Integrand size = 26, antiderivative size = 67 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}}+\frac {2 (4 b c-5 a d) \sqrt [4]{a+b x^2}}{5 a^2 e^3 \sqrt {e x}} \]
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Time = 0.02 (sec) , antiderivative size = 67, 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 = {464, 270} \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{a+b x^2} (4 b c-5 a d)}{5 a^2 e^3 \sqrt {e x}}-\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}} \]
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Rule 270
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}}-\frac {(4 b c-5 a d) \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx}{5 a e^2} \\ & = -\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}}+\frac {2 (4 b c-5 a d) \sqrt [4]{a+b x^2}}{5 a^2 e^3 \sqrt {e x}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.66 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {2 x \sqrt [4]{a+b x^2} \left (a c-4 b c x^2+5 a d x^2\right )}{5 a^2 (e x)^{7/2}} \]
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Time = 3.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.58
method | result | size |
gosper | \(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (5 a d \,x^{2}-4 c b \,x^{2}+a c \right )}{5 a^{2} \left (e x \right )^{\frac {7}{2}}}\) | \(39\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (5 a d \,x^{2}-4 c b \,x^{2}+a c \right )}{5 e^{3} \sqrt {e x}\, a^{2} x^{2}}\) | \(44\) |
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none
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx=\frac {2 \, {\left ({\left (4 \, b c - 5 \, a d\right )} x^{2} - a c\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {e x}}{5 \, a^{2} e^{4} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (60) = 120\).
Time = 32.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.81 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx=- \frac {\sqrt [4]{b} c \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{8 a e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {3}{4}\right )} + \frac {\sqrt [4]{b} d \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {1}{4}\right )}{2 a e^{\frac {7}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {b^{\frac {5}{4}} c \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{2 a^{2} e^{\frac {7}{2}} \Gamma \left (\frac {3}{4}\right )} \]
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\[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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Time = 5.62 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx=-\frac {\left (\frac {2\,c}{5\,a\,e^3}+\frac {x^2\,\left (10\,a\,d-8\,b\,c\right )}{5\,a^2\,e^3}\right )\,{\left (b\,x^2+a\right )}^{1/4}}{x^2\,\sqrt {e\,x}} \]
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