Integrand size = 19, antiderivative size = 108 \[ \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx=-\frac {32 b^3 \sqrt {b x+c x^2}}{35 c^4 \sqrt {x}}+\frac {16 b^2 \sqrt {x} \sqrt {b x+c x^2}}{35 c^3}-\frac {12 b x^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 x^{5/2} \sqrt {b x+c x^2}}{7 c} \]
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Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {670, 662} \[ \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx=-\frac {32 b^3 \sqrt {b x+c x^2}}{35 c^4 \sqrt {x}}+\frac {16 b^2 \sqrt {x} \sqrt {b x+c x^2}}{35 c^3}-\frac {12 b x^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 x^{5/2} \sqrt {b x+c x^2}}{7 c} \]
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Rule 662
Rule 670
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{5/2} \sqrt {b x+c x^2}}{7 c}-\frac {(6 b) \int \frac {x^{5/2}}{\sqrt {b x+c x^2}} \, dx}{7 c} \\ & = -\frac {12 b x^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 x^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {\left (24 b^2\right ) \int \frac {x^{3/2}}{\sqrt {b x+c x^2}} \, dx}{35 c^2} \\ & = \frac {16 b^2 \sqrt {x} \sqrt {b x+c x^2}}{35 c^3}-\frac {12 b x^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 x^{5/2} \sqrt {b x+c x^2}}{7 c}-\frac {\left (16 b^3\right ) \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx}{35 c^3} \\ & = -\frac {32 b^3 \sqrt {b x+c x^2}}{35 c^4 \sqrt {x}}+\frac {16 b^2 \sqrt {x} \sqrt {b x+c x^2}}{35 c^3}-\frac {12 b x^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 x^{5/2} \sqrt {b x+c x^2}}{7 c} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.49 \[ \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {x (b+c x)} \left (-16 b^3+8 b^2 c x-6 b c^2 x^2+5 c^3 x^3\right )}{35 c^4 \sqrt {x}} \]
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Time = 1.98 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.44
method | result | size |
default | \(-\frac {2 \sqrt {x \left (c x +b \right )}\, \left (-5 c^{3} x^{3}+6 b \,c^{2} x^{2}-8 b^{2} c x +16 b^{3}\right )}{35 \sqrt {x}\, c^{4}}\) | \(48\) |
risch | \(-\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (-5 c^{3} x^{3}+6 b \,c^{2} x^{2}-8 b^{2} c x +16 b^{3}\right )}{35 \sqrt {x \left (c x +b \right )}\, c^{4}}\) | \(53\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-5 c^{3} x^{3}+6 b \,c^{2} x^{2}-8 b^{2} c x +16 b^{3}\right ) \sqrt {x}}{35 c^{4} \sqrt {c \,x^{2}+b x}}\) | \(55\) |
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Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.45 \[ \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (5 \, c^{3} x^{3} - 6 \, b c^{2} x^{2} + 8 \, b^{2} c x - 16 \, b^{3}\right )} \sqrt {c x^{2} + b x}}{35 \, c^{4} \sqrt {x}} \]
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\[ \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx=\int \frac {x^{\frac {7}{2}}}{\sqrt {x \left (b + c x\right )}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.49 \[ \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (5 \, c^{4} x^{4} - b c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 8 \, b^{3} c x - 16 \, b^{4}\right )}}{35 \, \sqrt {c x + b} c^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.56 \[ \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx=-\frac {2 \, \sqrt {c x + b} b^{3}}{c^{4}} + \frac {32 \, b^{\frac {7}{2}}}{35 \, c^{4}} + \frac {2 \, {\left (5 \, {\left (c x + b\right )}^{\frac {7}{2}} - 21 \, {\left (c x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{2}\right )}}{35 \, c^{4}} \]
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Timed out. \[ \int \frac {x^{7/2}}{\sqrt {b x+c x^2}} \, dx=\int \frac {x^{7/2}}{\sqrt {c\,x^2+b\,x}} \,d x \]
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