Integrand size = 19, antiderivative size = 128 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=-\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {5 b^{3/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{7/2}} \]
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Time = 0.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {49, 52, 65, 223, 212} \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=-\frac {5 b^{3/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{7/2}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}-\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]
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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)^{5/2}}{3 d (c+d x)^{3/2}}+\frac {(5 b) \int \frac {(a+b x)^{3/2}}{(c+d x)^{3/2}} \, dx}{3 d} \\ & = -\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {\left (5 b^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{d^2} \\ & = -\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {\left (5 b^2 (b c-a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 d^3} \\ & = -\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {(5 b (b c-a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{d^3} \\ & = -\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {(5 b (b c-a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{d^3} \\ & = -\frac {2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}}-\frac {10 b (a+b x)^{3/2}}{3 d^2 \sqrt {c+d x}}+\frac {5 b^2 \sqrt {a+b x} \sqrt {c+d x}}{d^3}-\frac {5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{7/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (-2 a^2 d^2-2 a b d (5 c+7 d x)+b^2 \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{3 d^3 (c+d x)^{3/2}}-\frac {5 b^{3/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{d^{7/2}} \]
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\[\int \frac {\left (b x +a \right )^{\frac {5}{2}}}{\left (d x +c \right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (100) = 200\).
Time = 0.39 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.71 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\left [-\frac {15 \, {\left (b^{2} c^{3} - a b c^{2} d + {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d - a b c d^{2}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \, {\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}, \frac {15 \, {\left (b^{2} c^{3} - a b c^{2} d + {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d - a b c d^{2}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \, {\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}\right ] \]
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\[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (100) = 200\).
Time = 0.37 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.16 \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{6} c d^{4} - a b^{5} d^{5}\right )} {\left (b x + a\right )}}{b^{2} c d^{5} {\left | b \right |} - a b d^{6} {\left | b \right |}} + \frac {20 \, {\left (b^{7} c^{2} d^{3} - 2 \, a b^{6} c d^{4} + a^{2} b^{5} d^{5}\right )}}{b^{2} c d^{5} {\left | b \right |} - a b d^{6} {\left | b \right |}}\right )} + \frac {15 \, {\left (b^{8} c^{3} d^{2} - 3 \, a b^{7} c^{2} d^{3} + 3 \, a^{2} b^{6} c d^{4} - a^{3} b^{5} d^{5}\right )}}{b^{2} c d^{5} {\left | b \right |} - a b d^{6} {\left | b \right |}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {5 \, {\left (b^{4} c - a b^{3} d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{3} {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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