Integrand size = 21, antiderivative size = 172 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x \sqrt {a+b x^2}}{6 a^2 b^3}+\frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 (6 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {424, 540, 396, 223, 212} \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (c+d x^2\right ) (b c-a d) (5 a d+2 b c)}{3 a^2 b^2 \sqrt {a+b x^2}}-\frac {d x \sqrt {a+b x^2} \left (-15 a^2 d^2+8 a b c d+4 b^2 c^2\right )}{6 a^2 b^3}+\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) (6 b c-5 a d)}{2 b^{7/2}}+\frac {x \left (c+d x^2\right )^2 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]
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Rule 212
Rule 223
Rule 396
Rule 424
Rule 540
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\int \frac {\left (c+d x^2\right ) \left (c (2 b c+a d)-d (2 b c-5 a d) x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a b} \\ & = \frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {a c d (2 b c-5 a d)+d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x^2}{\sqrt {a+b x^2}} \, dx}{3 a^2 b^2} \\ & = -\frac {d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x \sqrt {a+b x^2}}{6 a^2 b^3}+\frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\left (d^2 (6 b c-5 a d)\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b^3} \\ & = -\frac {d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x \sqrt {a+b x^2}}{6 a^2 b^3}+\frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\left (d^2 (6 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b^3} \\ & = -\frac {d \left (4 b^2 c^2+8 a b c d-15 a^2 d^2\right ) x \sqrt {a+b x^2}}{6 a^2 b^3}+\frac {(b c-a d) (2 b c+5 a d) x \left (c+d x^2\right )}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{7/2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.83 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (15 a^4 d^3+4 b^4 c^3 x^2+3 a^2 b^2 d^2 x^2 \left (-8 c+d x^2\right )+6 a b^3 c^2 \left (c+d x^2\right )+2 a^3 b d^2 \left (-9 c+10 d x^2\right )\right )}{6 a^2 b^3 \left (a+b x^2\right )^{3/2}}+\frac {d^2 (-6 b c+5 a d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{7/2}} \]
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Time = 2.48 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.83
method | result | size |
pseudoelliptic | \(-\frac {5 \left (\left (b \,x^{2}+a \right )^{\frac {3}{2}} d^{2} \left (a d -\frac {6 b c}{5}\right ) a^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-x \left (-\frac {6 \left (-\frac {10 d \,x^{2}}{9}+c \right ) d^{2} a^{3} b^{\frac {3}{2}}}{5}-\frac {8 x^{2} \left (-\frac {d \,x^{2}}{8}+c \right ) d^{2} a^{2} b^{\frac {5}{2}}}{5}+\frac {2 a \,c^{2} \left (d \,x^{2}+c \right ) b^{\frac {7}{2}}}{5}+\sqrt {b}\, a^{4} d^{3}+\frac {4 b^{\frac {9}{2}} c^{3} x^{2}}{15}\right )\right )}{2 b^{\frac {7}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}\) | \(142\) |
default | \(c^{3} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+d^{3} \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+3 c \,d^{2} \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+3 c^{2} d \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )\) | \(246\) |
risch | \(\frac {d^{3} x \sqrt {b \,x^{2}+a}}{2 b^{3}}-\frac {\frac {d^{2} \left (5 a d -6 b c \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{2 b a}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{2 b a}-\frac {\left (5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{2 a^{2} b \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{2 a^{2} b \left (x -\frac {\sqrt {-a b}}{b}\right )}}{2 b^{3}}\) | \(589\) |
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Time = 0.30 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.83 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (6 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (6 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (6 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (3 \, a^{2} b^{3} d^{3} x^{5} + 2 \, {\left (2 \, b^{5} c^{3} + 3 \, a b^{4} c^{2} d - 12 \, a^{2} b^{3} c d^{2} + 10 \, a^{3} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (2 \, a b^{4} c^{3} - 6 \, a^{3} b^{2} c d^{2} + 5 \, a^{4} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{12 \, {\left (a^{2} b^{6} x^{4} + 2 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}}, -\frac {3 \, {\left (6 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (6 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (6 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, a^{2} b^{3} d^{3} x^{5} + 2 \, {\left (2 \, b^{5} c^{3} + 3 \, a b^{4} c^{2} d - 12 \, a^{2} b^{3} c d^{2} + 10 \, a^{3} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (2 \, a b^{4} c^{3} - 6 \, a^{3} b^{2} c d^{2} + 5 \, a^{4} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{2} b^{6} x^{4} + 2 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}}\right ] \]
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\[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.48 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {d^{3} x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - c d^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {5 \, a d^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{6 \, b} + \frac {2 \, c^{3} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c^{3} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {c^{2} d x}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {c^{2} d x}{\sqrt {b x^{2} + a} a b} - \frac {c d^{2} x}{\sqrt {b x^{2} + a} b^{2}} + \frac {5 \, a d^{3} x}{6 \, \sqrt {b x^{2} + a} b^{3}} + \frac {3 \, c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} - \frac {5 \, a d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {7}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (\frac {3 \, d^{3} x^{2}}{b} + \frac {2 \, {\left (2 \, b^{6} c^{3} + 3 \, a b^{5} c^{2} d - 12 \, a^{2} b^{4} c d^{2} + 10 \, a^{3} b^{3} d^{3}\right )}}{a^{2} b^{5}}\right )} x^{2} + \frac {3 \, {\left (2 \, a b^{5} c^{3} - 6 \, a^{3} b^{3} c d^{2} + 5 \, a^{4} b^{2} d^{3}\right )}}{a^{2} b^{5}}\right )} x}{6 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (6 \, b c d^{2} - 5 \, a d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^3}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
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