Integrand size = 21, antiderivative size = 255 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 b^4}-\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}} \]
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Time = 0.19 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {424, 540, 542, 396, 223, 212} \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right )}{8 b^{9/2}}+\frac {x \left (c+d x^2\right )^2 (b c-a d) (7 a d+2 b c)}{3 a^2 b^2 \sqrt {a+b x^2}}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{12 a^2 b^3}-\frac {d x \sqrt {a+b x^2} \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )}{24 a^2 b^4}+\frac {x \left (c+d x^2\right )^3 (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]
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Rule 212
Rule 223
Rule 396
Rule 424
Rule 540
Rule 542
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\int \frac {\left (c+d x^2\right )^2 \left (c (2 b c+a d)-d (4 b c-7 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+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {\left (c+d x^2\right ) \left (a c d (4 b c-7 a d)+d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x^2\right )}{\sqrt {a+b x^2}} \, dx}{3 a^2 b^2} \\ & = -\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {a c d \left (8 b^2 c^2-52 a b c d+35 a^2 d^2\right )+d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x^2}{\sqrt {a+b x^2}} \, dx}{12 a^2 b^3} \\ & = -\frac {d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 b^4}-\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\left (d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^4} \\ & = -\frac {d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 b^4}-\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\left (d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^4} \\ & = -\frac {d \left (16 b^3 c^3+40 a b^2 c^2 d-170 a^2 b c d^2+105 a^3 d^3\right ) x \sqrt {a+b x^2}}{24 a^2 b^4}-\frac {d \left (8 b^2 c^2+24 a b c d-35 a^2 d^2\right ) x \sqrt {a+b x^2} \left (c+d x^2\right )}{12 a^2 b^3}+\frac {(b c-a d) (2 b c+7 a d) x \left (c+d x^2\right )^2}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )^3}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{9/2}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.79 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (-105 a^5 d^4+16 b^5 c^4 x^2+20 a^4 b d^3 \left (12 c-7 d x^2\right )+8 a b^4 c^3 \left (3 c+4 d x^2\right )+a^3 b^2 d^2 \left (-144 c^2+320 c d x^2-21 d^2 x^4\right )+6 a^2 b^3 d^2 x^2 \left (-32 c^2+8 c d x^2+d^2 x^4\right )\right )}{24 a^2 b^4 \left (a+b x^2\right )^{3/2}}-\frac {d^2 \left (48 b^2 c^2-80 a b c d+35 a^2 d^2\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 b^{9/2}} \]
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Time = 2.55 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.78
method | result | size |
pseudoelliptic | \(\frac {\frac {35 \left (b \,x^{2}+a \right )^{\frac {3}{2}} d^{2} \left (a^{2} d^{2}-\frac {16}{7} a b c d +\frac {48}{35} b^{2} c^{2}\right ) a^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{8}-\frac {35 x \left (\frac {48 d^{2} a^{3} \left (\frac {7}{48} d^{2} x^{4}-\frac {20}{9} c d \,x^{2}+c^{2}\right ) b^{\frac {5}{2}}}{35}+\frac {64 x^{2} \left (-\frac {1}{32} d^{2} x^{4}-\frac {1}{4} c d \,x^{2}+c^{2}\right ) d^{2} a^{2} b^{\frac {7}{2}}}{35}-\frac {16 \left (-\frac {7 d \,x^{2}}{12}+c \right ) d^{3} a^{4} b^{\frac {3}{2}}}{7}-\frac {8 \left (\frac {4 d \,x^{2}}{3}+c \right ) c^{3} a \,b^{\frac {9}{2}}}{35}+\sqrt {b}\, a^{5} d^{4}-\frac {16 b^{\frac {11}{2}} c^{4} x^{2}}{105}\right )}{8}}{a^{2} b^{\frac {9}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(198\) |
default | \(c^{4} \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )+d^{4} \left (\frac {x^{7}}{4 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {7 a \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )}{4 b}\right )+4 c \,d^{3} \left (\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+6 c^{2} d^{2} \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}\right )+4 c^{3} d \left (-\frac {x}{2 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {b \,x^{2}+a}}\right )}{2 b}\right )\) | \(360\) |
