Integrand size = 19, antiderivative size = 204 \[ \int \frac {(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx=-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {105 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {105 b^{3/2} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{11/2}} \]
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Time = 0.08 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {49, 52, 65, 223, 212} \[ \int \frac {(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx=-\frac {105 b^{3/2} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{11/2}}+\frac {105 b^2 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 d^5}-\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}} \]
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Rule 49
Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}+\frac {(3 b) \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx}{d} \\ & = -\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {\left (21 b^2\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{d^2} \\ & = -\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {\left (35 b^2 (b c-a d)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{2 d^3} \\ & = -\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}+\frac {\left (105 b^2 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 d^4} \\ & = -\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {105 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {\left (105 b^2 (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 d^5} \\ & = -\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {105 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {\left (105 b (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 d^5} \\ & = -\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {105 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {\left (105 b (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 d^5} \\ & = -\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {105 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {105 b^{3/2} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{11/2}} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (-16 a^4 d^4-16 a^3 b d^3 (9 c+13 d x)+3 a^2 b^2 d^2 \left (231 c^2+318 c d x+55 d^2 x^2\right )-2 a b^3 d \left (420 c^3+567 c^2 d x+90 c d^2 x^2-25 d^3 x^3\right )+b^4 \left (315 c^4+420 c^3 d x+63 c^2 d^2 x^2-18 c d^3 x^3+8 d^4 x^4\right )\right )}{24 d^5 (c+d x)^{3/2}}-\frac {105 b^{3/2} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 d^{11/2}} \]
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\[\int \frac {\left (b x +a \right )^{\frac {9}{2}}}{\left (d x +c \right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (164) = 328\).
Time = 0.74 (sec) , antiderivative size = 879, normalized size of antiderivative = 4.31 \[ \int \frac {(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx=\left [-\frac {315 \, {\left (b^{4} c^{5} - 3 \, a b^{3} c^{4} d + 3 \, a^{2} b^{2} c^{3} d^{2} - a^{3} b c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{2} + 2 \, {\left (b^{4} c^{4} d - 3 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} - a^{3} b c d^{4}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (8 \, b^{4} d^{4} x^{4} + 315 \, b^{4} c^{4} - 840 \, a b^{3} c^{3} d + 693 \, a^{2} b^{2} c^{2} d^{2} - 144 \, a^{3} b c d^{3} - 16 \, a^{4} d^{4} - 2 \, {\left (9 \, b^{4} c d^{3} - 25 \, a b^{3} d^{4}\right )} x^{3} + 3 \, {\left (21 \, b^{4} c^{2} d^{2} - 60 \, a b^{3} c d^{3} + 55 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (210 \, b^{4} c^{3} d - 567 \, a b^{3} c^{2} d^{2} + 477 \, a^{2} b^{2} c d^{3} - 104 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}}, \frac {315 \, {\left (b^{4} c^{5} - 3 \, a b^{3} c^{4} d + 3 \, a^{2} b^{2} c^{3} d^{2} - a^{3} b c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{2} + 2 \, {\left (b^{4} c^{4} d - 3 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} - a^{3} b c d^{4}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, {\left (8 \, b^{4} d^{4} x^{4} + 315 \, b^{4} c^{4} - 840 \, a b^{3} c^{3} d + 693 \, a^{2} b^{2} c^{2} d^{2} - 144 \, a^{3} b c d^{3} - 16 \, a^{4} d^{4} - 2 \, {\left (9 \, b^{4} c d^{3} - 25 \, a b^{3} d^{4}\right )} x^{3} + 3 \, {\left (21 \, b^{4} c^{2} d^{2} - 60 \, a b^{3} c d^{3} + 55 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (210 \, b^{4} c^{3} d - 567 \, a b^{3} c^{2} d^{2} + 477 \, a^{2} b^{2} c d^{3} - 104 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}}\right ] \]
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\[ \int \frac {(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {9}{2}}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (164) = 328\).
Time = 0.44 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.45 \[ \int \frac {(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx=\frac {{\left ({\left ({\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b^{6} c d^{8} - a b^{5} d^{9}\right )} {\left (b x + a\right )}}{b^{2} c d^{9} {\left | b \right |} - a b d^{10} {\left | b \right |}} - \frac {9 \, {\left (b^{7} c^{2} d^{7} - 2 \, a b^{6} c d^{8} + a^{2} b^{5} d^{9}\right )}}{b^{2} c d^{9} {\left | b \right |} - a b d^{10} {\left | b \right |}}\right )} + \frac {63 \, {\left (b^{8} c^{3} d^{6} - 3 \, a b^{7} c^{2} d^{7} + 3 \, a^{2} b^{6} c d^{8} - a^{3} b^{5} d^{9}\right )}}{b^{2} c d^{9} {\left | b \right |} - a b d^{10} {\left | b \right |}}\right )} {\left (b x + a\right )} + \frac {420 \, {\left (b^{9} c^{4} d^{5} - 4 \, a b^{8} c^{3} d^{6} + 6 \, a^{2} b^{7} c^{2} d^{7} - 4 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )}}{b^{2} c d^{9} {\left | b \right |} - a b d^{10} {\left | b \right |}}\right )} {\left (b x + a\right )} + \frac {315 \, {\left (b^{10} c^{5} d^{4} - 5 \, a b^{9} c^{4} d^{5} + 10 \, a^{2} b^{8} c^{3} d^{6} - 10 \, a^{3} b^{7} c^{2} d^{7} + 5 \, a^{4} b^{6} c d^{8} - a^{5} b^{5} d^{9}\right )}}{b^{2} c d^{9} {\left | b \right |} - a b d^{10} {\left | b \right |}}\right )} \sqrt {b x + a}}{24 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {105 \, {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{8 \, \sqrt {b d} d^{5} {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{9/2}}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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