Integrand size = 21, antiderivative size = 257 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt {a+b x^2}}{48 a b^4}-\frac {d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}} \]
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Time = 0.18 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {424, 542, 396, 223, 212} \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (35 a^2 d^2-64 a b c d+24 b^2 c^2\right )}{24 a b^3}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (-35 a^3 d^3+120 a^2 b c d^2-144 a b^2 c^2 d+64 b^3 c^3\right )}{16 b^{9/2}}-\frac {d x \sqrt {a+b x^2} \left (-105 a^3 d^3+290 a^2 b c d^2-248 a b^2 c^2 d+48 b^3 c^3\right )}{48 a b^4}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right )^2 (6 b c-7 a d)}{6 a b^2}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{a b \sqrt {a+b x^2}} \]
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Rule 212
Rule 223
Rule 396
Rule 424
Rule 542
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {\int \frac {\left (c+d x^2\right )^2 \left (a c d-d (6 b c-7 a d) x^2\right )}{\sqrt {a+b x^2}} \, dx}{a b} \\ & = -\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {\int \frac {\left (c+d x^2\right ) \left (a c d (12 b c-7 a d)-d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x^2\right )}{\sqrt {a+b x^2}} \, dx}{6 a b^2} \\ & = -\frac {d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {\int \frac {a c d \left (72 b^2 c^2-92 a b c d+35 a^2 d^2\right )-d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x^2}{\sqrt {a+b x^2}} \, dx}{24 a b^3} \\ & = -\frac {d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt {a+b x^2}}{48 a b^4}-\frac {d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {\left (d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{16 b^4} \\ & = -\frac {d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt {a+b x^2}}{48 a b^4}-\frac {d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {\left (d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 b^4} \\ & = -\frac {d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt {a+b x^2}}{48 a b^4}-\frac {d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac {d (6 b c-7 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt {a+b x^2}}+\frac {d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 b^{9/2}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.77 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {b} x \left (48 b^4 c^4+105 a^4 d^4+5 a^3 b d^3 \left (-72 c+7 d x^2\right )-2 a^2 b^2 d^2 \left (-216 c^2+60 c d x^2+7 d^2 x^4\right )+8 a b^3 d \left (-24 c^3+18 c^2 d x^2+6 c d^2 x^4+d^3 x^6\right )\right )}{a \sqrt {a+b x^2}}+3 d \left (-64 b^3 c^3+144 a b^2 c^2 d-120 a^2 b c d^2+35 a^3 d^3\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{48 b^{9/2}} \]
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Time = 2.48 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.75
method | result | size |
pseudoelliptic | \(-\frac {35 \left (\sqrt {b \,x^{2}+a}\, a d \left (a^{3} d^{3}-\frac {24}{7} a^{2} b c \,d^{2}+\frac {144}{35} a \,b^{2} c^{2} d -\frac {64}{35} b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-x \left (-\frac {64 d \left (-\frac {1}{24} d^{3} x^{6}-\frac {1}{4} c \,d^{2} x^{4}-\frac {3}{4} c^{2} d \,x^{2}+c^{3}\right ) a \,b^{\frac {7}{2}}}{35}+\frac {144 d^{2} \left (-\frac {7}{216} d^{2} x^{4}-\frac {5}{18} c d \,x^{2}+c^{2}\right ) a^{2} b^{\frac {5}{2}}}{35}-\frac {24 d^{3} \left (-\frac {7 d \,x^{2}}{72}+c \right ) a^{3} b^{\frac {3}{2}}}{7}+\sqrt {b}\, a^{4} d^{4}+\frac {16 b^{\frac {9}{2}} c^{4}}{35}\right )\right )}{16 b^{\frac {9}{2}} \sqrt {b \,x^{2}+a}\, a}\) | \(192\) |
risch | \(\frac {x \,d^{2} \left (8 b^{2} d^{2} x^{4}-22 x^{2} a b \,d^{2}+48 x^{2} b^{2} c d +57 a^{2} d^{2}-168 a b c d +144 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}}{48 b^{4}}-\frac {\frac {19 a^{3} d^{4} x}{\sqrt {b \,x^{2}+a}}-\frac {16 b^{4} c^{4} x}{a \sqrt {b \,x^{2}+a}}-\frac {56 a^{2} b c \,d^{3} x}{\sqrt {b \,x^{2}+a}}+\frac {48 a \,b^{2} c^{2} d^{2} x}{\sqrt {b \,x^{2}+a}}+\left (35 d^{4} a^{3} b -120 a^{2} b^{2} c \,d^{3}+144 a \,b^{3} c^{2} d^{2}-64 b^{4} c^{3} d \right ) \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{16 b^{4}}\) | \(241\) |
