Integrand size = 22, antiderivative size = 169 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {3 (b c-3 a d) (b c-a d)^2 x}{2 b^5}+\frac {d \left (5 b^2 c^2-7 a b c d+3 a^2 d^2\right ) x^3}{2 b^4}+\frac {3 d^2 (7 b c-3 a d) x^5}{10 b^3}+\frac {9 d^3 x^7}{14 b^2}-\frac {x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {3 \sqrt {a} (b c-3 a d) (b c-a d)^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}} \]
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Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {478, 584, 211} \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {d x^3 \left (3 a^2 d^2-7 a b c d+5 b^2 c^2\right )}{2 b^4}-\frac {3 \sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-3 a d) (b c-a d)^2}{2 b^{11/2}}+\frac {3 x (b c-3 a d) (b c-a d)^2}{2 b^5}+\frac {3 d^2 x^5 (7 b c-3 a d)}{10 b^3}-\frac {x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {9 d^3 x^7}{14 b^2} \]
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Rule 211
Rule 478
Rule 584
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\int \frac {x^2 \left (c+d x^2\right )^2 \left (3 c+9 d x^2\right )}{a+b x^2} \, dx}{2 b} \\ & = -\frac {x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\int \left (\frac {3 (b c-3 a d) (b c-a d)^2}{b^4}+\frac {3 d \left (5 b^2 c^2-7 a b c d+3 a^2 d^2\right ) x^2}{b^3}+\frac {3 d^2 (7 b c-3 a d) x^4}{b^2}+\frac {9 d^3 x^6}{b}+\frac {3 \left (-a b^3 c^3+5 a^2 b^2 c^2 d-7 a^3 b c d^2+3 a^4 d^3\right )}{b^4 \left (a+b x^2\right )}\right ) \, dx}{2 b} \\ & = \frac {3 (b c-3 a d) (b c-a d)^2 x}{2 b^5}+\frac {d \left (5 b^2 c^2-7 a b c d+3 a^2 d^2\right ) x^3}{2 b^4}+\frac {3 d^2 (7 b c-3 a d) x^5}{10 b^3}+\frac {9 d^3 x^7}{14 b^2}-\frac {x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {\left (3 a (b c-3 a d) (b c-a d)^2\right ) \int \frac {1}{a+b x^2} \, dx}{2 b^5} \\ & = \frac {3 (b c-3 a d) (b c-a d)^2 x}{2 b^5}+\frac {d \left (5 b^2 c^2-7 a b c d+3 a^2 d^2\right ) x^3}{2 b^4}+\frac {3 d^2 (7 b c-3 a d) x^5}{10 b^3}+\frac {9 d^3 x^7}{14 b^2}-\frac {x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {3 \sqrt {a} (b c-3 a d) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {(b c-4 a d) (b c-a d)^2 x}{b^5}+\frac {d (b c-a d)^2 x^3}{b^4}+\frac {d^2 (3 b c-2 a d) x^5}{5 b^3}+\frac {d^3 x^7}{7 b^2}+\frac {a (b c-a d)^3 x}{2 b^5 \left (a+b x^2\right )}+\frac {3 \sqrt {a} (b c-a d)^2 (-b c+3 a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}} \]
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Time = 2.74 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.34
method | result | size |
default | \(-\frac {-\frac {1}{7} b^{3} d^{3} x^{7}+\frac {2}{5} a \,b^{2} d^{3} x^{5}-\frac {3}{5} b^{3} c \,d^{2} x^{5}-a^{2} b \,d^{3} x^{3}+2 a \,b^{2} c \,d^{2} x^{3}-b^{3} c^{2} d \,x^{3}+4 a^{3} d^{3} x -9 a^{2} b c \,d^{2} x +6 a \,b^{2} c^{2} d x -b^{3} c^{3} x}{b^{5}}+\frac {a \left (\frac {\left (-\frac {1}{2} a^{3} d^{3}+\frac {3}{2} a^{2} b c \,d^{2}-\frac {3}{2} a \,b^{2} c^{2} d +\frac {1}{2} b^{3} c^{3}\right ) x}{b \,x^{2}+a}+\frac {3 \left (3 a^{3} d^{3}-7 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{5}}\) | \(227\) |
risch | \(\frac {d^{3} x^{7}}{7 b^{2}}-\frac {2 a \,d^{3} x^{5}}{5 b^{3}}+\frac {3 c \,d^{2} x^{5}}{5 b^{2}}+\frac {a^{2} d^{3} x^{3}}{b^{4}}-\frac {2 a c \,d^{2} x^{3}}{b^{3}}+\frac {c^{2} d \,x^{3}}{b^{2}}-\frac {4 a^{3} d^{3} x}{b^{5}}+\frac {9 a^{2} c \,d^{2} x}{b^{4}}-\frac {6 a \,c^{2} d x}{b^{3}}+\frac {c^{3} x}{b^{2}}+\frac {\left (-\frac {1}{2} a^{4} d^{3}+\frac {3}{2} a^{3} b c \,d^{2}-\frac {3}{2} a^{2} b^{2} c^{2} d +\frac {1}{2} a \,b^{3} c^{3}\right ) x}{b^{5} \left (b \,x^{2}+a \right )}+\frac {9 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a^{3} d^{3}}{4 b^{6}}-\frac {21 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a^{2} c \,d^{2}}{4 b^{5}}+\frac {15 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a \,c^{2} d}{4 b^{4}}-\frac {3 \sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) c^{3}}{4 b^{3}}-\frac {9 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a^{3} d^{3}}{4 b^{6}}+\frac {21 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a^{2} c \,d^{2}}{4 b^{5}}-\frac {15 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a \,c^{2} d}{4 b^{4}}+\frac {3 \sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) c^{3}}{4 b^{3}}\) | \(394\) |
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Time = 0.25 (sec) , antiderivative size = 580, normalized size of antiderivative = 3.43 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\left [\frac {20 \, b^{4} d^{3} x^{9} + 12 \, {\left (7 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{7} + 28 \, {\left (5 \, b^{4} c^{2} d - 7 \, a b^{3} c d^{2} + 3 \, a^{2} b^{2} d^{3}\right )} x^{5} + 140 \, {\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{3} - 105 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 210 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} x}{140 \, {\left (b^{6} x^{2} + a b^{5}\right )}}, \frac {10 \, b^{4} d^{3} x^{9} + 6 \, {\left (7 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{7} + 14 \, {\left (5 \, b^{4} c^{2} d - 7 \, a b^{3} c d^{2} + 3 \, a^{2} b^{2} d^{3}\right )} x^{5} + 70 \, {\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{3} - 105 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 105 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} x}{70 \, {\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (167) = 334\).
