Integrand size = 28, antiderivative size = 818 \[ \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx=\frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {8 \sqrt {-a} c^{3/2} f g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} e g \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 e^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \]
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Time = 1.00 (sec) , antiderivative size = 818, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {972, 759, 849, 858, 733, 435, 430, 21, 947, 174, 552, 551} \[ \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \operatorname {EllipticPi}\left (\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {-a}}+e},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {-a} g}{\sqrt {c} f+\sqrt {-a} g}\right ) e^2}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right ) e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}+\frac {2 g^2 \sqrt {c x^2+a} e}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {8 \sqrt {-a} c^{3/2} f g \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}-\frac {2 \sqrt {-a} \sqrt {c} g \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{3 (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {8 c f g^2 \sqrt {c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 g^2 \sqrt {c x^2+a}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}} \]
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Rule 21
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 759
Rule 849
Rule 858
Rule 947
Rule 972
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {g}{(e f-d g) (f+g x)^{5/2} \sqrt {a+c x^2}}-\frac {e g}{(e f-d g)^2 (f+g x)^{3/2} \sqrt {a+c x^2}}+\frac {e^2}{(e f-d g)^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx \\ & = \frac {e^2 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{(e f-d g)^2}-\frac {(e g) \int \frac {1}{(f+g x)^{3/2} \sqrt {a+c x^2}} \, dx}{(e f-d g)^2}-\frac {g \int \frac {1}{(f+g x)^{5/2} \sqrt {a+c x^2}} \, dx}{e f-d g} \\ & = \frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {(2 c e g) \int \frac {-\frac {f}{2}-\frac {g x}{2}}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2+a g^2\right )}+\frac {(2 c g) \int \frac {-\frac {3 f}{2}+\frac {g x}{2}}{(f+g x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )}+\frac {\left (e^2 \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 f-d g)^2 \sqrt {a+c x^2}} \\ & = \frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}-\frac {(4 c g) \int \frac {\frac {1}{4} \left (3 c f^2-a g^2\right )+c f g x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )^2}-\frac {(c e g) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{(e f-d g)^2 \left (c f^2+a g^2\right )}-\frac {\left (2 e^2 \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 f-d g)^2 \sqrt {a+c x^2}} \\ & = \frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}-\frac {\left (4 c^2 f g\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )^2}+\frac {(c g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{3 (e f-d g) \left (c f^2+a g^2\right )}-\frac {\left (2 e^2 \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 f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (2 a \sqrt {c} e 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} (e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}} \\ & = \frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {2 \sqrt {-a} \sqrt {c} e 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 )}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 e^2 \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 )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (8 a c^{3/2} 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} (e f-d g) \left (c f^2+a g^2\right )^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 \sqrt {c} g \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} (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = \frac {2 g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right ) (f+g x)^{3/2}}+\frac {8 c f g^2 \sqrt {a+c x^2}}{3 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}+\frac {2 e g^2 \sqrt {a+c x^2}}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {f+g x}}+\frac {8 \sqrt {-a} c^{3/2} 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 (e f-d g) \left (c f^2+a g^2\right )^2 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} e 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 )}{(e f-d g)^2 \left (c f^2+a g^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} \sqrt {c} g \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 (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {2 e^2 \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 )}{\left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) (e f-d g)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 27.37 (sec) , antiderivative size = 1917, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx=\frac {2 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-d g) \left (a+c x^2\right ) \left (a g^2 (4 e f-d g+3 e g x)+c f (-d g (5 f+4 g x)+e f (8 f+7 g x))\right )-(f+g x) \left (7 c^2 e^2 f^5 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-11 c^2 d e f^4 g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+4 c^2 d^2 f^3 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+10 a c e^2 f^3 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-14 a c d e f^2 g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+4 a c d^2 f g^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}+3 a^2 e^2 f g^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-3 a^2 d e g^5 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}-14 c^2 e^2 f^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)+22 c^2 d e f^3 g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)-8 c^2 d^2 f^2 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)-6 a c e^2 f^2 g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)+6 a c d e f g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)+7 c^2 e^2 f^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2-11 c^2 d e f^2 g \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2+4 c^2 d^2 f g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2+3 a c e^2 f g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2-3 a c d e g^3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)^2+\sqrt {c} \left (-i \sqrt {c} f+\sqrt {a} g\right ) (e f-d g) \left (3 a e g^2+c f (7 e f-4 d g)\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} 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 )+\left (\sqrt {c} f+i \sqrt {a} g\right ) \left (3 a^{3/2} e^2 g^3+3 i a \sqrt {c} e g^2 (2 e f-d g)+\sqrt {a} c g \left (2 e^2 f^2+2 d e f g-d^2 g^2\right )+3 i c^{3/2} f \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \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 )-3 i c^2 e^2 f^4 \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \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 )-6 i a c e^2 f^2 g^2 \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \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 )-3 i a^2 e^2 g^4 \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \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 )\right )\right )}{3 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-d g)^3 \left (c f^2+a g^2\right )^2 (f+g x)^{3/2} \sqrt {a+c x^2}} \]
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Time = 3.18 (sec) , antiderivative size = 1079, normalized size of antiderivative = 1.32
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1079\) |
default | \(\text {Expression too large to display}\) | \(9415\) |
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Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right ) \left (f + g x\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^{5/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^{5/2}\,\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
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