default | \(\frac {c^{4} x}{a \sqrt {b \,x^{2}+a}}+d^{4} \left (\frac {x^{7}}{6 b \sqrt {b \,x^{2}+a}}-\frac {7 a \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )}{6 b}\right )+4 c \,d^{3} \left (\frac {x^{5}}{4 b \sqrt {b \,x^{2}+a}}-\frac {5 a \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )}{4 b}\right )+6 c^{2} d^{2} \left (\frac {x^{3}}{2 b \sqrt {b \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )}{2 b}\right )+4 c^{3} d \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )\) | \(331\) |
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Time = 0.35 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.27 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (64 \, a^{2} b^{3} c^{3} d - 144 \, a^{3} b^{2} c^{2} d^{2} + 120 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4} + {\left (64 \, a b^{4} c^{3} d - 144 \, a^{2} b^{3} c^{2} d^{2} + 120 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (8 \, a b^{4} d^{4} x^{7} + 2 \, {\left (24 \, a b^{4} c d^{3} - 7 \, a^{2} b^{3} d^{4}\right )} x^{5} + {\left (144 \, a b^{4} c^{2} d^{2} - 120 \, a^{2} b^{3} c d^{3} + 35 \, a^{3} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (16 \, b^{5} c^{4} - 64 \, a b^{4} c^{3} d + 144 \, a^{2} b^{3} c^{2} d^{2} - 120 \, a^{3} b^{2} c d^{3} + 35 \, a^{4} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, {\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}, -\frac {3 \, {\left (64 \, a^{2} b^{3} c^{3} d - 144 \, a^{3} b^{2} c^{2} d^{2} + 120 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4} + {\left (64 \, a b^{4} c^{3} d - 144 \, a^{2} b^{3} c^{2} d^{2} + 120 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, a b^{4} d^{4} x^{7} + 2 \, {\left (24 \, a b^{4} c d^{3} - 7 \, a^{2} b^{3} d^{4}\right )} x^{5} + {\left (144 \, a b^{4} c^{2} d^{2} - 120 \, a^{2} b^{3} c d^{3} + 35 \, a^{3} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (16 \, b^{5} c^{4} - 64 \, a b^{4} c^{3} d + 144 \, a^{2} b^{3} c^{2} d^{2} - 120 \, a^{3} b^{2} c d^{3} + 35 \, a^{4} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, {\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \]
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\[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{4}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.21 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {d^{4} x^{7}}{6 \, \sqrt {b x^{2} + a} b} + \frac {c d^{3} x^{5}}{\sqrt {b x^{2} + a} b} - \frac {7 \, a d^{4} x^{5}}{24 \, \sqrt {b x^{2} + a} b^{2}} + \frac {3 \, c^{2} d^{2} x^{3}}{\sqrt {b x^{2} + a} b} - \frac {5 \, a c d^{3} x^{3}}{2 \, \sqrt {b x^{2} + a} b^{2}} + \frac {35 \, a^{2} d^{4} x^{3}}{48 \, \sqrt {b x^{2} + a} b^{3}} + \frac {c^{4} x}{\sqrt {b x^{2} + a} a} - \frac {4 \, c^{3} d x}{\sqrt {b x^{2} + a} b} + \frac {9 \, a c^{2} d^{2} x}{\sqrt {b x^{2} + a} b^{2}} - \frac {15 \, a^{2} c d^{3} x}{2 \, \sqrt {b x^{2} + a} b^{3}} + \frac {35 \, a^{3} d^{4} x}{16 \, \sqrt {b x^{2} + a} b^{4}} + \frac {4 \, c^{3} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {9 \, a c^{2} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} + \frac {15 \, a^{2} c d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {7}{2}}} - \frac {35 \, a^{3} d^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {9}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.91 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (2 \, {\left (\frac {4 \, d^{4} x^{2}}{b} + \frac {24 \, a b^{6} c d^{3} - 7 \, a^{2} b^{5} d^{4}}{a b^{7}}\right )} x^{2} + \frac {144 \, a b^{6} c^{2} d^{2} - 120 \, a^{2} b^{5} c d^{3} + 35 \, a^{3} b^{4} d^{4}}{a b^{7}}\right )} x^{2} + \frac {3 \, {\left (16 \, b^{7} c^{4} - 64 \, a b^{6} c^{3} d + 144 \, a^{2} b^{5} c^{2} d^{2} - 120 \, a^{3} b^{4} c d^{3} + 35 \, a^{4} b^{3} d^{4}\right )}}{a b^{7}}\right )} x}{48 \, \sqrt {b x^{2} + a}} - \frac {{\left (64 \, b^{3} c^{3} d - 144 \, a b^{2} c^{2} d^{2} + 120 \, a^{2} b c d^{3} - 35 \, a^{3} d^{4}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{16 \, b^{\frac {9}{2}}} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^4}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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