risch | \(-\frac {d^{3} x \left (-2 b d \,x^{2}+11 a d -16 b c \right ) \sqrt {b \,x^{2}+a}}{8 b^{4}}+\frac {\frac {d^{2} \left (35 a^{2} d^{2}-80 a b c d +48 b^{2} c^{2}\right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}-\frac {2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{b a}-\frac {2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \left (\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{b a}-\frac {2 \left (7 a^{4} d^{4}-20 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{a^{2} b \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {2 \left (7 a^{4} d^{4}-20 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{a^{2} b \left (x -\frac {\sqrt {-a b}}{b}\right )}}{8 b^{4}}\) | \(675\) |
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Time = 0.40 (sec) , antiderivative size = 684, normalized size of antiderivative = 2.68 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (48 \, a^{4} b^{2} c^{2} d^{2} - 80 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} + {\left (48 \, a^{2} b^{4} c^{2} d^{2} - 80 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (48 \, a^{3} b^{3} c^{2} d^{2} - 80 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (6 \, a^{2} b^{4} d^{4} x^{7} + 3 \, {\left (16 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 4 \, {\left (4 \, b^{6} c^{4} + 8 \, a b^{5} c^{3} d - 48 \, a^{2} b^{4} c^{2} d^{2} + 80 \, a^{3} b^{3} c d^{3} - 35 \, a^{4} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (8 \, a b^{5} c^{4} - 48 \, a^{3} b^{3} c^{2} d^{2} + 80 \, a^{4} b^{2} c d^{3} - 35 \, a^{5} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{48 \, {\left (a^{2} b^{7} x^{4} + 2 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}, -\frac {3 \, {\left (48 \, a^{4} b^{2} c^{2} d^{2} - 80 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} + {\left (48 \, a^{2} b^{4} c^{2} d^{2} - 80 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (48 \, a^{3} b^{3} c^{2} d^{2} - 80 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (6 \, a^{2} b^{4} d^{4} x^{7} + 3 \, {\left (16 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 4 \, {\left (4 \, b^{6} c^{4} + 8 \, a b^{5} c^{3} d - 48 \, a^{2} b^{4} c^{2} d^{2} + 80 \, a^{3} b^{3} c d^{3} - 35 \, a^{4} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (8 \, a b^{5} c^{4} - 48 \, a^{3} b^{3} c^{2} d^{2} + 80 \, a^{4} b^{2} c d^{3} - 35 \, a^{5} b d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{24 \, {\left (a^{2} b^{7} x^{4} + 2 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}\right ] \]
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\[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{4}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.54 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {d^{4} x^{7}}{4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, c d^{3} x^{5}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {7 \, a d^{4} x^{5}}{8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} - 2 \, c^{2} d^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {10 \, a c d^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b} - \frac {35 \, a^{2} d^{4} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{24 \, b^{2}} + \frac {2 \, c^{4} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c^{4} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {4 \, c^{3} d x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {4 \, c^{3} d x}{3 \, \sqrt {b x^{2} + a} a b} - \frac {2 \, c^{2} d^{2} x}{\sqrt {b x^{2} + a} b^{2}} + \frac {10 \, a c d^{3} x}{3 \, \sqrt {b x^{2} + a} b^{3}} - \frac {35 \, a^{2} d^{4} x}{24 \, \sqrt {b x^{2} + a} b^{4}} + \frac {6 \, c^{2} d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} - \frac {10 \, a c d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {7}{2}}} + \frac {35 \, a^{2} d^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {9}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.93 \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (\frac {2 \, d^{4} x^{2}}{b} + \frac {16 \, a^{2} b^{6} c d^{3} - 7 \, a^{3} b^{5} d^{4}}{a^{2} b^{7}}\right )} x^{2} + \frac {4 \, {\left (4 \, b^{8} c^{4} + 8 \, a b^{7} c^{3} d - 48 \, a^{2} b^{6} c^{2} d^{2} + 80 \, a^{3} b^{5} c d^{3} - 35 \, a^{4} b^{4} d^{4}\right )}}{a^{2} b^{7}}\right )} x^{2} + \frac {3 \, {\left (8 \, a b^{7} c^{4} - 48 \, a^{3} b^{5} c^{2} d^{2} + 80 \, a^{4} b^{4} c d^{3} - 35 \, a^{5} b^{3} d^{4}\right )}}{a^{2} b^{7}}\right )} x}{24 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {{\left (48 \, b^{2} c^{2} d^{2} - 80 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {9}{2}}} \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^4}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
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