Integrand size = 22, antiderivative size = 185 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{9/2}} \, dx=-\frac {2 (b e-a f)}{7 f (d e-c f) (e+f x)^{7/2}}-\frac {2 (b c-a d)}{5 (d e-c f)^2 (e+f x)^{5/2}}-\frac {2 d (b c-a d)}{3 (d e-c f)^3 (e+f x)^{3/2}}-\frac {2 d^2 (b c-a d)}{(d e-c f)^4 \sqrt {e+f x}}+\frac {2 d^{5/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{9/2}} \]
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Time = 0.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {79, 53, 65, 214} \[ \int \frac {a+b x}{(c+d x) (e+f x)^{9/2}} \, dx=\frac {2 d^{5/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{9/2}}-\frac {2 d^2 (b c-a d)}{\sqrt {e+f x} (d e-c f)^4}-\frac {2 d (b c-a d)}{3 (e+f x)^{3/2} (d e-c f)^3}-\frac {2 (b c-a d)}{5 (e+f x)^{5/2} (d e-c f)^2}-\frac {2 (b e-a f)}{7 f (e+f x)^{7/2} (d e-c f)} \]
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Rule 53
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b e-a f)}{7 f (d e-c f) (e+f x)^{7/2}}-\frac {(b c-a d) \int \frac {1}{(c+d x) (e+f x)^{7/2}} \, dx}{d e-c f} \\ & = -\frac {2 (b e-a f)}{7 f (d e-c f) (e+f x)^{7/2}}-\frac {2 (b c-a d)}{5 (d e-c f)^2 (e+f x)^{5/2}}-\frac {(d (b c-a d)) \int \frac {1}{(c+d x) (e+f x)^{5/2}} \, dx}{(d e-c f)^2} \\ & = -\frac {2 (b e-a f)}{7 f (d e-c f) (e+f x)^{7/2}}-\frac {2 (b c-a d)}{5 (d e-c f)^2 (e+f x)^{5/2}}-\frac {2 d (b c-a d)}{3 (d e-c f)^3 (e+f x)^{3/2}}-\frac {\left (d^2 (b c-a d)\right ) \int \frac {1}{(c+d x) (e+f x)^{3/2}} \, dx}{(d e-c f)^3} \\ & = -\frac {2 (b e-a f)}{7 f (d e-c f) (e+f x)^{7/2}}-\frac {2 (b c-a d)}{5 (d e-c f)^2 (e+f x)^{5/2}}-\frac {2 d (b c-a d)}{3 (d e-c f)^3 (e+f x)^{3/2}}-\frac {2 d^2 (b c-a d)}{(d e-c f)^4 \sqrt {e+f x}}-\frac {\left (d^3 (b c-a d)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{(d e-c f)^4} \\ & = -\frac {2 (b e-a f)}{7 f (d e-c f) (e+f x)^{7/2}}-\frac {2 (b c-a d)}{5 (d e-c f)^2 (e+f x)^{5/2}}-\frac {2 d (b c-a d)}{3 (d e-c f)^3 (e+f x)^{3/2}}-\frac {2 d^2 (b c-a d)}{(d e-c f)^4 \sqrt {e+f x}}-\frac {\left (2 d^3 (b c-a d)\right ) \text {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{f (d e-c f)^4} \\ & = -\frac {2 (b e-a f)}{7 f (d e-c f) (e+f x)^{7/2}}-\frac {2 (b c-a d)}{5 (d e-c f)^2 (e+f x)^{5/2}}-\frac {2 d (b c-a d)}{3 (d e-c f)^3 (e+f x)^{3/2}}-\frac {2 d^2 (b c-a d)}{(d e-c f)^4 \sqrt {e+f x}}+\frac {2 d^{5/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{9/2}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.43 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{9/2}} \, dx=\frac {-2 b \left (15 d^3 e^4+3 c^3 f^3 (2 e+7 f x)-c^2 d f^2 \left (32 e^2+112 e f x+35 f^2 x^2\right )+c d^2 f \left (116 e^3+406 e^2 f x+350 e f^2 x^2+105 f^3 x^3\right )\right )+2 a f \left (-15 c^3 f^3+3 c^2 d f^2 (22 e+7 f x)-c d^2 f \left (122 e^2+112 e f x+35 f^2 x^2\right )+d^3 \left (176 e^3+406 e^2 f x+350 e f^2 x^2+105 f^3 x^3\right )\right )}{105 f (d e-c f)^4 (e+f x)^{7/2}}+\frac {2 d^{5/2} (-b c+a d) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{(-d e+c f)^{9/2}} \]
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Time = 3.01 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {\frac {2 d^{3} f \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a f -b e \right )}{7 \left (c f -d e \right ) \left (f x +e \right )^{\frac {7}{2}}}-\frac {2 f \left (a d -b c \right ) d}{3 \left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 f \left (a d -b c \right )}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}}}+\frac {2 f \left (a d -b c \right ) d^{2}}{\left (c f -d e \right )^{4} \sqrt {f x +e}}}{f}\) | \(178\) |
default | \(\frac {\frac {2 d^{3} f \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a f -b e \right )}{7 \left (c f -d e \right ) \left (f x +e \right )^{\frac {7}{2}}}-\frac {2 f \left (a d -b c \right ) d}{3 \left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 f \left (a d -b c \right )}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}}}+\frac {2 f \left (a d -b c \right ) d^{2}}{\left (c f -d e \right )^{4} \sqrt {f x +e}}}{f}\) | \(178\) |
pseudoelliptic | \(-\frac {2 \left (-7 d^{3} f \left (f x +e \right )^{\frac {7}{2}} \left (a d -b c \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\sqrt {\left (c f -d e \right ) d}\, \left (\left (-7 a \,d^{3} x^{3}+\frac {7 c \,x^{2} \left (3 b x +a \right ) d^{2}}{3}-\frac {7 \left (\frac {5 b x}{3}+a \right ) x \,c^{2} d}{5}+c^{3} \left (a +\frac {7 b x}{5}\right )\right ) f^{4}-\frac {22 e \left (\frac {175 a \,d^{3} x^{2}}{33}-\frac {56 x \left (\frac {25 b x}{8}+a \right ) c \,d^{2}}{33}+c^{2} \left (\frac {56 b x}{33}+a \right ) d -\frac {b \,c^{3}}{11}\right ) f^{3}}{5}+\frac {122 e^{2} \left (-\frac {203 x a \,d^{2}}{61}+c \left (\frac {203 b x}{61}+a \right ) d -\frac {16 b \,c^{2}}{61}\right ) d \,f^{2}}{15}-\frac {176 e^{3} \left (a d -\frac {29 b c}{44}\right ) d^{2} f}{15}+b \,d^{3} e^{4}\right )\right )}{7 \sqrt {\left (c f -d e \right ) d}\, \left (f x +e \right )^{\frac {7}{2}} f \left (c f -d e \right )^{4}}\) | \(248\) |
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Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (161) = 322\).
