Optimal. Leaf size=1246 \[ -\frac {\left (a g e^2+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {c x^2+a} e}{4 \left (c d^2+a e^2\right )^2 (e f-d g) (d+e x)}-\frac {\sqrt {f+g x} \sqrt {c x^2+a} e}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {\sqrt {-a} \sqrt {c} \left (a g e^2+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} 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 \left (c d^2+a e^2\right )^2 (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 \left (a g e^2+c d (6 e f-5 d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} 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 \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {\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} 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 \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {c x^2+a} e}-\frac {\sqrt {-a} \sqrt {c} d g \left (a g e^2+c d (6 e f-5 d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} 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 \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a} e}+\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} \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 ) \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {c x^2+a} e}+\frac {\left (a g e^2+c d (6 e f-5 d g)\right ) \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 {\frac {c x^2}{a}+1} \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 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a} e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 4.40, antiderivative size = 1246, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {946, 6742, 719, 419, 940, 844, 424, 933, 168, 538, 537} \[ -\frac {\left (a g e^2+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {c x^2+a} e}{4 \left (c d^2+a e^2\right )^2 (e f-d g) (d+e x)}-\frac {\sqrt {f+g x} \sqrt {c x^2+a} e}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {\sqrt {-a} \sqrt {c} \left (a g e^2+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {\frac {c x^2}{a}+1} 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 \left (c d^2+a e^2\right )^2 (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 \left (a g e^2+c d (6 e f-5 d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} 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 \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a}}+\frac {\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} 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 \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {c x^2+a} e}-\frac {\sqrt {-a} \sqrt {c} d g \left (a g e^2+c d (6 e f-5 d g)\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {\frac {c x^2}{a}+1} 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 \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a} e}+\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} \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 ) \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {c x^2+a} e}+\frac {\left (a g e^2+c d (6 e f-5 d g)\right ) \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 {\frac {c x^2}{a}+1} \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 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {f+g x} \sqrt {c x^2+a} e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 844
Rule 933
Rule 940
Rule 946
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sqrt {f+g x}}{(d+e x)^3 \sqrt {a+c x^2}} \, dx &=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {\int \frac {-4 c d f-a e g+2 c (e f-2 d g) x+c e g x^2}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 \left (c d^2+a e^2\right )}\\ &=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {\int \left (\frac {c g}{e \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {-a e^2 g-c d (6 e f-5 d g)}{e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {2 c (e f-3 d g)}{e (d+e x) \sqrt {f+g x} \sqrt {a+c x^2}}\right ) \, dx}{4 \left (c d^2+a e^2\right )}\\ &=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {(c g) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 e \left (c d^2+a e^2\right )}-\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 \left (c d^2+a e^2\right )}+\frac {\left (a e^2 g+c d (6 e f-5 d g)\right ) \int \frac {1}{(d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{4 e \left (c d^2+a e^2\right )}\\ &=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {e \left (a e^2 g+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g) (d+e x)}-\frac {\left (a e^2 g+c d (6 e f-5 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 \left (c d^2+a e^2\right )^2 (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 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {\left (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 ) \operatorname {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 \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {e \left (a e^2 g+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g) (d+e x)}+\frac {\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 \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\left (a e^2 g+c d (6 e f-5 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 \left (c d^2+a e^2\right )^2 (e f-d g)}+\frac {\left (c (e f-3 d g) \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {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 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}\\ &=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {e \left (a e^2 g+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g) (d+e x)}+\frac {\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 \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (c g \left (a e^2 g+c d (6 e f-5 d g)\right )\right ) \int \frac {x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)}+\frac {\left (c d g \left (a e^2 g+c d (6 e f-5 d g)\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e \left (c d^2+a e^2\right )^2 (e f-d g)}-\frac {\left (\left (a e^2 g+c d (6 e f-5 d g)\right ) \left (a e^2 g-c d (2 e f-3 d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 e \left (c d^2+a e^2\right )^2 (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 ) \operatorname {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 \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {e \left (a e^2 g+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g) (d+e x)}+\frac {\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 \left (c d^2+a e^2\right ) \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 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (c \left (a e^2 g+c d (6 e f-5 d g)\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)}-\frac {\left (c f \left (a e^2 g+c d (6 e f-5 d g)\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{8 \left (c d^2+a e^2\right )^2 (e f-d g)}-\frac {\left (\left (a e^2 g+c d (6 e f-5 d g)\right ) \left (a e^2 g-c d (2 e f-3 d g)\right ) \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 \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {a+c x^2}}+\frac {\left (a \sqrt {c} d g \left (a e^2 g+c d (6 e f-5 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 ) \operatorname {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 \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {e \left (a e^2 g+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g) (d+e x)}+\frac {\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 \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}-\frac {\sqrt {-a} \sqrt {c} d g \left (a e^2 g+c d (6 e f-5 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 \left (c d^2+a e^2\right )^2 (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 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (\left (a e^2 g+c d (6 e f-5 d g)\right ) \left (a e^2 g-c d (2 e f-3 d g)\right ) \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {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 \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {a+c x^2}}+\frac {\left (a \sqrt {c} \left (a e^2 g+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \operatorname {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} \left (c d^2+a e^2\right )^2 (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 (6 e f-5 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 ) \operatorname {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} \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {e \left (a e^2 g+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \left (a e^2 g+c d (6 e f-5 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 \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\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 \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \left (a e^2 g+c d (6 e f-5 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 \left (c d^2+a e^2\right )^2 (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 (6 e f-5 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 \left (c d^2+a e^2\right )^2 (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 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (\left (a e^2 g+c d (6 e f-5 d g)\right ) \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}}\right ) \operatorname {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 \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ &=-\frac {e \sqrt {f+g x} \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {e \left (a e^2 g+c d (6 e f-5 d g)\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{4 \left (c d^2+a e^2\right )^2 (e f-d g) (d+e x)}-\frac {\sqrt {-a} \sqrt {c} \left (a e^2 g+c d (6 e f-5 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 \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {\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 \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\sqrt {-a} \sqrt {c} f \left (a e^2 g+c d (6 e f-5 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 \left (c d^2+a e^2\right )^2 (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 (6 e f-5 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 \left (c d^2+a e^2\right )^2 (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 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right ) \sqrt {f+g x} \sqrt {a+c x^2}}+\frac {\left (a e^2 g+c d (6 e f-5 d g)\right ) \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}} \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 \left (\frac {\sqrt {c} d}{\sqrt {-a}}+e\right ) \left (c d^2+a e^2\right )^2 (e f-d g) \sqrt {f+g x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 10.97, size = 2450, normalized size = 1.97 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {g x + f}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 20359, normalized size = 16.34 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {g x + f}}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {f+g\,x}}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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