Integrand size = 26, antiderivative size = 104 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}+\frac {8 (8 b c-7 a d) \left (a+b x^2\right )^{3/4}}{21 a^3 e^3 (e x)^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {464, 279, 270} \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=\frac {8 \left (a+b x^2\right )^{3/4} (8 b c-7 a d)}{21 a^3 e^3 (e x)^{3/2}}-\frac {2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac {2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}} \]
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Rule 270
Rule 279
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {(8 b c-7 a d) \int \frac {1}{(e x)^{5/2} \left (a+b x^2\right )^{5/4}} \, dx}{7 a e^2} \\ & = -\frac {2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac {(4 (8 b c-7 a d)) \int \frac {1}{(e x)^{5/2} \sqrt [4]{a+b x^2}} \, dx}{7 a^2 e^2} \\ & = -\frac {2 c}{7 a e (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {2 (8 b c-7 a d)}{7 a^2 e^3 (e x)^{3/2} \sqrt [4]{a+b x^2}}+\frac {8 (8 b c-7 a d) \left (a+b x^2\right )^{3/4}}{21 a^3 e^3 (e x)^{3/2}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {2 x \left (3 a^2 c-8 a b c x^2+7 a^2 d x^2-32 b^2 c x^4+28 a b d x^4\right )}{21 a^3 (e x)^{9/2} \sqrt [4]{a+b x^2}} \]
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Time = 3.20 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(-\frac {2 x \left (28 a b d \,x^{4}-32 b^{2} c \,x^{4}+7 a^{2} d \,x^{2}-8 a b c \,x^{2}+3 a^{2} c \right )}{21 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3} \left (e x \right )^{\frac {9}{2}}}\) | \(62\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (7 a d \,x^{2}-11 c b \,x^{2}+3 a c \right )}{21 a^{3} x^{3} e^{4} \sqrt {e x}}-\frac {2 b x \left (a d -b c \right )}{a^{3} e^{4} \sqrt {e x}\, \left (b \,x^{2}+a \right )^{\frac {1}{4}}}\) | \(78\) |
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Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.77 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=\frac {2 \, {\left (4 \, {\left (8 \, b^{2} c - 7 \, a b d\right )} x^{4} - 3 \, a^{2} c + {\left (8 \, a b c - 7 \, a^{2} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{21 \, {\left (a^{3} b e^{5} x^{6} + a^{4} e^{5} x^{4}\right )}} \]
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Timed out. \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {9}{2}}} \,d x } \]
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Time = 5.75 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx=-\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {2\,c}{7\,a\,b\,e^4}+\frac {x^2\,\left (14\,a^2\,d-16\,a\,b\,c\right )}{21\,a^3\,b\,e^4}-\frac {x^4\,\left (64\,b^2\,c-56\,a\,b\,d\right )}{21\,a^3\,b\,e^4}\right )}{x^5\,\sqrt {e\,x}+\frac {a\,x^3\,\sqrt {e\,x}}{b}} \]
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