Integrand size = 22, antiderivative size = 152 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^9} \, dx=\frac {5 b^2 (A b-8 a B) \sqrt {a+b x^2}}{128 a x^2}+\frac {5 b (A b-8 a B) \left (a+b x^2\right )^{3/2}}{192 a x^4}+\frac {(A b-8 a B) \left (a+b x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{7/2}}{8 a x^8}+\frac {5 b^3 (A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{3/2}} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {457, 79, 43, 65, 214} \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^9} \, dx=\frac {5 b^3 (A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{3/2}}+\frac {5 b^2 \sqrt {a+b x^2} (A b-8 a B)}{128 a x^2}+\frac {\left (a+b x^2\right )^{5/2} (A b-8 a B)}{48 a x^6}+\frac {5 b \left (a+b x^2\right )^{3/2} (A b-8 a B)}{192 a x^4}-\frac {A \left (a+b x^2\right )^{7/2}}{8 a x^8} \]
[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)^{5/2} (A+B x)}{x^5} \, dx,x,x^2\right ) \\ & = -\frac {A \left (a+b x^2\right )^{7/2}}{8 a x^8}+\frac {\left (-\frac {A b}{2}+4 a B\right ) \text {Subst}\left (\int \frac {(a+b x)^{5/2}}{x^4} \, dx,x,x^2\right )}{8 a} \\ & = \frac {(A b-8 a B) \left (a+b x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{7/2}}{8 a x^8}-\frac {(5 b (A b-8 a B)) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,x^2\right )}{96 a} \\ & = \frac {5 b (A b-8 a B) \left (a+b x^2\right )^{3/2}}{192 a x^4}+\frac {(A b-8 a B) \left (a+b x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{7/2}}{8 a x^8}-\frac {\left (5 b^2 (A b-8 a B)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,x^2\right )}{128 a} \\ & = \frac {5 b^2 (A b-8 a B) \sqrt {a+b x^2}}{128 a x^2}+\frac {5 b (A b-8 a B) \left (a+b x^2\right )^{3/2}}{192 a x^4}+\frac {(A b-8 a B) \left (a+b x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{7/2}}{8 a x^8}-\frac {\left (5 b^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} \\ & = \frac {5 b^2 (A b-8 a B) \sqrt {a+b x^2}}{128 a x^2}+\frac {5 b (A b-8 a B) \left (a+b x^2\right )^{3/2}}{192 a x^4}+\frac {(A b-8 a B) \left (a+b x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{7/2}}{8 a x^8}-\frac {\left (5 b^2 (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} \\ & = \frac {5 b^2 (A b-8 a B) \sqrt {a+b x^2}}{128 a x^2}+\frac {5 b (A b-8 a B) \left (a+b x^2\right )^{3/2}}{192 a x^4}+\frac {(A b-8 a B) \left (a+b x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+b x^2\right )^{7/2}}{8 a x^8}+\frac {5 b^3 (A b-8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{3/2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^9} \, dx=-\frac {\sqrt {a+b x^2} \left (15 A b^3 x^6+16 a^3 \left (3 A+4 B x^2\right )+8 a^2 b x^2 \left (17 A+26 B x^2\right )+2 a b^2 x^4 \left (59 A+132 B x^2\right )\right )}{384 a x^8}+\frac {5 b^3 (A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{3/2}} \]
[In]
[Out]
Time = 2.91 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.74
method | result | size |
pseudoelliptic | \(-\frac {17 \left (-\frac {15 b^{3} x^{8} \left (A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{136}+\sqrt {b \,x^{2}+a}\, \left (\frac {59 x^{4} b^{2} \left (\frac {132 x^{2} B}{59}+A \right ) a^{\frac {3}{2}}}{68}+b \,x^{2} \left (\frac {26 x^{2} B}{17}+A \right ) a^{\frac {5}{2}}+\frac {2 \left (4 x^{2} B +3 A \right ) a^{\frac {7}{2}}}{17}+\frac {15 A \sqrt {a}\, b^{3} x^{6}}{136}\right )\right )}{48 a^{\frac {3}{2}} x^{8}}\) | \(113\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (15 x^{6} b^{3} A +264 x^{6} a \,b^{2} B +118 A a \,b^{2} x^{4}+208 B \,a^{2} b \,x^{4}+136 A \,a^{2} b \,x^{2}+64 B \,a^{3} x^{2}+48 a^{3} A \right )}{384 x^{8} a}+\frac {5 \left (A b -8 B a \right ) b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{128 a^{\frac {3}{2}}}\) | \(123\) |
