Integrand size = 23, antiderivative size = 470 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}-\frac {2 \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.42 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {752, 832, 846, 857, 729, 113, 111, 118, 117} \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {8 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{7/2} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {8 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (b^3 e^3-6 b c^2 d^2 e+4 c^3 d^3\right )}{3 b^4 c^2}+\frac {2 (d+e x)^{3/2} \left (x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b c d^2 (8 c d-11 b e)\right )}{3 b^4 c \sqrt {b x+c x^2}} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 752
Rule 832
Rule 846
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {(d+e x)^{5/2} \left (\frac {1}{2} d (8 c d-11 b e)-\frac {3}{2} e (2 c d-b e) x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {4 \int \frac {\sqrt {d+e x} \left (\frac {3}{4} b d e \left (8 c^2 d^2-13 b c d e+b^2 e^2\right )+3 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}-\frac {8 \int \frac {\frac {3}{8} b d e \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3\right )+\frac {3}{8} e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{9 b^4 c^2} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}+\frac {\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c^2}-\frac {\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c^2} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}+\frac {\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {\left (\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^4 c^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}-\frac {\left (\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{3 b^4 c^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{3 b^4 c^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{7/2} (b d+(2 c d-b e) x)}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 (d+e x)^{3/2} \left (b c d^2 (8 c d-11 b e)+(2 c d-b e) \left (8 c^2 d^2-8 b c d e-3 b^2 e^2\right ) x\right )}{3 b^4 c \sqrt {b x+c x^2}}-\frac {8 e \left (4 c^3 d^3-6 b c^2 d^2 e+b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^4 c^2}-\frac {2 \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {8 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 24.93 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (b (d+e x) \left (b (c d-b e)^4 x^2+(c d-b e)^3 (8 c d+5 b e) x^2 (b+c x)-b c^2 d^4 (b+c x)^2+c^2 d^3 (8 c d-13 b e) x (b+c x)^2\right )-\sqrt {\frac {b}{c}} x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) (b+c x) (d+e x)+i b e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^4 d^4-17 b c^3 d^3 e+6 b^2 c^2 d^2 e^2+11 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 c^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(894\) vs. \(2(410)=820\).
Time = 2.55 (sec) , antiderivative size = 895, normalized size of antiderivative = 1.90
method | result | size |
elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 d^{4} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} x^{2}}-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) d^{3} \left (13 b e -8 c d \right )}{3 b^{4} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c^{4} b^{3} \left (\frac {b}{c}+x \right )^{2}}-\frac {2 \left (c e \,x^{2}+c d x \right ) \left (5 b^{4} e^{4}-7 b^{3} c d \,e^{3}-9 b^{2} c^{2} d^{2} e^{2}+19 b \,c^{3} d^{3} e -8 c^{4} d^{4}\right )}{3 b^{4} c^{3} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (-\frac {e^{4} \left (2 b e -5 c d \right )}{c^{3}}-\frac {d^{4} c e}{3 b^{3}}+\frac {\left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right ) e}{3 c^{3} b^{3}}+\frac {\left (5 b^{4} e^{4}-7 b^{3} c d \,e^{3}-9 b^{2} c^{2} d^{2} e^{2}+19 b \,c^{3} d^{3} e -8 c^{4} d^{4}\right ) \left (b e -c d \right )}{3 c^{3} b^{4}}+\frac {d \left (5 b^{4} e^{4}-7 b^{3} c d \,e^{3}-9 b^{2} c^{2} d^{2} e^{2}+19 b \,c^{3} d^{3} e -8 c^{4} d^{4}\right )}{3 c^{2} b^{4}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {e^{5}}{c^{2}}+\frac {d^{3} e c \left (13 b e -8 c d \right )}{3 b^{4}}+\frac {\left (5 b^{4} e^{4}-7 b^{3} c d \,e^{3}-9 b^{2} c^{2} d^{2} e^{2}+19 b \,c^{3} d^{3} e -8 c^{4} d^{4}\right ) e}{3 c^{2} b^{4}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(895\) |
default | \(\text {Expression too large to display}\) | \(2086\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.31 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.99 \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (16 \, c^{7} d^{5} - 40 \, b c^{6} d^{4} e + 22 \, b^{2} c^{5} d^{3} e^{2} + 7 \, b^{3} c^{4} d^{2} e^{3} + 11 \, b^{4} c^{3} d e^{4} - 8 \, b^{5} c^{2} e^{5}\right )} x^{4} + 2 \, {\left (16 \, b c^{6} d^{5} - 40 \, b^{2} c^{5} d^{4} e + 22 \, b^{3} c^{4} d^{3} e^{2} + 7 \, b^{4} c^{3} d^{2} e^{3} + 11 \, b^{5} c^{2} d e^{4} - 8 \, b^{6} c e^{5}\right )} x^{3} + {\left (16 \, b^{2} c^{5} d^{5} - 40 \, b^{3} c^{4} d^{4} e + 22 \, b^{4} c^{3} d^{3} e^{2} + 7 \, b^{5} c^{2} d^{2} e^{3} + 11 \, b^{6} c d e^{4} - 8 \, b^{7} e^{5}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left ({\left (16 \, c^{7} d^{4} e - 32 \, b c^{6} d^{3} e^{2} + 9 \, b^{2} c^{5} d^{2} e^{3} + 7 \, b^{3} c^{4} d e^{4} - 8 \, b^{4} c^{3} e^{5}\right )} x^{4} + 2 \, {\left (16 \, b c^{6} d^{4} e - 32 \, b^{2} c^{5} d^{3} e^{2} + 9 \, b^{3} c^{4} d^{2} e^{3} + 7 \, b^{4} c^{3} d e^{4} - 8 \, b^{5} c^{2} e^{5}\right )} x^{3} + {\left (16 \, b^{2} c^{5} d^{4} e - 32 \, b^{3} c^{4} d^{3} e^{2} + 9 \, b^{4} c^{3} d^{2} e^{3} + 7 \, b^{5} c^{2} d e^{4} - 8 \, b^{6} c e^{5}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (b^{3} c^{4} d^{4} e - {\left (16 \, c^{7} d^{4} e - 32 \, b c^{6} d^{3} e^{2} + 9 \, b^{2} c^{5} d^{2} e^{3} + 7 \, b^{3} c^{4} d e^{4} - 5 \, b^{4} c^{3} e^{5}\right )} x^{3} - {\left (24 \, b c^{6} d^{4} e - 49 \, b^{2} c^{5} d^{3} e^{2} + 15 \, b^{3} c^{4} d^{2} e^{3} + 3 \, b^{4} c^{3} d e^{4} - 4 \, b^{5} c^{2} e^{5}\right )} x^{2} - {\left (6 \, b^{2} c^{5} d^{4} e - 13 \, b^{3} c^{4} d^{3} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (b^{4} c^{6} e x^{4} + 2 \, b^{5} c^{5} e x^{3} + b^{6} c^{4} e x^{2}\right )}} \]
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Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]
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