Integrand size = 19, antiderivative size = 66 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {2 \sqrt {a+b x}}{3 (b c-a d) (c+d x)^{3/2}}+\frac {4 b \sqrt {a+b x}}{3 (b c-a d)^2 \sqrt {c+d x}} \]
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Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {4 b \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^2}+\frac {2 \sqrt {a+b x}}{3 (c+d x)^{3/2} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {a+b x}}{3 (b c-a d) (c+d x)^{3/2}}+\frac {(2 b) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)} \\ & = \frac {2 \sqrt {a+b x}}{3 (b c-a d) (c+d x)^{3/2}}+\frac {4 b \sqrt {a+b x}}{3 (b c-a d)^2 \sqrt {c+d x}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {2 \sqrt {a+b x} (3 b c-a d+2 b d x)}{3 (b c-a d)^2 (c+d x)^{3/2}} \]
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Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-2 b d x +a d -3 b c \right )}{3 \left (d x +c \right )^{\frac {3}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(53\) |
default | \(-\frac {2 \sqrt {b x +a}}{3 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 \left (a d -b c \right )^{2} \sqrt {d x +c}}\) | \(55\) |
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (54) = 108\).
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {2 \, {\left (2 \, b d x + 3 \, b c - a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \]
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\[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {1}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (54) = 108\).
Time = 0.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.91 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {2 \, {\left (b x + a\right )} b^{4} d^{2}}{b^{2} c^{2} d {\left | b \right |} - 2 \, a b c d^{2} {\left | b \right |} + a^{2} d^{3} {\left | b \right |}} + \frac {3 \, {\left (b^{5} c d - a b^{4} d^{2}\right )}}{b^{2} c^{2} d {\left | b \right |} - 2 \, a b c d^{2} {\left | b \right |} + a^{2} d^{3} {\left | b \right |}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} \]
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Time = 0.84 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.92 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {x\,\left (6\,c\,b^2+2\,a\,d\,b\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,a^2\,d-6\,a\,b\,c}{3\,d^2\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,b^2\,x^2}{3\,d\,{\left (a\,d-b\,c\right )}^2}\right )}{x^2\,\sqrt {a+b\,x}+\frac {c^2\,\sqrt {a+b\,x}}{d^2}+\frac {2\,c\,x\,\sqrt {a+b\,x}}{d}} \]
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