Integrand size = 19, antiderivative size = 66 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx=-\frac {2 (c+d x)^{3/2}}{5 (b c-a d) (a+b x)^{5/2}}+\frac {4 d (c+d x)^{3/2}}{15 (b c-a d)^2 (a+b x)^{3/2}} \]
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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} \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx=\frac {4 d (c+d x)^{3/2}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac {2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c+d x)^{3/2}}{5 (b c-a d) (a+b x)^{5/2}}-\frac {(2 d) \int \frac {\sqrt {c+d x}}{(a+b x)^{5/2}} \, dx}{5 (b c-a d)} \\ & = -\frac {2 (c+d x)^{3/2}}{5 (b c-a d) (a+b x)^{5/2}}+\frac {4 d (c+d x)^{3/2}}{15 (b c-a d)^2 (a+b x)^{3/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx=-\frac {2 (c+d x)^{3/2} (3 b c-5 a d-2 b d x)}{15 (b c-a d)^2 (a+b x)^{5/2}} \]
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Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (2 b d x +5 a d -3 b c \right )}{15 \left (b x +a \right )^{\frac {5}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(54\) |
default | \(-\frac {\sqrt {d x +c}}{2 b \left (b x +a \right )^{\frac {5}{2}}}+\frac {\left (a d -b c \right ) \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{4 b}\) | \(128\) |
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Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (54) = 108\).
Time = 0.38 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.65 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx=\frac {2 \, {\left (2 \, b d^{2} x^{2} - 3 \, b c^{2} + 5 \, a c d - {\left (b c d - 5 \, a d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} + {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{3} + 3 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x\right )}} \]
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\[ \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx=\int \frac {\sqrt {c + d x}}{\left (a + b x\right )^{\frac {7}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (54) = 108\).
Time = 0.40 (sec) , antiderivative size = 447, normalized size of antiderivative = 6.77 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx=\frac {8 \, {\left (\sqrt {b d} b^{7} c^{3} d^{2} - 3 \, \sqrt {b d} a b^{6} c^{2} d^{3} + 3 \, \sqrt {b d} a^{2} b^{5} c d^{4} - \sqrt {b d} a^{3} b^{4} d^{5} - 5 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{5} c^{2} d^{2} + 10 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} c d^{3} - 5 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{3} d^{4} - 5 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{3} c d^{2} + 5 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{2} d^{3} - 15 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b d^{2}\right )} {\left | b \right |}}{15 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{5} b^{2}} \]
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Time = 0.96 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.92 \[ \int \frac {\sqrt {c+d x}}{(a+b x)^{7/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {x\,\left (10\,a\,d^2-2\,b\,c\,d\right )}{15\,b^2\,{\left (a\,d-b\,c\right )}^2}-\frac {6\,b\,c^2-10\,a\,c\,d}{15\,b^2\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,d^2\,x^2}{15\,b\,{\left (a\,d-b\,c\right )}^2}\right )}{x^2\,\sqrt {a+b\,x}+\frac {a^2\,\sqrt {a+b\,x}}{b^2}+\frac {2\,a\,x\,\sqrt {a+b\,x}}{b}} \]
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