Integrand size = 46, antiderivative size = 360 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {(2 c d-b e)^{3/2} (5 c e f-9 c d g+2 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \]
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Time = 0.40 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {806, 678, 674, 214} \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {(2 c d-b e)^{3/2} \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right ) (2 b e g-9 c d g+5 c e f)}{e^2}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (2 b e g-9 c d g+5 c e f)}{5 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-9 c d g+5 c e f)}{3 e^2 (d+e x)^{3/2}}-\frac {(2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-9 c d g+5 c e f)}{e^2 \sqrt {d+e x}} \]
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Rule 214
Rule 674
Rule 678
Rule 806
Rubi steps \begin{align*} \text {integral}& = -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(5 c e f-9 c d g+2 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx}{2 e (2 c d-b e)} \\ & = -\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {((-2 c d+b e) (5 c e f-9 c d g+2 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{2 e (2 c d-b e)} \\ & = -\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {((2 c d-b e) (5 c e f-9 c d g+2 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{2 e} \\ & = -\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {\left ((2 c d-b e)^2 (5 c e f-9 c d g+2 b e g)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{2 e} \\ & = -\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\left ((2 c d-b e)^2 (5 c e f-9 c d g+2 b e g)\right ) \text {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right ) \\ & = -\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {(2 c d-b e)^{3/2} (5 c e f-9 c d g+2 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.71 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {b^2 e^2 (-15 e f+61 d g+46 e g x)+2 b c e \left (-146 d^2 g+5 d e (13 f-21 g x)+e^2 x (35 f+11 g x)\right )+2 c^2 \left (168 d^3 g+e^3 x^2 (5 f+3 g x)-6 d e^2 x (10 f+3 g x)+d^2 e (-95 f+117 g x)\right )}{(d+e x) (-c d+b e+c e x)^2}+\frac {15 (-2 c d+b e)^{3/2} (-5 c e f+9 c d g-2 b e g) \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-b e+c (d-e x))^{5/2}}\right )}{15 e^2 (d+e x)^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1127\) vs. \(2(332)=664\).
Time = 0.36 (sec) , antiderivative size = 1128, normalized size of antiderivative = 3.13
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Time = 0.38 (sec) , antiderivative size = 990, normalized size of antiderivative = 2.75 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\left [-\frac {15 \, {\left ({\left (5 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (18 \, c^{2} d^{2} e^{2} - 13 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (18 \, c^{2} d^{4} - 13 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (5 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (18 \, c^{2} d^{3} e - 13 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (6 \, c^{2} e^{3} g x^{3} + 2 \, {\left (5 \, c^{2} e^{3} f - {\left (18 \, c^{2} d e^{2} - 11 \, b c e^{3}\right )} g\right )} x^{2} - 5 \, {\left (38 \, c^{2} d^{2} e - 26 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + {\left (336 \, c^{2} d^{3} - 292 \, b c d^{2} e + 61 \, b^{2} d e^{2}\right )} g - 2 \, {\left (5 \, {\left (12 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (117 \, c^{2} d^{2} e - 105 \, b c d e^{2} + 23 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{30 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac {15 \, {\left ({\left (5 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (18 \, c^{2} d^{2} e^{2} - 13 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (18 \, c^{2} d^{4} - 13 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (5 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (18 \, c^{2} d^{3} e - 13 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) + {\left (6 \, c^{2} e^{3} g x^{3} + 2 \, {\left (5 \, c^{2} e^{3} f - {\left (18 \, c^{2} d e^{2} - 11 \, b c e^{3}\right )} g\right )} x^{2} - 5 \, {\left (38 \, c^{2} d^{2} e - 26 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + {\left (336 \, c^{2} d^{3} - 292 \, b c d^{2} e + 61 \, b^{2} d e^{2}\right )} g - 2 \, {\left (5 \, {\left (12 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (117 \, c^{2} d^{2} e - 105 \, b c d e^{2} + 23 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \]
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\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
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\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
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Time = 0.54 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.61 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {120 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - 60 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 240 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g + 180 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g - 30 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c e^{2} g + 10 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} e f - 30 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 10 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c e g - 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c g + \frac {15 \, {\left (20 \, c^{4} d^{2} e f - 20 \, b c^{3} d e^{2} f + 5 \, b^{2} c^{2} e^{3} f - 36 \, c^{4} d^{3} g + 44 \, b c^{3} d^{2} e g - 17 \, b^{2} c^{2} d e^{2} g + 2 \, b^{3} c e^{3} g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {15 \, {\left (4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{2} e f - 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d e^{2} f + \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} e^{3} f - 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{3} g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d^{2} e g - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} d e^{2} g\right )}}{{\left (e x + d\right )} c}}{15 \, c e^{2}} \]
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Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]
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