Integrand size = 22, antiderivative size = 120 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {b (A b-6 a B) \sqrt {a+b x^2}}{16 a x^2}+\frac {(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}+\frac {b^2 (A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 120, 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 = {457, 79, 43, 65, 214} \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {b^2 (A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}}+\frac {b \sqrt {a+b x^2} (A b-6 a B)}{16 a x^2}+\frac {\left (a+b x^2\right )^{3/2} (A b-6 a B)}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6} \]
[In]
[Out]
Rule 43
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^4} \, dx,x,x^2\right ) \\ & = -\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}+\frac {\left (-\frac {A b}{2}+3 a B\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,x^2\right )}{6 a} \\ & = \frac {(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac {(b (A b-6 a B)) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,x^2\right )}{16 a} \\ & = \frac {b (A b-6 a B) \sqrt {a+b x^2}}{16 a x^2}+\frac {(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac {\left (b^2 (A b-6 a B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{32 a} \\ & = \frac {b (A b-6 a B) \sqrt {a+b x^2}}{16 a x^2}+\frac {(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac {(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 )}{16 a} \\ & = \frac {b (A b-6 a B) \sqrt {a+b x^2}}{16 a x^2}+\frac {(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}+\frac {b^2 (A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {\sqrt {a+b x^2} \left (-8 a^2 A-14 a A b x^2-12 a^2 B x^2-3 A b^2 x^4-30 a b B x^4\right )}{48 a x^6}-\frac {b^2 (-A b+6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}} \]
[In]
[Out]
Time = 2.84 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.77
method | result | size |
pseudoelliptic | \(-\frac {-\frac {3 b^{2} x^{6} \left (A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{8}+\sqrt {b \,x^{2}+a}\, \left (\frac {7 x^{2} \left (\frac {15 x^{2} B}{7}+A \right ) b \,a^{\frac {3}{2}}}{4}+\left (\frac {3 x^{2} B}{2}+A \right ) a^{\frac {5}{2}}+\frac {3 A \sqrt {a}\, b^{2} x^{4}}{8}\right )}{6 a^{\frac {3}{2}} x^{6}}\) | \(92\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (3 A \,b^{2} x^{4}+30 B a b \,x^{4}+14 a A b \,x^{2}+12 a^{2} B \,x^{2}+8 a^{2} A \right )}{48 x^{6} a}+\frac {\left (A b -6 B a \right ) b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{16 a^{\frac {3}{2}}}\) | \(99\) |
default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )\) | \(230\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\left [-\frac {3 \, {\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt {a} x^{6} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (10 \, B a^{2} b + A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, a^{2} x^{6}}, \frac {3 \, {\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, {\left (10 \, B a^{2} b + A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, a^{2} x^{6}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (105) = 210\).
Time = 60.40 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.11 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=- \frac {A a^{2}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {11 A a \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 A b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {A b^{\frac {5}{2}}}{16 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {3}{2}}} - \frac {B a^{2}}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 B a \sqrt {b}}{8 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {B b^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 B b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 \sqrt {a}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (103) = 206\).
Time = 0.22 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=-\frac {3 \, B b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B b^{2}}{8 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} A b^{3}}{16 \, a^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{8 \, a^{2} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{4 \, a x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{6 \, a x^{6}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {\frac {3 \, {\left (6 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {30 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a b^{3} - 48 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} b^{3} + 18 \, \sqrt {b x^{2} + a} B a^{3} b^{3} + 3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{4} + 8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b^{4} - 3 \, \sqrt {b x^{2} + a} A a^{2} b^{4}}{a b^{3} x^{6}}}{48 \, b} \]
[In]
[Out]
Time = 7.93 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx=\frac {A\,a\,\sqrt {b\,x^2+a}}{16\,x^6}-\frac {5\,B\,{\left (b\,x^2+a\right )}^{3/2}}{8\,x^4}-\frac {3\,B\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,\sqrt {a}}-\frac {A\,{\left (b\,x^2+a\right )}^{3/2}}{6\,x^6}+\frac {3\,B\,a\,\sqrt {b\,x^2+a}}{8\,x^4}-\frac {A\,{\left (b\,x^2+a\right )}^{5/2}}{16\,a\,x^6}-\frac {A\,b^3\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{16\,a^{3/2}} \]
[In]
[Out]