Integrand size = 22, antiderivative size = 221 \[ \int x^4 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=-\frac {a^4 (12 A b-5 a B) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a^3 (12 A b-5 a B) x^3 \sqrt {a+b x^2}}{1536 b^2}+\frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {a^5 (12 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {470, 285, 327, 223, 212} \[ \int x^4 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {a^5 (12 A b-5 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}}-\frac {a^4 x \sqrt {a+b x^2} (12 A b-5 a B)}{1024 b^3}+\frac {a^3 x^3 \sqrt {a+b x^2} (12 A b-5 a B)}{1536 b^2}+\frac {a^2 x^5 \sqrt {a+b x^2} (12 A b-5 a B)}{384 b}+\frac {a x^5 \left (a+b x^2\right )^{3/2} (12 A b-5 a B)}{192 b}+\frac {x^5 \left (a+b x^2\right )^{5/2} (12 A b-5 a B)}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b} \]
[In]
[Out]
Rule 212
Rule 223
Rule 285
Rule 327
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}-\frac {(-12 A b+5 a B) \int x^4 \left (a+b x^2\right )^{5/2} \, dx}{12 b} \\ & = \frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {(a (12 A b-5 a B)) \int x^4 \left (a+b x^2\right )^{3/2} \, dx}{24 b} \\ & = \frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {\left (a^2 (12 A b-5 a B)\right ) \int x^4 \sqrt {a+b x^2} \, dx}{64 b} \\ & = \frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {\left (a^3 (12 A b-5 a B)\right ) \int \frac {x^4}{\sqrt {a+b x^2}} \, dx}{384 b} \\ & = \frac {a^3 (12 A b-5 a B) x^3 \sqrt {a+b x^2}}{1536 b^2}+\frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}-\frac {\left (a^4 (12 A b-5 a B)\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{512 b^2} \\ & = -\frac {a^4 (12 A b-5 a B) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a^3 (12 A b-5 a B) x^3 \sqrt {a+b x^2}}{1536 b^2}+\frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {\left (a^5 (12 A b-5 a B)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{1024 b^3} \\ & = -\frac {a^4 (12 A b-5 a B) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a^3 (12 A b-5 a B) x^3 \sqrt {a+b x^2}}{1536 b^2}+\frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {\left (a^5 (12 A b-5 a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{1024 b^3} \\ & = -\frac {a^4 (12 A b-5 a B) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a^3 (12 A b-5 a B) x^3 \sqrt {a+b x^2}}{1536 b^2}+\frac {a^2 (12 A b-5 a B) x^5 \sqrt {a+b x^2}}{384 b}+\frac {a (12 A b-5 a B) x^5 \left (a+b x^2\right )^{3/2}}{192 b}+\frac {(12 A b-5 a B) x^5 \left (a+b x^2\right )^{5/2}}{120 b}+\frac {B x^5 \left (a+b x^2\right )^{7/2}}{12 b}+\frac {a^5 (12 A b-5 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.78 \[ \int x^4 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {\sqrt {b} x \sqrt {a+b x^2} \left (75 a^5 B+40 a^3 b^2 x^2 \left (3 A+B x^2\right )+256 b^5 x^8 \left (6 A+5 B x^2\right )-10 a^4 b \left (18 A+5 B x^2\right )+48 a^2 b^3 x^4 \left (62 A+45 B x^2\right )+64 a b^4 x^6 \left (63 A+50 B x^2\right )\right )+30 a^5 (-12 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}-\sqrt {a+b x^2}}\right )}{15360 b^{7/2}} \]
[In]
[Out]
Time = 2.94 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.67
method | result | size |
pseudoelliptic | \(\frac {\left (\frac {3}{2} A \,a^{5} b -\frac {5}{8} B \,a^{6}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+x \left (\frac {64 x^{8} \left (\frac {5 x^{2} B}{6}+A \right ) b^{\frac {11}{2}}}{5}+\left (-\frac {3 \left (\frac {5 x^{2} B}{18}+A \right ) a^{3} b^{\frac {3}{2}}}{2}+a^{2} x^{2} \left (\frac {x^{2} B}{3}+A \right ) b^{\frac {5}{2}}+\frac {124 x^{4} a \left (\frac {45 x^{2} B}{62}+A \right ) b^{\frac {7}{2}}}{5}+\left (\frac {80}{3} B \,x^{8}+\frac {168}{5} A \,x^{6}\right ) b^{\frac {9}{2}}+\frac {5 B \,a^{4} \sqrt {b}}{8}\right ) a \right ) \sqrt {b \,x^{2}+a}}{128 b^{\frac {7}{2}}}\) | \(147\) |
risch | \(-\frac {x \left (-1280 b^{5} B \,x^{10}-1536 b^{5} A \,x^{8}-3200 a \,b^{4} B \,x^{8}-4032 a \,b^{4} A \,x^{6}-2160 a^{2} b^{3} B \,x^{6}-2976 a^{2} b^{3} A \,x^{4}-40 a^{3} b^{2} B \,x^{4}-120 a^{3} A \,b^{2} x^{2}+50 B \,a^{4} b \,x^{2}+180 a^{4} b A -75 a^{5} B \right ) \sqrt {b \,x^{2}+a}}{15360 b^{3}}+\frac {a^{5} \left (12 A b -5 B a \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{1024 b^{\frac {7}{2}}}\) | \(160\) |
default | \(B \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{12 b}-\frac {5 a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )}{12 b}\right )+A \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )\) | \(256\) |
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.61 \[ \int x^4 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\left [-\frac {15 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (1280 \, B b^{6} x^{11} + 128 \, {\left (25 \, B a b^{5} + 12 \, A b^{6}\right )} x^{9} + 144 \, {\left (15 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} x^{7} + 8 \, {\left (5 \, B a^{3} b^{3} + 372 \, A a^{2} b^{4}\right )} x^{5} - 10 \, {\left (5 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} + 15 \, {\left (5 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{30720 \, b^{4}}, \frac {15 \, {\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (1280 \, B b^{6} x^{11} + 128 \, {\left (25 \, B a b^{5} + 12 \, A b^{6}\right )} x^{9} + 144 \, {\left (15 \, B a^{2} b^{4} + 28 \, A a b^{5}\right )} x^{7} + 8 \, {\left (5 \, B a^{3} b^{3} + 372 \, A a^{2} b^{4}\right )} x^{5} - 10 \, {\left (5 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} + 15 \, {\left (5 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{15360 \, b^{4}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (204) = 408\).
