Integrand size = 28, antiderivative size = 1205 \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=-\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {3 \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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \left (a e^2 g-c d (2 e f-3 d g)\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 {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 d g)\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 {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (e f-3 d g) \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 )}{e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (a e^2 g-c d (2 e f-3 d g)\right )^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 )}{4 e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \]
[Out]
Time = 3.22 (sec) , antiderivative size = 1205, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {931, 6874, 733, 430, 954, 858, 435, 947, 174, 552, 551} \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\frac {\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 ) \left (a e^2 g-c d (2 e f-3 d g)\right )^2}{4 e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {\sqrt {-a} \sqrt {c} \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 ) \left (a e^2 g-c d (2 e f-3 d g)\right )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {c x^2+a}}+\frac {\sqrt {-a} \sqrt {c} f \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 ) \left (a e^2 g-c d (2 e f-3 d g)\right )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {\sqrt {-a} \sqrt {c} d 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 ) \left (a e^2 g-c d (2 e f-3 d g)\right )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {\sqrt {f+g x} \sqrt {c x^2+a} \left (a e^2 g-c d (2 e f-3 d g)\right )}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {3 \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 )}{2 e^3 \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {c (e f-3 d g) \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^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \sqrt {f+g x} \sqrt {c x^2+a}}-\frac {\sqrt {f+g x} \sqrt {c x^2+a}}{2 e (d+e x)^2} \]
[In]
[Out]
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 858
Rule 931
Rule 947
Rule 954
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}+\frac {\int \frac {a g+2 c f x+3 c g x^2}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 e} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}+\frac {\int \left (\frac {3 c g}{e^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {a e^2 g-c d (2 e f-3 d g)}{e^2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {2 c (e f-3 d g)}{e^2 (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx}{4 e} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}+\frac {(3 c g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 e^3}+\frac {(c (e f-3 d g)) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{2 e^3}+\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 e^3} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \frac {a e^2 g-2 c d (e f-d g)-2 c d e g x-c e^2 g x^2}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e^3 \left (c d^2+a e^2\right ) (e f-d g)}+\frac {\left (c (e f-3 d g) \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}{2 e^3 \sqrt {a+c x^2}}+\frac {\left (3 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 )}{2 \sqrt {-a} e^3 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {3 \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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \int \left (-\frac {c d g}{\sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c e g x}{\sqrt {f+g x} \sqrt {a+c x^2}}+\frac {a e^2 g-c d (2 e f-3 d g)}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx}{8 e^3 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (c (e f-3 d 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 {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 {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {3 \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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (c d g \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e^3 \left (c d^2+a e^2\right ) (e f-d g)}+\frac {\left (c g \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac {x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e^2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right )^2 \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e^3 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (c (e f-3 d g) \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 {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {3 \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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (e f-3 d g) \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}}+\frac {\left (c \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{8 e^2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (c f \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e^2 \left (c d^2+a e^2\right ) (e f-d g)}-\frac {\left (\left (a e^2 g-c d (2 e f-3 d g)\right )^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}{8 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {a+c x^2}}+\frac {\left (a \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 d g)\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 )}{4 \sqrt {-a} e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {3 \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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 d g)\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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (e f-3 d g) \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}}+\frac {\left (\left (a e^2 g-c d (2 e f-3 d g)\right )^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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {a+c x^2}}+\frac {\left (a \sqrt {c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 \sqrt {-a} e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (a \sqrt {c} f \left (a e^2 g-c d (2 e f-3 d g)\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 )}{4 \sqrt {-a} e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {3 \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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \left (a e^2 g-c d (2 e f-3 d g)\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 )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 d g)\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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (e f-3 d g) \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}}+\frac {\left (\left (a e^2 g-c d (2 e f-3 d g)\right )^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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {\sqrt {f+g x} \sqrt {a+c x^2}}{2 e (d+e x)^2}-\frac {\left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 e \left (c d^2+a e^2\right ) (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {3 \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 )}{2 e^3 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \left (a e^2 g-c d (2 e f-3 d g)\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 )}{4 e^2 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \left (a e^2 g-c d (2 e f-3 d g)\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 )}{4 e^3 \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {c (e f-3 d g) \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}}+\frac {\left (a e^2 g-c d (2 e f-3 d g)\right )^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 )}{4 e^3 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 29.36 (sec) , antiderivative size = 2526, normalized size of antiderivative = 2.10 \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\text {Result too large to show} \]
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Time = 0.89 (sec) , antiderivative size = 1161, normalized size of antiderivative = 0.96
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1161\) |
default | \(\text {Expression too large to display}\) | \(19181\) |
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Timed out. \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {a + c x^{2}} \sqrt {f + g x}}{\left (d + e x\right )^{3}}\, dx \]
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\[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\int { \frac {\sqrt {c x^{2} + a} \sqrt {g x + f}}{{\left (e x + d\right )}^{3}} \,d x } \]
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\[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\int { \frac {\sqrt {c x^{2} + a} \sqrt {g x + f}}{{\left (e x + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {f+g x} \sqrt {a+c x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^3} \,d x \]
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