Integrand size = 19, antiderivative size = 120 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=-\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {49, 65, 223, 212} \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2}}-\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}} \]
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Rule 49
Rule 65
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {d \int \frac {(c+d x)^{3/2}}{(a+b x)^{5/2}} \, dx}{b} \\ & = -\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {d^2 \int \frac {\sqrt {c+d x}}{(a+b x)^{3/2}} \, dx}{b^2} \\ & = -\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {d^3 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{b^3} \\ & = -\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {\left (2 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 )}{b^4} \\ & = -\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {\left (2 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 )}{b^4} \\ & = -\frac {2 d^2 \sqrt {c+d x}}{b^3 \sqrt {a+b x}}-\frac {2 d (c+d x)^{3/2}}{3 b^2 (a+b x)^{3/2}}-\frac {2 (c+d x)^{5/2}}{5 b (a+b x)^{5/2}}+\frac {2 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{7/2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=-\frac {2 \sqrt {c+d x} \left (15 a^2 d^2+5 a b d (c+7 d x)+b^2 \left (3 c^2+11 c d x+23 d^2 x^2\right )\right )}{15 b^3 (a+b x)^{5/2}}+\frac {2 d^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{7/2}} \]
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\[\int \frac {\left (d x +c \right )^{\frac {5}{2}}}{\left (b x +a \right )^{\frac {7}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (92) = 184\).
Time = 0.49 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.86 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=\left [\frac {15 \, {\left (b^{3} d^{2} x^{3} + 3 \, a b^{2} d^{2} x^{2} + 3 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (23 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + {\left (11 \, b^{2} c d + 35 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, -\frac {15 \, {\left (b^{3} d^{2} x^{3} + 3 \, a b^{2} d^{2} x^{2} + 3 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + 2 \, {\left (23 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 5 \, a b c d + 15 \, a^{2} d^{2} + {\left (11 \, b^{2} c d + 35 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \]
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\[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {7}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1025 vs. \(2 (92) = 184\).
Time = 0.57 (sec) , antiderivative size = 1025, normalized size of antiderivative = 8.54 \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=-\frac {\sqrt {b d} d^{2} {\left | b \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 )}^{2}\right )}{b^{5}} - \frac {4 \, {\left (23 \, \sqrt {b d} b^{9} c^{5} d^{2} {\left | b \right |} - 115 \, \sqrt {b d} a b^{8} c^{4} d^{3} {\left | b \right |} + 230 \, \sqrt {b d} a^{2} b^{7} c^{3} d^{4} {\left | b \right |} - 230 \, \sqrt {b d} a^{3} b^{6} c^{2} d^{5} {\left | b \right |} + 115 \, \sqrt {b d} a^{4} b^{5} c d^{6} {\left | b \right |} - 23 \, \sqrt {b d} a^{5} b^{4} d^{7} {\left | b \right |} - 70 \, \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^{7} c^{4} d^{2} {\left | b \right |} + 280 \, \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^{6} c^{3} d^{3} {\left | b \right |} - 420 \, \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^{5} c^{2} d^{4} {\left | b \right |} + 280 \, \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^{3} b^{4} c d^{5} {\left | b \right |} - 70 \, \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^{4} b^{3} d^{6} {\left | b \right |} + 140 \, \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^{5} c^{3} d^{2} {\left | b \right |} - 420 \, \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^{4} c^{2} d^{3} {\left | b \right |} + 420 \, \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^{2} b^{3} c d^{4} {\left | b \right |} - 140 \, \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^{3} b^{2} d^{5} {\left | b \right |} - 90 \, \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^{3} c^{2} d^{2} {\left | b \right |} + 180 \, \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} a b^{2} c d^{3} {\left | b \right |} - 90 \, \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} a^{2} b d^{4} {\left | b \right |} + 45 \, \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 )}^{8} b c d^{2} {\left | b \right |} - 45 \, \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 )}^{8} a d^{3} {\left | b \right |}\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^{4}} \]
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Timed out. \[ \int \frac {(c+d x)^{5/2}}{(a+b x)^{7/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{7/2}} \,d x \]
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