Time = 0.77 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.30 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=x^{5} \left (- \frac {2 a d^{3}}{5 b^{3}} + \frac {3 c d^{2}}{5 b^{2}}\right ) + x^{3} \left (\frac {a^{2} d^{3}}{b^{4}} - \frac {2 a c d^{2}}{b^{3}} + \frac {c^{2} d}{b^{2}}\right ) + x \left (- \frac {4 a^{3} d^{3}}{b^{5}} + \frac {9 a^{2} c d^{2}}{b^{4}} - \frac {6 a c^{2} d}{b^{3}} + \frac {c^{3}}{b^{2}}\right ) + \frac {x \left (- a^{4} d^{3} + 3 a^{3} b c d^{2} - 3 a^{2} b^{2} c^{2} d + a b^{3} c^{3}\right )}{2 a b^{5} + 2 b^{6} x^{2}} - \frac {3 \sqrt {- \frac {a}{b^{11}}} \left (a d - b c\right )^{2} \cdot \left (3 a d - b c\right ) \log {\left (- \frac {3 b^{5} \sqrt {- \frac {a}{b^{11}}} \left (a d - b c\right )^{2} \cdot \left (3 a d - b c\right )}{9 a^{3} d^{3} - 21 a^{2} b c d^{2} + 15 a b^{2} c^{2} d - 3 b^{3} c^{3}} + x \right )}}{4} + \frac {3 \sqrt {- \frac {a}{b^{11}}} \left (a d - b c\right )^{2} \cdot \left (3 a d - b c\right ) \log {\left (\frac {3 b^{5} \sqrt {- \frac {a}{b^{11}}} \left (a d - b c\right )^{2} \cdot \left (3 a d - b c\right )}{9 a^{3} d^{3} - 21 a^{2} b c d^{2} + 15 a b^{2} c^{2} d - 3 b^{3} c^{3}} + x \right )}}{4} + \frac {d^{3} x^{7}}{7 b^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.35 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} - \frac {3 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {5 \, b^{3} d^{3} x^{7} + 7 \, {\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{5} + 35 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 35 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} x}{35 \, b^{5}} \]
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Time = 0.30 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.43 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=-\frac {3 \, {\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {a b^{3} c^{3} x - 3 \, a^{2} b^{2} c^{2} d x + 3 \, a^{3} b c d^{2} x - a^{4} d^{3} x}{2 \, {\left (b x^{2} + a\right )} b^{5}} + \frac {5 \, b^{12} d^{3} x^{7} + 21 \, b^{12} c d^{2} x^{5} - 14 \, a b^{11} d^{3} x^{5} + 35 \, b^{12} c^{2} d x^{3} - 70 \, a b^{11} c d^{2} x^{3} + 35 \, a^{2} b^{10} d^{3} x^{3} + 35 \, b^{12} c^{3} x - 210 \, a b^{11} c^{2} d x + 315 \, a^{2} b^{10} c d^{2} x - 140 \, a^{3} b^{9} d^{3} x}{35 \, b^{14}} \]
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Time = 4.78 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.94 \[ \int \frac {x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx=x\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )-x^5\,\left (\frac {2\,a\,d^3}{5\,b^3}-\frac {3\,c\,d^2}{5\,b^2}\right )+x^3\,\left (\frac {c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{3\,b}-\frac {a^2\,d^3}{3\,b^4}\right )-\frac {x\,\left (\frac {a^4\,d^3}{2}-\frac {3\,a^3\,b\,c\,d^2}{2}+\frac {3\,a^2\,b^2\,c^2\,d}{2}-\frac {a\,b^3\,c^3}{2}\right )}{b^6\,x^2+a\,b^5}+\frac {d^3\,x^7}{7\,b^2}+\frac {3\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d-b\,c\right )}{3\,a^4\,d^3-7\,a^3\,b\,c\,d^2+5\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d-b\,c\right )}{2\,b^{11/2}} \]
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