Integrand size = 26, antiderivative size = 104 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (8 b c-5 a d)}{15 a^2 e^3 \sqrt {e x} \left (a+b x^2\right )^{3/4}}+\frac {8 (8 b c-5 a d) \sqrt [4]{a+b x^2}}{15 a^3 e^3 \sqrt {e x}} \]
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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} \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx=\frac {8 \sqrt [4]{a+b x^2} (8 b c-5 a d)}{15 a^3 e^3 \sqrt {e x}}-\frac {2 (8 b c-5 a d)}{15 a^2 e^3 \sqrt {e x} \left (a+b x^2\right )^{3/4}}-\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{3/4}} \]
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Rule 270
Rule 279
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{3/4}}-\frac {(8 b c-5 a d) \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^{7/4}} \, dx}{5 a e^2} \\ & = -\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (8 b c-5 a d)}{15 a^2 e^3 \sqrt {e x} \left (a+b x^2\right )^{3/4}}-\frac {(4 (8 b c-5 a d)) \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx}{15 a^2 e^2} \\ & = -\frac {2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{3/4}}-\frac {2 (8 b c-5 a d)}{15 a^2 e^3 \sqrt {e x} \left (a+b x^2\right )^{3/4}}+\frac {8 (8 b c-5 a d) \sqrt [4]{a+b x^2}}{15 a^3 e^3 \sqrt {e x}} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {2 x \left (3 a^2 c-24 a b c x^2+15 a^2 d x^2-32 b^2 c x^4+20 a b d x^4\right )}{15 a^3 (e x)^{7/2} \left (a+b x^2\right )^{3/4}} \]
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Time = 3.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.60
method | result | size |
gosper | \(-\frac {2 x \left (20 a b d \,x^{4}-32 b^{2} c \,x^{4}+15 a^{2} d \,x^{2}-24 a b c \,x^{2}+3 a^{2} c \right )}{15 \left (b \,x^{2}+a \right )^{\frac {3}{4}} a^{3} \left (e x \right )^{\frac {7}{2}}}\) | \(62\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (5 a d \,x^{2}-9 c b \,x^{2}+a c \right )}{5 a^{3} x^{2} e^{3} \sqrt {e x}}-\frac {2 b \left (a d -b c \right ) x^{2}}{3 a^{3} e^{3} \sqrt {e x}\, \left (b \,x^{2}+a \right )^{\frac {3}{4}}}\) | \(79\) |
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none
Time = 0.31 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.78 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx=\frac {2 \, {\left (4 \, {\left (8 \, b^{2} c - 5 \, a b d\right )} x^{4} - 3 \, a^{2} c + 3 \, {\left (8 \, a b c - 5 \, a^{2} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {e x}}{15 \, {\left (a^{3} b e^{4} x^{5} + a^{4} e^{4} x^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (99) = 198\).
Time = 138.65 (sec) , antiderivative size = 469, normalized size of antiderivative = 4.51 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx=c \left (- \frac {3 a^{3} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{32 a^{5} b^{4} e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {7}{4}\right ) + 64 a^{4} b^{5} e^{\frac {7}{2}} x^{4} \Gamma \left (\frac {7}{4}\right ) + 32 a^{3} b^{6} e^{\frac {7}{2}} x^{6} \Gamma \left (\frac {7}{4}\right )} + \frac {21 a^{2} b^{\frac {21}{4}} x^{2} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{32 a^{5} b^{4} e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {7}{4}\right ) + 64 a^{4} b^{5} e^{\frac {7}{2}} x^{4} \Gamma \left (\frac {7}{4}\right ) + 32 a^{3} b^{6} e^{\frac {7}{2}} x^{6} \Gamma \left (\frac {7}{4}\right )} + \frac {56 a b^{\frac {25}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{32 a^{5} b^{4} e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {7}{4}\right ) + 64 a^{4} b^{5} e^{\frac {7}{2}} x^{4} \Gamma \left (\frac {7}{4}\right ) + 32 a^{3} b^{6} e^{\frac {7}{2}} x^{6} \Gamma \left (\frac {7}{4}\right )} + \frac {32 b^{\frac {29}{4}} x^{6} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{32 a^{5} b^{4} e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {7}{4}\right ) + 64 a^{4} b^{5} e^{\frac {7}{2}} x^{4} \Gamma \left (\frac {7}{4}\right ) + 32 a^{3} b^{6} e^{\frac {7}{2}} x^{6} \Gamma \left (\frac {7}{4}\right )}\right ) + d \left (\frac {3 \Gamma \left (- \frac {1}{4}\right )}{8 a b^{\frac {3}{4}} e^{\frac {7}{2}} x^{2} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt [4]{b} \Gamma \left (- \frac {1}{4}\right )}{2 a^{2} e^{\frac {7}{2}} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )}\right ) \]
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\[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx=\int { \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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Time = 5.76 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx=-\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2\,c}{5\,a\,b\,e^3}+\frac {x^2\,\left (30\,a^2\,d-48\,a\,b\,c\right )}{15\,a^3\,b\,e^3}-\frac {x^4\,\left (64\,b^2\,c-40\,a\,b\,d\right )}{15\,a^3\,b\,e^3}\right )}{x^4\,\sqrt {e\,x}+\frac {a\,x^2\,\sqrt {e\,x}}{b}} \]
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