Time = 0.63 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.06 \[ \int x^4 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\begin {cases} \frac {3 a^{2} \left (A a^{3} - \frac {5 a \left (3 A a^{2} b + B a^{3} - \frac {7 a \left (3 A a b^{2} + 3 B a^{2} b - \frac {9 a \left (A b^{3} + \frac {25 B a b^{2}}{12}\right )}{10 b}\right )}{8 b}\right )}{6 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b^{2}} + \sqrt {a + b x^{2}} \left (\frac {B b^{2} x^{11}}{12} - \frac {3 a x \left (A a^{3} - \frac {5 a \left (3 A a^{2} b + B a^{3} - \frac {7 a \left (3 A a b^{2} + 3 B a^{2} b - \frac {9 a \left (A b^{3} + \frac {25 B a b^{2}}{12}\right )}{10 b}\right )}{8 b}\right )}{6 b}\right )}{8 b^{2}} + \frac {x^{9} \left (A b^{3} + \frac {25 B a b^{2}}{12}\right )}{10 b} + \frac {x^{7} \cdot \left (3 A a b^{2} + 3 B a^{2} b - \frac {9 a \left (A b^{3} + \frac {25 B a b^{2}}{12}\right )}{10 b}\right )}{8 b} + \frac {x^{5} \cdot \left (3 A a^{2} b + B a^{3} - \frac {7 a \left (3 A a b^{2} + 3 B a^{2} b - \frac {9 a \left (A b^{3} + \frac {25 B a b^{2}}{12}\right )}{10 b}\right )}{8 b}\right )}{6 b} + \frac {x^{3} \left (A a^{3} - \frac {5 a \left (3 A a^{2} b + B a^{3} - \frac {7 a \left (3 A a b^{2} + 3 B a^{2} b - \frac {9 a \left (A b^{3} + \frac {25 B a b^{2}}{12}\right )}{10 b}\right )}{8 b}\right )}{6 b}\right )}{4 b}\right ) & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{5}}{5} + \frac {B x^{7}}{7}\right ) & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.10 \[ \int x^4 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B x^{5}}{12 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B a x^{3}}{24 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A x^{3}}{10 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2} x}{64 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{3} x}{384 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{4} x}{1536 \, b^{3}} - \frac {5 \, \sqrt {b x^{2} + a} B a^{5} x}{1024 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a x}{80 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A a^{2} x}{160 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A a^{3} x}{128 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} A a^{4} x}{256 \, b^{2}} - \frac {5 \, B a^{6} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{1024 \, b^{\frac {7}{2}}} + \frac {3 \, A a^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.88 \[ \int x^4 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B b^{2} x^{2} + \frac {25 \, B a b^{11} + 12 \, A b^{12}}{b^{10}}\right )} x^{2} + \frac {9 \, {\left (15 \, B a^{2} b^{10} + 28 \, A a b^{11}\right )}}{b^{10}}\right )} x^{2} + \frac {5 \, B a^{3} b^{9} + 372 \, A a^{2} b^{10}}{b^{10}}\right )} x^{2} - \frac {5 \, {\left (5 \, B a^{4} b^{8} - 12 \, A a^{3} b^{9}\right )}}{b^{10}}\right )} x^{2} + \frac {15 \, {\left (5 \, B a^{5} b^{7} - 12 \, A a^{4} b^{8}\right )}}{b^{10}}\right )} \sqrt {b x^{2} + a} x + \frac {{\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{1024 \, b^{\frac {7}{2}}} \]
[In]
[Out]
Timed out. \[ \int x^4 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx=\int x^4\,\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{5/2} \,d x \]
[In]
[Out]