16 a^4 \sqrt {a+b x^2}}-\frac {A}{6 a x^6 \sqrt {a+b x^2}}+\frac {7 A b-6 a B}{24 a^2 x^4 \sqrt {a+b x^2}}-\frac {5 b (7 A b-6 a B)}{48 a^3 x^2 \sqrt {a+b x^2}}-\frac {(5 b (7 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 )}{16 a^4} \\ & = -\frac {5 b^2 (7 A b-6 a B)}{16 a^4 \sqrt {a+b x^2}}-\frac {A}{6 a x^6 \sqrt {a+b x^2}}+\frac {7 A b-6 a B}{24 a^2 x^4 \sqrt {a+b x^2}}-\frac {5 b (7 A b-6 a B)}{48 a^3 x^2 \sqrt {a+b x^2}}+\frac {5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{9/2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=\frac {-8 a^3 A+14 a^2 A b x^2-12 a^3 B x^2-35 a A b^2 x^4+30 a^2 b B x^4-105 A b^3 x^6+90 a b^2 B x^6}{48 a^4 x^6 \sqrt {a+b x^2}}-\frac {5 b^2 (-7 A b+6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{9/2}} \]
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Time = 2.88 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.78
method | result | size |
pseudoelliptic | \(\frac {\frac {35 x^{6} \left (A b -\frac {6 B a}{7}\right ) b^{2} \sqrt {b \,x^{2}+a}\, \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{16}-\frac {35 x^{4} \left (-\frac {18 x^{2} B}{7}+A \right ) b^{2} a^{\frac {3}{2}}}{48}+\frac {7 b \,x^{2} \left (\frac {15 x^{2} B}{7}+A \right ) a^{\frac {5}{2}}}{24}+\frac {\left (-3 x^{2} B -2 A \right ) a^{\frac {7}{2}}}{12}-\frac {35 A \sqrt {a}\, b^{3} x^{6}}{16}}{x^{6} a^{\frac {9}{2}} \sqrt {b \,x^{2}+a}}\) | \(120\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (57 A \,b^{2} x^{4}-42 B a b \,x^{4}-22 a A b \,x^{2}+12 a^{2} B \,x^{2}+8 a^{2} A \right )}{48 a^{4} x^{6}}-\frac {b^{2} \left (-\frac {19 A b -14 B a}{\sqrt {b \,x^{2}+a}}+5 a \left (7 A b -6 B a \right ) \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )\right )}{16 a^{4}}\) | \(143\) |
default | \(A \left (-\frac {1}{6 a \,x^{6} \sqrt {b \,x^{2}+a}}-\frac {7 b \left (-\frac {1}{4 a \,x^{4} \sqrt {b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+B \left (-\frac {1}{4 a \,x^{4} \sqrt {b \,x^{2}+a}}-\frac {5 b \left (-\frac {1}{2 a \,x^{2} \sqrt {b \,x^{2}+a}}-\frac {3 b \left (\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{2 a}\right )}{4 a}\right )\) | \(210\) |
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Time = 0.28 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.23 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {15 \, {\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (15 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} - 8 \, A a^{4} + 5 \, {\left (6 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} - 2 \, {\left (6 \, B a^{4} - 7 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, {\left (a^{5} b x^{8} + a^{6} x^{6}\right )}}, \frac {15 \, {\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (15 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} - 8 \, A a^{4} + 5 \, {\left (6 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} - 2 \, {\left (6 \, B a^{4} - 7 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, {\left (a^{5} b x^{8} + a^{6} x^{6}\right )}}\right ] \]
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Time = 54.82 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.54 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=A \left (- \frac {1}{6 a \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {7 \sqrt {b}}{24 a^{2} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 b^{\frac {3}{2}}}{48 a^{3} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 b^{\frac {5}{2}}}{16 a^{4} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {35 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {9}{2}}}\right ) + B \left (- \frac {1}{4 a \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {5 \sqrt {b}}{8 a^{2} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {15 b^{\frac {3}{2}}}{8 a^{3} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {15 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {7}{2}}}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {15 \, B b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {7}{2}}} + \frac {35 \, A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {9}{2}}} + \frac {15 \, B b^{2}}{8 \, \sqrt {b x^{2} + a} a^{3}} - \frac {35 \, A b^{3}}{16 \, \sqrt {b x^{2} + a} a^{4}} + \frac {5 \, B b}{8 \, \sqrt {b x^{2} + a} a^{2} x^{2}} - \frac {35 \, A b^{2}}{48 \, \sqrt {b x^{2} + a} a^{3} x^{2}} - \frac {B}{4 \, \sqrt {b x^{2} + a} a x^{4}} + \frac {7 \, A b}{24 \, \sqrt {b x^{2} + a} a^{2} x^{4}} - \frac {A}{6 \, \sqrt {b x^{2} + a} a x^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=\frac {5 \, {\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{16 \, \sqrt {-a} a^{4}} + \frac {B a b^{2} - A b^{3}}{\sqrt {b x^{2} + a} a^{4}} + \frac {42 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a b^{2} - 96 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} b^{2} + 54 \, \sqrt {b x^{2} + a} B a^{3} b^{2} - 57 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{3} + 136 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b^{3} - 87 \, \sqrt {b x^{2} + a} A a^{2} b^{3}}{48 \, a^{4} b^{3} x^{6}} \]
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Time = 6.57 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^{3/2}} \, dx=\frac {35\,A\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{16\,a^{9/2}}-\frac {15\,B\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{7/2}}-\frac {35\,A\,b^3}{16\,a^4\,\sqrt {b\,x^2+a}}+\frac {15\,B\,b^2}{8\,a^3\,\sqrt {b\,x^2+a}}-\frac {A}{6\,a\,x^6\,\sqrt {b\,x^2+a}}-\frac {B}{4\,a\,x^4\,\sqrt {b\,x^2+a}}+\frac {7\,A\,b}{24\,a^2\,x^4\,\sqrt {b\,x^2+a}}+\frac {5\,B\,b}{8\,a^2\,x^2\,\sqrt {b\,x^2+a}}-\frac {35\,A\,b^2}{48\,a^3\,x^2\,\sqrt {b\,x^2+a}} \]
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