Integrand size = 26, antiderivative size = 79 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{9/4}} \, dx=\frac {2 (b c-a d) \sqrt {e x}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {2 (4 b c+a d) \sqrt {e x}}{5 a^2 b e \sqrt [4]{a+b x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {468, 270} \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{9/4}} \, dx=\frac {2 \sqrt {e x} (a d+4 b c)}{5 a^2 b e \sqrt [4]{a+b x^2}}+\frac {2 \sqrt {e x} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]
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Rule 270
Rule 468
Rubi steps \begin{align*} \text {integral}& = \frac {2 (b c-a d) \sqrt {e x}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {\left (2 \left (2 b c+\frac {a d}{2}\right )\right ) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{5/4}} \, dx}{5 a b} \\ & = \frac {2 (b c-a d) \sqrt {e x}}{5 a b e \left (a+b x^2\right )^{5/4}}+\frac {2 (4 b c+a d) \sqrt {e x}}{5 a^2 b e \sqrt [4]{a+b x^2}} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.56 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{9/4}} \, dx=\frac {2 x \left (5 a c+4 b c x^2+a d x^2\right )}{5 a^2 \sqrt {e x} \left (a+b x^2\right )^{5/4}} \]
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Time = 3.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.49
method | result | size |
gosper | \(\frac {2 x \left (a d \,x^{2}+4 c b \,x^{2}+5 a c \right )}{5 \left (b \,x^{2}+a \right )^{\frac {5}{4}} a^{2} \sqrt {e x}}\) | \(39\) |
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none
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{9/4}} \, dx=\frac {2 \, {\left ({\left (4 \, b c + a d\right )} x^{2} + 5 \, a c\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{5 \, {\left (a^{2} b^{2} e x^{4} + 2 \, a^{3} b e x^{2} + a^{4} e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (71) = 142\).
Time = 69.40 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.91 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{9/4}} \, dx=c \left (\frac {5 a \Gamma \left (\frac {1}{4}\right )}{8 a^{3} \sqrt [4]{b} \sqrt {e} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {9}{4}\right ) + 8 a^{2} b^{\frac {5}{4}} \sqrt {e} x^{2} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {9}{4}\right )} + \frac {4 b x^{2} \Gamma \left (\frac {1}{4}\right )}{8 a^{3} \sqrt [4]{b} \sqrt {e} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {9}{4}\right ) + 8 a^{2} b^{\frac {5}{4}} \sqrt {e} x^{2} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {9}{4}\right )}\right ) + \frac {d \Gamma \left (\frac {5}{4}\right )}{\frac {2 a^{2} \sqrt [4]{b} \sqrt {e} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {9}{4}\right )}{x^{2}} + 2 a b^{\frac {5}{4}} \sqrt {e} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {9}{4}\right )} \]
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\[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \sqrt {e x}} \,d x } \]
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\[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{9/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \sqrt {e x}} \,d x } \]
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Time = 5.86 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{9/4}} \, dx=\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {x^3\,\left (2\,a\,d+8\,b\,c\right )}{5\,a^2\,b^2}+\frac {2\,c\,x}{a\,b^2}\right )}{x^4\,\sqrt {e\,x}+\frac {a^2\,\sqrt {e\,x}}{b^2}+\frac {2\,a\,x^2\,\sqrt {e\,x}}{b}} \]
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