Integrand size = 21, antiderivative size = 498 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {16 \sqrt {-a} c^{5/2} d \left (32 c d^2+29 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{21 e^6 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} c^{3/2} \left (32 c d^2+5 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{21 e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]
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Time = 0.31 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {747, 825, 827, 858, 733, 435, 430} \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {16 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \left (5 a e^2+32 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{21 e^6 \sqrt {a+c x^2} \sqrt {d+e x}}+\frac {16 \sqrt {-a} c^{5/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (29 a e^2+32 c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{21 e^6 \sqrt {a+c x^2} \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {8 c^2 \sqrt {a+c x^2} \left (e x \left (5 a e^2+8 c d^2\right )+d \left (29 a e^2+32 c d^2\right )\right )}{21 e^5 \sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {4 c \left (a+c x^2\right )^{3/2} \left (e x \left (5 a e^2+11 c d^2\right )+2 d \left (a e^2+4 c d^2\right )\right )}{21 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac {2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}} \]
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Rule 430
Rule 435
Rule 733
Rule 747
Rule 825
Rule 827
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {(10 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx}{7 e} \\ & = -\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {(4 c) \int \frac {\left (3 a c d e-c \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx}{7 e^3 \left (c d^2+a e^2\right )} \\ & = \frac {8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {(8 c) \int \frac {a c e \left (8 c d^2+5 a e^2\right )-c^2 d \left (32 c d^2+29 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{21 e^5 \left (c d^2+a e^2\right )} \\ & = \frac {8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {\left (8 c^2 \left (32 c d^2+5 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{21 e^6}-\frac {\left (8 c^3 d \left (32 c d^2+29 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{21 e^6 \left (c d^2+a e^2\right )} \\ & = \frac {8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\left (16 a c^{5/2} d \left (32 c d^2+29 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{21 \sqrt {-a} e^6 \left (c d^2+a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (16 a c^{3/2} \left (32 c d^2+5 a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{21 \sqrt {-a} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = \frac {8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt {a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt {d+e x}}-\frac {4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {16 \sqrt {-a} c^{5/2} d \left (32 c d^2+29 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{21 e^6 \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} c^{3/2} \left (32 c d^2+5 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{21 e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 13.42 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (7 c^2-\frac {3 \left (c d^2+a e^2\right )^2}{(d+e x)^4}+\frac {18 c d \left (c d^2+a e^2\right )}{(d+e x)^3}-\frac {4 c \left (13 c d^2+4 a e^2\right )}{(d+e x)^2}+\frac {2 c^2 d \left (79 c d^2+67 a e^2\right )}{\left (c d^2+a e^2\right ) (d+e x)}\right )}{e^5}-\frac {16 c^2 \left (d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c d^2+29 a e^2\right ) \left (a+c x^2\right )+\sqrt {c} d \left (-32 i c^{3/2} d^3+32 \sqrt {a} c d^2 e-29 i a \sqrt {c} d e^2+29 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} e \left (32 c^{3/2} d^3+8 i \sqrt {a} c d^2 e+29 a \sqrt {c} d e^2+5 i a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^7 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2+a e^2\right ) (d+e x)}\right )}{21 \sqrt {a+c x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(941\) vs. \(2(420)=840\).
Time = 8.77 (sec) , antiderivative size = 942, normalized size of antiderivative = 1.89
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{7 e^{9} \left (x +\frac {d}{e}\right )^{4}}+\frac {12 \left (e^{2} a +c \,d^{2}\right ) c d \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{7 e^{8} \left (x +\frac {d}{e}\right )^{3}}-\frac {8 \left (4 e^{2} a +13 c \,d^{2}\right ) c \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{21 e^{7} \left (x +\frac {d}{e}\right )^{2}}+\frac {4 \left (c e \,x^{2}+a e \right ) c^{2} d \left (67 e^{2} a +79 c \,d^{2}\right )}{21 e^{6} \left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 c^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{5}}+\frac {2 \left (\frac {c^{2} \left (3 e^{2} a +10 c \,d^{2}\right )}{e^{6}}-\frac {4 c^{2} \left (4 e^{2} a +13 c \,d^{2}\right )}{21 e^{6}}-\frac {2 c^{3} d^{2} \left (67 e^{2} a +79 c \,d^{2}\right )}{21 e^{6} \left (e^{2} a +c \,d^{2}\right )}-\frac {c^{2} a}{3 e^{4}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (-\frac {14 c^{3} d}{3 e^{5}}-\frac {2 c^{3} d \left (67 e^{2} a +79 c \,d^{2}\right )}{21 e^{5} \left (e^{2} a +c \,d^{2}\right )}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(942\) |
risch | \(\text {Expression too large to display}\) | \(3611\) |
default | \(\text {Expression too large to display}\) | \(5303\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.28 (sec) , antiderivative size = 772, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (8 \, {\left (32 \, c^{3} d^{8} + 53 \, a c^{2} d^{6} e^{2} + 15 \, a^{2} c d^{4} e^{4} + {\left (32 \, c^{3} d^{4} e^{4} + 53 \, a c^{2} d^{2} e^{6} + 15 \, a^{2} c e^{8}\right )} x^{4} + 4 \, {\left (32 \, c^{3} d^{5} e^{3} + 53 \, a c^{2} d^{3} e^{5} + 15 \, a^{2} c d e^{7}\right )} x^{3} + 6 \, {\left (32 \, c^{3} d^{6} e^{2} + 53 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6}\right )} x^{2} + 4 \, {\left (32 \, c^{3} d^{7} e + 53 \, a c^{2} d^{5} e^{3} + 15 \, a^{2} c d^{3} e^{5}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 24 \, {\left (32 \, c^{3} d^{7} e + 29 \, a c^{2} d^{5} e^{3} + {\left (32 \, c^{3} d^{3} e^{5} + 29 \, a c^{2} d e^{7}\right )} x^{4} + 4 \, {\left (32 \, c^{3} d^{4} e^{4} + 29 \, a c^{2} d^{2} e^{6}\right )} x^{3} + 6 \, {\left (32 \, c^{3} d^{5} e^{3} + 29 \, a c^{2} d^{3} e^{5}\right )} x^{2} + 4 \, {\left (32 \, c^{3} d^{6} e^{2} + 29 \, a c^{2} d^{4} e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (128 \, c^{3} d^{6} e^{2} + 100 \, a c^{2} d^{4} e^{4} - 7 \, a^{2} c d^{2} e^{6} - 3 \, a^{3} e^{8} + 7 \, {\left (c^{3} d^{2} e^{6} + a c^{2} e^{8}\right )} x^{4} + 6 \, {\left (31 \, c^{3} d^{3} e^{5} + 27 \, a c^{2} d e^{7}\right )} x^{3} + 8 \, {\left (58 \, c^{3} d^{4} e^{4} + 47 \, a c^{2} d^{2} e^{6} - 2 \, a^{2} c e^{8}\right )} x^{2} + 2 \, {\left (208 \, c^{3} d^{5} e^{3} + 165 \, a c^{2} d^{3} e^{5} - 7 \, a^{2} c d e^{7}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{63 \, {\left (c d^{6} e^{7} + a d^{4} e^{9} + {\left (c d^{2} e^{11} + a e^{13}\right )} x^{4} + 4 \, {\left (c d^{3} e^{10} + a d e^{12}\right )} x^{3} + 6 \, {\left (c d^{4} e^{9} + a d^{2} e^{11}\right )} x^{2} + 4 \, {\left (c d^{5} e^{8} + a d^{3} e^{10}\right )} x\right )}} \]
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\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
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\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]
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