Time = 0.30 (sec) , antiderivative size = 1396, normalized size of antiderivative = 7.55 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{9/2}} \, dx=\text {Too large to display} \]
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Time = 4.63 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.10 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{9/2}} \, dx=\begin {cases} \frac {2 \left (\frac {d^{2} f \left (a d - b c\right )}{\sqrt {e + f x} \left (c f - d e\right )^{4}} + \frac {d^{2} f \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{\sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )^{4}} - \frac {d f \left (a d - b c\right )}{3 \left (e + f x\right )^{\frac {3}{2}} \left (c f - d e\right )^{3}} + \frac {f \left (a d - b c\right )}{5 \left (e + f x\right )^{\frac {5}{2}} \left (c f - d e\right )^{2}} - \frac {a f - b e}{7 \left (e + f x\right )^{\frac {7}{2}} \left (c f - d e\right )}\right )}{f} & \text {for}\: f \neq 0 \\\frac {\frac {b x}{d} + \frac {\left (a d - b c\right ) \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d}}{e^{\frac {9}{2}}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {a+b x}{(c+d x) (e+f x)^{9/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (161) = 322\).
Time = 0.30 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.37 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{9/2}} \, dx=-\frac {2 \, {\left (b c d^{3} - a d^{4}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (d^{4} e^{4} - 4 \, c d^{3} e^{3} f + 6 \, c^{2} d^{2} e^{2} f^{2} - 4 \, c^{3} d e f^{3} + c^{4} f^{4}\right )} \sqrt {-d^{2} e + c d f}} - \frac {2 \, {\left (15 \, b d^{3} e^{4} + 105 \, {\left (f x + e\right )}^{3} b c d^{2} f - 105 \, {\left (f x + e\right )}^{3} a d^{3} f + 35 \, {\left (f x + e\right )}^{2} b c d^{2} e f - 35 \, {\left (f x + e\right )}^{2} a d^{3} e f + 21 \, {\left (f x + e\right )} b c d^{2} e^{2} f - 21 \, {\left (f x + e\right )} a d^{3} e^{2} f - 45 \, b c d^{2} e^{3} f - 15 \, a d^{3} e^{3} f - 35 \, {\left (f x + e\right )}^{2} b c^{2} d f^{2} + 35 \, {\left (f x + e\right )}^{2} a c d^{2} f^{2} - 42 \, {\left (f x + e\right )} b c^{2} d e f^{2} + 42 \, {\left (f x + e\right )} a c d^{2} e f^{2} + 45 \, b c^{2} d e^{2} f^{2} + 45 \, a c d^{2} e^{2} f^{2} + 21 \, {\left (f x + e\right )} b c^{3} f^{3} - 21 \, {\left (f x + e\right )} a c^{2} d f^{3} - 15 \, b c^{3} e f^{3} - 45 \, a c^{2} d e f^{3} + 15 \, a c^{3} f^{4}\right )}}{105 \, {\left (d^{4} e^{4} f - 4 \, c d^{3} e^{3} f^{2} + 6 \, c^{2} d^{2} e^{2} f^{3} - 4 \, c^{3} d e f^{4} + c^{4} f^{5}\right )} {\left (f x + e\right )}^{\frac {7}{2}}} \]
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Time = 1.65 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.18 \[ \int \frac {a+b x}{(c+d x) (e+f x)^{9/2}} \, dx=\frac {2\,d^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,\left (c^4\,f^4-4\,c^3\,d\,e\,f^3+6\,c^2\,d^2\,e^2\,f^2-4\,c\,d^3\,e^3\,f+d^4\,e^4\right )}{{\left (c\,f-d\,e\right )}^{9/2}}\right )\,\left (a\,d-b\,c\right )}{{\left (c\,f-d\,e\right )}^{9/2}}-\frac {\frac {2\,\left (a\,f-b\,e\right )}{7\,\left (c\,f-d\,e\right )}-\frac {2\,\left (e+f\,x\right )\,\left (a\,d\,f-b\,c\,f\right )}{5\,{\left (c\,f-d\,e\right )}^2}-\frac {2\,d^2\,{\left (e+f\,x\right )}^3\,\left (a\,d\,f-b\,c\,f\right )}{{\left (c\,f-d\,e\right )}^4}+\frac {2\,d\,{\left (e+f\,x\right )}^2\,\left (a\,d\,f-b\,c\,f\right )}{3\,{\left (c\,f-d\,e\right )}^3}}{f\,{\left (e+f\,x\right )}^{7/2}} \]
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