Integrand size = 19, antiderivative size = 118 \[ \int \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {a (6 A b-a B) x \sqrt {a+b x^2}}{16 b}+\frac {(6 A b-a B) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {a^2 (6 A b-a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {396, 201, 223, 212} \[ \int \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {a^2 (6 A b-a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2}}+\frac {x \left (a+b x^2\right )^{3/2} (6 A b-a B)}{24 b}+\frac {a x \sqrt {a+b x^2} (6 A b-a B)}{16 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b} \]
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Rule 201
Rule 212
Rule 223
Rule 396
Rubi steps \begin{align*} \text {integral}& = \frac {B x \left (a+b x^2\right )^{5/2}}{6 b}-\frac {(-6 A b+a B) \int \left (a+b x^2\right )^{3/2} \, dx}{6 b} \\ & = \frac {(6 A b-a B) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {(a (6 A b-a B)) \int \sqrt {a+b x^2} \, dx}{8 b} \\ & = \frac {a (6 A b-a B) x \sqrt {a+b x^2}}{16 b}+\frac {(6 A b-a B) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {\left (a^2 (6 A b-a B)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{16 b} \\ & = \frac {a (6 A b-a B) x \sqrt {a+b x^2}}{16 b}+\frac {(6 A b-a B) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {\left (a^2 (6 A b-a B)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 b} \\ & = \frac {a (6 A b-a B) x \sqrt {a+b x^2}}{16 b}+\frac {(6 A b-a B) x \left (a+b x^2\right )^{3/2}}{24 b}+\frac {B x \left (a+b x^2\right )^{5/2}}{6 b}+\frac {a^2 (6 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{3/2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.84 \[ \int \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {x \sqrt {a+b x^2} \left (30 a A b+3 a^2 B+12 A b^2 x^2+14 a b B x^2+8 b^2 B x^4\right )}{48 b}+\frac {a^2 (-6 A b+a B) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{16 b^{3/2}} \]
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Time = 2.85 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {x \left (8 b^{2} B \,x^{4}+12 A \,b^{2} x^{2}+14 B a b \,x^{2}+30 a b A +3 a^{2} B \right ) \sqrt {b \,x^{2}+a}}{48 b}+\frac {a^{2} \left (6 A b -B a \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{16 b^{\frac {3}{2}}}\) | \(88\) |
pseudoelliptic | \(\frac {\left (\frac {3}{2} a^{2} b A -\frac {1}{4} a^{3} B \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )+x \sqrt {b \,x^{2}+a}\, \left (\frac {5 \left (\frac {7 x^{2} B}{15}+A \right ) a \,b^{\frac {3}{2}}}{2}+x^{2} \left (\frac {2 x^{2} B}{3}+A \right ) b^{\frac {5}{2}}+\frac {B \,a^{2} \sqrt {b}}{4}\right )}{4 b^{\frac {3}{2}}}\) | \(89\) |
default | \(A \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+B \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )\) | \(130\) |
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Time = 0.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.75 \[ \int \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\left [-\frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8 \, B b^{3} x^{5} + 2 \, {\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} x^{3} + 3 \, {\left (B a^{2} b + 10 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, b^{2}}, \frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (8 \, B b^{3} x^{5} + 2 \, {\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} x^{3} + 3 \, {\left (B a^{2} b + 10 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, b^{2}}\right ] \]
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Time = 0.36 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.48 \[ \int \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\begin {cases} \sqrt {a + b x^{2}} \left (\frac {B b x^{5}}{6} + \frac {x^{3} \left (A b^{2} + \frac {7 B a b}{6}\right )}{4 b} + \frac {x \left (2 A a b + B a^{2} - \frac {3 a \left (A b^{2} + \frac {7 B a b}{6}\right )}{4 b}\right )}{2 b}\right ) + \left (A a^{2} - \frac {a \left (2 A a b + B a^{2} - \frac {3 a \left (A b^{2} + \frac {7 B a b}{6}\right )}{4 b}\right )}{2 b}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (A x + \frac {B x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.98 \[ \int \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A x + \frac {3}{8} \, \sqrt {b x^{2} + a} A a x + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B x}{6 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B a x}{24 \, b} - \frac {\sqrt {b x^{2} + a} B a^{2} x}{16 \, b} - \frac {B a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} + \frac {3 \, A a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} \]
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Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86 \[ \int \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {1}{48} \, {\left (2 \, {\left (4 \, B b x^{2} + \frac {7 \, B a b^{4} + 6 \, A b^{5}}{b^{4}}\right )} x^{2} + \frac {3 \, {\left (B a^{2} b^{3} + 10 \, A a b^{4}\right )}}{b^{4}}\right )} \sqrt {b x^{2} + a} x + \frac {{\left (B a^{3} - 6 \, A a^{2} b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {3}{2}}} \]
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Timed out. \[ \int \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\int \left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{3/2} \,d x \]
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