Integrand size = 28, antiderivative size = 600 \[ \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {2 \sqrt {-a} g (7 e f-3 d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {a \sqrt {c} x}{(-a)^{3/2}}}}{\sqrt {2}}\right )|\frac {2 a g}{-\sqrt {-a} \sqrt {c} f+a g}\right )}{3 \sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} g \left (a e^2 g^2+c \left (-2 e^2 f^2+6 d e f g-3 d^2 g^2\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {a \sqrt {c} x}{(-a)^{3/2}}}}{\sqrt {2}}\right ),\frac {2 a g}{-\sqrt {-a} \sqrt {c} f+a g}\right )}{3 c^{3/2} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 (e f-d g)^2 \sqrt {\frac {g \left (\sqrt {-a}-\sqrt {c} x\right )}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {-\frac {g \left (\sqrt {-a}+\sqrt {c} x\right )}{\sqrt {c} f-\sqrt {-a} g}} \operatorname {EllipticPi}\left (\frac {e \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}{e f-d g},\arcsin \left (\sqrt {\frac {c}{c f+\sqrt {-a} \sqrt {c} g}} \sqrt {f+g x}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{e^3 \sqrt {\frac {c}{c f+\sqrt {-a} \sqrt {c} g}} \sqrt {a+c x^2}} \]
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Time = 1.07 (sec) , antiderivative size = 808, normalized size of antiderivative = 1.35, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {972, 733, 430, 947, 174, 552, 551, 435, 757, 858} \[ \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right ) (e f-d g)^3}{e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {2 \sqrt {-a} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right ) (e f-d g)^2}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {2 \sqrt {-a} g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right ) (e f-d g)}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}-\frac {8 \sqrt {-a} f g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}+\frac {2 \sqrt {-a} g \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 c^{3/2} e \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {2 g^2 \sqrt {f+g x} \sqrt {c x^2+a}}{3 c e} \]
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Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 757
Rule 858
Rule 947
Rule 972
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {g (e f-d g)^2}{e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {(e f-d g)^3}{e^3 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {g (e f-d g) \sqrt {f+g x}}{e^2 \sqrt {a+c x^2}}+\frac {g (f+g x)^{3/2}}{e \sqrt {a+c x^2}}\right ) \, dx \\ & = \frac {g \int \frac {(f+g x)^{3/2}}{\sqrt {a+c x^2}} \, dx}{e}+\frac {(g (e f-d g)) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{e^2}+\frac {\left (g (e f-d g)^2\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^3}+\frac {(e f-d g)^3 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{e^3} \\ & = \frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}+\frac {(2 g) \int \frac {\frac {1}{2} \left (3 c f^2-a g^2\right )+2 c f g x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 c e}+\frac {\left ((e f-d g)^3 \sqrt {1+\frac {c x^2}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}} \sqrt {1+\frac {\sqrt {c} x}{\sqrt {-a}}} (d+e x) \sqrt {f+g x}} \, dx}{e^3 \sqrt {a+c x^2}}+\frac {\left (2 a g (e f-d g) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} e^2 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (2 a g (e f-d g)^2 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\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} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{\sqrt {-a} \sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = \frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {f+g 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 g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \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 g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {(4 f g) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{3 e}-\frac {\left (g \left (c f^2+a g^2\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 c e}-\frac {\left (2 (e f-d g)^3 \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {f+\frac {\sqrt {-a} g}{\sqrt {c}}-\frac {\sqrt {-a} g x^2}{\sqrt {c}}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{e^3 \sqrt {a+c x^2}} \\ & = \frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {f+g 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 g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \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 g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 (e f-d g)^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2} \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e-e x^2\right ) \sqrt {1-\frac {\sqrt {-a} g x^2}{\sqrt {c} \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}}} \, dx,x,\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}\right )}{e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (8 a f g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} \sqrt {c} e \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (2 a g \left (c f^2+a g^2\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\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} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} c^{3/2} e \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = \frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}-\frac {8 \sqrt {-a} f g \sqrt {f+g 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 g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 \sqrt {c} e \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g) \sqrt {f+g 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 g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} g (e f-d g)^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \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 g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{\sqrt {c} e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} g \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \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 g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 c^{3/2} e \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 (e f-d g)^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \Pi \left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e};\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 29.46 (sec) , antiderivative size = 1440, normalized size of antiderivative = 2.40 \[ \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {2 g^2 \sqrt {f+g x} \sqrt {a+c x^2}}{3 c e}+\frac {2 (f+g x)^{3/2} \left (7 c e^2 f \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-3 c d e g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+\frac {7 c e^2 f^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{(f+g x)^2}-\frac {3 c d e f^2 g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{(f+g x)^2}+\frac {7 a e^2 f g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{(f+g x)^2}-\frac {3 a d e g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{(f+g x)^2}-\frac {14 c e^2 f^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{f+g x}+\frac {6 c d e f g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{f+g x}+\frac {\sqrt {c} e \left (-i \sqrt {c} f+\sqrt {a} g\right ) (7 e f-3 d g) \sqrt {1-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \sqrt {1-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {i e \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (i \sqrt {a} e g+\sqrt {c} (6 e f-3 d g)\right ) \sqrt {1-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \sqrt {1-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {3 i c e^2 f^2 \sqrt {1-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \sqrt {1-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \operatorname {EllipticPi}\left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )},i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}-\frac {6 i c d e f g \sqrt {1-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \sqrt {1-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \operatorname {EllipticPi}\left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )},i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {3 i c d^2 g^2 \sqrt {1-\frac {f}{f+g x}-\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \sqrt {1-\frac {f}{f+g x}+\frac {i \sqrt {a} g}{\sqrt {c} (f+g x)}} \operatorname {EllipticPi}\left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )},i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}\right )}{3 c e^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \sqrt {a+\frac {c (f+g x)^2 \left (-1+\frac {f}{f+g x}\right )^2}{g^2}}} \]
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Time = 4.53 (sec) , antiderivative size = 948, normalized size of antiderivative = 1.58
method | result | size |
elliptic | \(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 g^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 e c}+\frac {2 \left (\frac {g \left (d^{2} g^{2}-3 d e f g +3 e^{2} f^{2}\right )}{e^{3}}-\frac {g^{3} a}{3 e c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (-\frac {g^{2} \left (d g -3 e f \right )}{e^{2}}-\frac {2 g^{2} f}{3 e}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}-\frac {2 \left (d^{3} g^{3}-3 d^{2} e f \,g^{2}+3 d \,e^{2} f^{2} g -e^{3} f^{3}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \Pi \left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{e^{4} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\) | \(948\) |
risch | \(\text {Expression too large to display}\) | \(1564\) |
default | \(\text {Expression too large to display}\) | \(3164\) |
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Timed out. \[ \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {\left (f + g x\right )^{\frac {5}{2}}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
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\[ \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}} \,d x } \]
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\[ \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}} \,d x } \]
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Timed out. \[ \int \frac {(f+g x)^{5/2}}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^{5/2}}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
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