default | \(B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{6 a \,x^{6}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 a \,x^{8}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{6 a \,x^{6}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )\) | \(306\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^9} \, dx=\left [-\frac {15 \, {\left (8 \, B a b^{3} - A b^{4}\right )} \sqrt {a} x^{8} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (88 \, B a^{2} b^{2} + 5 \, A a b^{3}\right )} x^{6} + 48 \, A a^{4} + 2 \, {\left (104 \, B a^{3} b + 59 \, A a^{2} b^{2}\right )} x^{4} + 8 \, {\left (8 \, B a^{4} + 17 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{768 \, a^{2} x^{8}}, \frac {15 \, {\left (8 \, B a b^{3} - A b^{4}\right )} \sqrt {-a} x^{8} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, {\left (88 \, B a^{2} b^{2} + 5 \, A a b^{3}\right )} x^{6} + 48 \, A a^{4} + 2 \, {\left (104 \, B a^{3} b + 59 \, A a^{2} b^{2}\right )} x^{4} + 8 \, {\left (8 \, B a^{4} + 17 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{384 \, a^{2} x^{8}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (139) = 278\).
Time = 135.74 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.08 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^9} \, dx=- \frac {A a^{3}}{8 \sqrt {b} x^{9} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {23 A a^{2} \sqrt {b}}{48 x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {127 A a b^{\frac {3}{2}}}{192 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {133 A b^{\frac {5}{2}}}{384 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 A b^{\frac {7}{2}}}{128 a x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {5 A b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{128 a^{\frac {3}{2}}} - \frac {B a^{3}}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {17 B a^{2} \sqrt {b}}{24 x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {35 B a b^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {B b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} - \frac {3 B b^{\frac {5}{2}}}{16 x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 B b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 \sqrt {a}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (132) = 264\).
Time = 0.20 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^9} \, dx=-\frac {5 \, B b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, \sqrt {a}} + \frac {5 \, A b^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{3}}{16 \, a^{3}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{3}}{48 \, a^{2}} + \frac {5 \, \sqrt {b x^{2} + a} B b^{3}}{16 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{4}}{128 \, a^{4}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{4}}{384 \, a^{3}} - \frac {5 \, \sqrt {b x^{2} + a} A b^{4}}{128 \, a^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B b^{2}}{16 \, a^{3} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{3}}{128 \, a^{4} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B b}{24 \, a^{2} x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{2}}{192 \, a^{3} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B}{6 \, a x^{6}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A b}{48 \, a^{2} x^{6}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{8 \, a x^{8}} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^9} \, dx=\frac {\frac {15 \, {\left (8 \, B a b^{4} - A b^{5}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {264 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a b^{4} - 584 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} b^{4} + 440 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{3} b^{4} - 120 \, \sqrt {b x^{2} + a} B a^{4} b^{4} + 15 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b^{5} + 73 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a b^{5} - 55 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{2} b^{5} + 15 \, \sqrt {b x^{2} + a} A a^{3} b^{5}}{a b^{4} x^{8}}}{384 \, b} \]
[In]
[Out]
Time = 8.39 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^9} \, dx=\frac {55\,A\,a\,{\left (b\,x^2+a\right )}^{3/2}}{384\,x^8}-\frac {11\,B\,{\left (b\,x^2+a\right )}^{5/2}}{16\,x^6}-\frac {73\,A\,{\left (b\,x^2+a\right )}^{5/2}}{384\,x^8}+\frac {5\,B\,a\,{\left (b\,x^2+a\right )}^{3/2}}{6\,x^6}-\frac {5\,A\,a^2\,\sqrt {b\,x^2+a}}{128\,x^8}-\frac {5\,A\,{\left (b\,x^2+a\right )}^{7/2}}{128\,a\,x^8}-\frac {5\,B\,a^2\,\sqrt {b\,x^2+a}}{16\,x^6}-\frac {A\,b^4\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{128\,a^{3/2}}+\frac {B\,b^3\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{16\,\sqrt {a}} \]
[In]
[Out]